Abstruse Goose

Abstruse Goose 1.0rc1 Attack of the Angry Monkeys

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For any reuse or distribution, you must make clear to others the license terms of this work. Any of the above conditions can be waived if you get permission from the copyright holder. If you are in doubt about any proposed reuse, you should contact the author via: http://abstrusegoose.com/contact .

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Be sure to visit http://abstrusegoose.com for the latest comics.

...and please feel free to make a donation at http://abstrusegoose.com/feedthegoose to gain major karma points.

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Acknowledgements There is nobody that I’d like to thank because I basically did everything by myself. Well, actually, I guess my readers deserve a mention. “Thank you” to all my readers and to all my supporters.

About the Author The author draws cartoons in his spare time and he wears a blue baseball cap a lot. He currently lives near Philadelphia, PA with his imaginary girlfriend.

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CONTENTS 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Introduction Convergent Subsequence Penis Size and IQ SETI Finally Receives a Signal LOLCAT Backlash There is No Spoon Schrödinger’s Infinitesimal Miscalculation Schrödinger’s Miscalculation - Part 2 Arguing With a String Theorist Blind Date The Red Button Ubuntu Sucks Math Text Ask Me Why I Never The Opportunist dckx The Birth of ENIAC I.I. Rabi’s Question Answered? That Annoying Friend Make a Wish Life Imitates Art Reality vs Fantasy Pi Out of the Closet At the Driving Range Calc-zilla Secrets and Lies Math vs Physics NSFW The Hottie Veritas Vos Liberabit Real Life The Inequivalence Principle Blind Date - Part 2 Particle in a Box An Elegant Weapon… Closed Timelike Curveball Best Friends The Exception LiveCommentJournal Today I Learned That… So Many Questions The Curve http://abstrusegoose.com

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44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 8

The Most Popular Girl in School I never lose this game What’s in a Name? A Wise Man Once Said… It could’ve been worse Dear CERN The Alpha Male 936 Little Blobs Frequently Asked E-Mail Question Answered Qapla’! Fun with Open-Ended Meta-Gödelian Statistics Fun with Statistics - Part 2 Family Reunion say what you mean Proof All Good Things… Free Pass stop me if you’ve heard this one Science is Supercool NUM63R5 2008: A Server Space Odyssey The Cantor Madness Cupcakes you’re not as cool as you think Darmok Blind Date - Part 3 I, Computer Yo, Adrian, we did it!!! The Belt Trick True Things Schrödinger’s (emotional) Miscalculation - Part 3 OCD - Obsessive Compulsive Don A Simple Puzzle A Simple Puzzle - SOLUTION Popular Science In the Beginning All You Zombies Batteries Included A Simple Request The Bionic Woman Hand Turkey The Mind of God You’re a Good Man… Ripoff The Adventures of Buckaroo Banzai… http://abstrusegoose.com

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The Purposeful Life Gift Horse The Butterfly Effect Young George Moment of Clarity Holiday Tradition Scientific Verification ask a silly question… Happy Zeno Year Computer Programming 101 Lie of Omission The Pantheon

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Introduction Welcome to my little book. Within these pages you will find a compilation of the first 100 Abstruse Goose cartoons that I posted on my perpetually server-challenged web site http://abstrusegoose.com. I must admit that I feel a little funny about putting together a book of my comics. After all, who the hell am I? What kind of narcissistic prick would publish a book of his crappy doodles and expect other people to read it? Aren't there, like, a gazillion other webcomics out there that are better than Abstruse Goose that never got compiled into a book? Sadly, that may be true, but potential accusations of narcissism notwithstanding, here is my book and here you are reading it. Now I'm sure you got this book to read comics and not to read some long boring exposition, so please feel free to skip this introduction entirely and proceed directly to the comics. I won't be offended... honestly. In fact, I insist that you skip this introduction and go to the comics right now. Go.

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WTF? You're still here? OK, read this intro if you want to but don't say I didn't warn you. I have always been fascinated by the concept of cartoons. In particular, I am fascinated by the way cartoons seem to be able to convey ideas to the reader in a manner that regular prose cannot do. A simple four-panel comic can sometimes be more effective at transmitting an idea than can a ten-page essay. In my opinion, Charles M. Schultz is the master in this regard. If you were a reader of the Peanuts comics as a kid and have not read it lately, I recommend going back and re-reading them. You may find that you will pick up some subtleties that you missed the first time around. I believe that the reason for the effectiveness of cartoons as a vehicle for information is not just a simple case of “a picture is worth a thousand words”. When people read comics, I believe that they often do so with a certain mindset that allows them to be more open to receiving new ideas. Generally, people don't want to be “preached at” or to have someone else's opinions or ideas rammed down their throats (I know that this is certainly the case with me). For this reason, it seems as if most people tend to live their everyday lives with certain mental filters activated by default; filters designed to catch and discard such unwanted detritus. However, people read comics to be entertained and not to learn new things or to garner new insight into the world around them. When reading comics, people http://abstrusegoose.com

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seem to turn off much of their mental filters, hence allowing them to be introduced to new ideas (and possibly new ways of thinking) that otherwise might have been dismissed. However, for the cartoonist (I use that term loosely when referring to myself), this attitude of the reader can be a double-edged sword. My readers check my cartoons seeking a good laugh (I assume) and sometimes find that I mysteriously seemed to have forgotten to include a punchline. I sometimes post a cartoon simply because I want to say something and not because I want to make people laugh. Some cartoonists go for “cute” or “clever” and a reader expecting a side-splitting, rim-shot-invoking punchline might be sorely disappointed by such a comic. Webcomics, as a sub-genre of comics, holds a particular fascination with me for several reasons. With the advent of the phenomenon of webcomics, anyone with a computer and an internet connection can be a cartoonist and have a potential audience of millions. Moreover, the response from the online community to any cartoon posted can be almost immediate. It still freaks me out when I draw a cartoon at my desk in my dusty little bedroom and within hours (sometimes minutes) of posting it online, I find that same cartoon reposted on blogs and web sites across the internets. When I first started posting, I actually used to read some of the online comments about my comics since I thought it would be a great way to get feedback and possibly to receive some constructive criticism. While it is true that, in the beginning, I did indeed get a lot of helpful advice and suggestions from the online comments, it turns out that there also happen to be a lot of... well, the only word to describe them would be “f*ckwads”, who have nothing useful to say. These f*ckwads certainly have every right to say whatever they want but unfortunately, it had a less than desirable effect on my ego. Therefore, towards the goal of maintaining the illusion of my awesomeness in my own mind, I have since given up the practice of reading online comments. Another reason that webcomics as a genre fascinates me is the amazing flexibility it affords the cartoonist. For the webcartoonist, there are is no longer any need for large evil syndicates (not that there's anything wrong with that) to distribute the comics to the audience. Also, there seems to be no limitation on the format (size, color, style, etc.) that can be adopted by the webcartoonist. I have seen some webcomics that have truly stretched the limits of what a comic can become. In that regard, I think that Randall Munroe, creator of the insanely popular xkcd comic, deserves a mention. He may not have been the first person to start a webcomic, but he certainly stretched the limits of the format and helped to define what we have come to think of as webcomics today. I started posting comics online regularly beginning in May 2008 as a fun little diversion and it remains exactly that today: a fun little diversion. However, don't let that fool you into thinking that I don't put a lot of time and energy into it. Some of the comics in this book actually took several hours to produce from conception to completion. Also, don't let the minimalist art style of my comics fool you into thinking that I can't draw. I am actually an exceptional artist but I simply choose to hide my world-class talent out of some misguided sense of modesty. Yeah, OK,... that previous statement is a total lie, but the truth is that my drawing style suits my purposes just fine. As I stated above, I believe that cartoons are a means of conveying ideas. Once the reader is in possession of the idea, the pictures and words are no longer needed, so why distract the reader with extraneous visual details? 12

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I draw cartoons about my observations about life and about things that interest me. Because of this, my cartoons tend to be very personal for me in the sense that they may only appeal to me and to people with similar interests. In this case, that means that many of the cartoons in this book are geared towards very science-y or math-y subject matter. For many people, the world of science is a bewildering jungle of arcane jargon and strange concepts; a world occupied by an even more bewildering collection of intellectual elitists and absent-minded professors. However, by including a lot of science in the comics, my intention is not to alienate or to confuse. Even if your background does not include much science, I think you'll find that having an understanding of the math-y/physics-y references that are sprinkled into the individual comics is usually not essential to grasping the gist of that strip as a whole. If you do happen to come from a science background, my hope is that even you may encounter in my comics a concept or two with which you may not be familiar. In fact, I do a little victory dance every time someone emails me to tell me that one of my comics prompted him/her to look up blah blah blah under Wikipedia and that he/she learned something new. However, for those of you who may be too busy to bother with Google searches or Wikipedia research, along with each comic in this book, I have included some additional author comments which I sometimes used to give simple explanations of some of the science references. Well, OK,... this introduction is already 1384 words longer than I would have liked (and much too serious), so without further ado, I hope you enjoy reading this book as much as I enjoyed creating it blah blah blah...

March, 2009

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Convergent Subsequence

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originally posted May 16, 2008 Many of you might recognize this as the cartoon version of the visual representation of the proof of the Bolzano-Weierstrass theorem. For the interested, one version of the theorem can be stated as: If a compact set C in R n contains a sequence, then that sequence has a convergent subsequence whose limit is in C. continued on page 145 16

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Penis Size and IQ

originally posted May 16, 2008 Obviously, there is no such study showing that penis size is inversely correlated with IQ. However, if such a study did exist, I imagine that many of the fellas out there would go to great lengths to convince everyone of his sub-average IQ. In panel five of the comic, the character mentions a polynomial time factoring algorithm so let's explain what that means First of all, what the hell is an algorithm? Informally, we can say that an algorithm is a collection of instructions for carrying out some computational task. For our purposes, that definition is good enough. Usually an algorithm works by taking an input, performing continued on page 147 http://abstrusegoose.com

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SETI Finally Receives a Signal

originally posted May 16, 2008 When I posted this comic, many people were already starting to get sick of the Rick Rolling phenomenon, but c’mon, admit it; you enjoyed getting Rick Rolled by the Pleiadians. Are we alone in the universe? For me, that is somewhat of a depressing prospect. I would much rather believe that our galaxy is teeming with intelligent life; but how likely is that to be true? Surely, the immense age of the Milky Way and the sheer number of stars in our galaxy demand that we at least accept the possibility. That is the motivation behind SETI – the Search for Extraterrestrial Intelligence. continued on page 150

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LOLCAT Backlash

originally posted May 16, 2008 LOLCATS may have been funny for a while but the cats are pissed. Is it just me or does the cat look like an ROUS (Rodent Of Unusual Size) in panel 3?

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There is No Spoon

originally posted May 16, 2008 This comic was inspired by a true event. After watching The Matrix, I actually held a spoon in my hand for five minutes and attempted to bend it with my mind. Obviously, it didn't work but a couple of days later, I walked into my kitchen and saw that same spoon in my utensils rack. Much to my surprise, I noticed that the handle of the spoon was bent by almost (I would estimate) 15 degrees. It took me a few seconds to realize what had happened. The day before, I was using that spoon to eat ice cream and every time I took a scoop, I apparently bent the spoon by a small increment; a small enough increment such that I wasn't even aware that I was bending the spoon. The cumulative effect of all of those scoops resulted in a spoon with a continued on page 151 20

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Schrödinger’s Infinitesimal Miscalculation

originally posted May 16, 2008 I'm sure that many of you are familiar with Schrödinger’s cat but I still feel compelled to give a short description here. However, please note that I will only be explaining enough to make sense of the comic and that many details will be omitted. The idea of “Schrödinger’s cat (paradox)” was put forth by the Austrian physicist Erwin Schrödinger in 1935 as a thought experiment to illustrate the absurdity of (what has come to be known as) the Copenhagen interpretation of quantum mechanics when applied to the “real” world of common sense and macroscopic objects. Imagine a box that is so perfectly sealed that no physical influence can get in or out. Now imagine that a cat is inside the box along with a device that can kill the cat when triggered by some “quantum event”. That is the setting for Schrödinger's cat. In Schrödinger's original version, the quantum event was the decay of a radioactive atom. Schrödinger asserted that the Copenhagen interpretation implies that the cat remains in a "superposition" of states: (both alive and dead) until the box is opened. I used a bit of jargon in the previous paragraphs so let's backtrack a little with a miniscience lesson. First of all what is quantum mechanics? To put it simply, quantum mechanics is the theoretical framework that describes the universe at the “smallest” scales: atoms, electrons, protons, quarks, etc. At such small scales, the “rules” are very different continued on page 153 http://abstrusegoose.com

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Schrödinger’s Miscalculation - Part 2

originally posted May 16, 2008 OK, I just couldn’t resist getting all meme-y again, so I gave you more LOLCATS. A clever reader emailed me about this comic and suggested that I should have titled it LOLKETS (thanks Vinnie), which is actually much funnier than the comic itself. Be sure to read the comments for comic #6 on page 21 for an explanation of the equations in the first and last panels. There WILL be a pop quiz. POP QUIZ: What was the probability of finding a LOLCAT in the box?

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Arguing with a String Theorist

originally posted May 16, 2008 This one caused a bit of discussion among some physicists. In particular, a blog post by physicist Luboš Motl amused me immensely. I don't think anyone has ever so thoroughly deconstructed a single comic in the history of... of... well, EVER!. If you want to read the blog post in its entirety see “Comments” on page 155.

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Blind Date

originally posted on May 16, 2008 It's an unfortunate reality that revealing too much of your inner geek on a first date will most likely scare your date away. One time, I was on a blind date at a restaurant and I did my level best to resist talking to (at) her about geek things. I was doing a good job until the very end when I looked at my watch and said something like, “Can we get out of here? Star Trek is on in 15 minutes.” In panel 2 of the comic, there is mention of something called symmetry groups. To learn more about symmetry groups, see “Comments” for comic #96 on page 201.

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The Red Button

originally posted on May 19, 2008 This comic is based on the idea of a fine-tuned universe. The notion of a fine-tuned universe, in turn, is based on the idea that the universe is the way it is because of the precise values of certain fundamental constants and that even slight variations of any of these constants would result in a universe that is radically different. To be specific, it is implied that any such universe would most likely not be conducive to the existence of matter or elements as we know them and hence to the existence of life as we know it. Many people (so I hear) consider this remarkable “fine-tuning” of the universe to be evidence that suggests the existence of a divine being of some sort managing a cosmic fine-tuning machine. I leave you, the reader, to draw your own conclusions, but whatever your opinion may be, I hope that you will learn two very important lessons from this comic: 1.) Always obey the Lord, and 2.) beware of angry monkeys.

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Ubuntu Sucks

originally posted on May 21, 2008 I’ve always kept separate computers for different operating systems. However, when I finally decided to install Linux on my Windows machine for dual booting, I couldn’t help but feel a little guilty, as if I was taking away hard disk space from Windows. FUN FACTS: The code illustrated in the first panel is an actual snippet of code from the master boot record (MBR) that is part of the boot process on most computers running Windows. The music depicted in panel 2 accurately represents the default startup music played by Windows Vista. The BCD store (mentioned in panel 4) is a database that contains boot configuration data and controls how the operating system starts up for computers running Window Vista. 26

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Math Text

originally posted on May 22, 2008 The theorem in the comic is actually taken from Walter Rudin’s text, Principles of Mathematical Analysis, which many math students affectionately refer to as Baby Rudin. Baby Rudin is (in)famous for offering very sparse (yet clean and elegant) proofs and, sometimes, for not offering any proof at all. The theorem in the comic is not actually stated as a theorem in the text, but (in true Rudin fashion) is left as an end-of-chapter exercise for the student.

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Ask Me Why

originally posted on May 25, 2008 Yes, I'm superstitious. I don't like the number 13, I make a wish every time I see 11:11 on my watch, and I don't like it when a black cat crosses my path. However, my reason for this seemingly irrational behavior has nothing to do with any kind of belief in the supernatural. When I originally posted this “comic”, I had intended also to post an accompanying essay explaining why I still hold these seemingly irrational beliefs. I still have not written that essay but if you want to read a simplified version, then read the comments for comic #5 regarding the unconscious mind and goal-priming. By walking under a ladder, is it possible to inadvertently prime yourself towards behavior that increases your probability of experiencing an unfortunate outcome? Who knows?... but why take the chance? continued on page 163

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I Never

originally posted on May 25, 2008 Just for the record, I never snorted a line of coke off of a stripper’s ass at Chucky’s bachelor party. Chucky never had a bachelor party. However, it is true that I have, on occasion, revealed too much information while playing ‘I Never’. The Riemann Hypothesis (mentioned in panel 4) is one of the most celebrated unsolved problems in all of mathematics. It was first proposed by mathematician Bernard Riemann in 1859. It is one of the so-called Millennium Prize Problems as designated by the Clay Mathematics Institute which offers a $1,000,000 prize for its solution. [I will include more about the Riemann Hypothesis in later drafts of this book.]

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The Opportunist

originally posted on May 25, 2008 Important Lesson: If life gives you an opportunity, pull the goddamn trigger.

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dckx

originally posted on May 28, 2008 original blog post: original: http://xkcd.com/139 This is my homage to Randall Munroe, creator of xkcd. I still haven’t asked for Randall 's permission to publish this yet.

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The Birth of ENIAC

originally posted on May 31, 2008 The University of Pennsylvania displays part of the original ENIAC computer in its ENIAC museum where it is usually safely protected from the public behind a display window. During one homecoming weekend, the display glass was removed and visitors were allowed to touch it. It was a bit of a thrill when I reached out my hand and touched this important part of computer history. That must have been how Picard felt when he touched the Phoenix.

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I.I. Rabi’s Question Answered?

originally posted on June 2, 2008 This is one of the few comics that I drew where an understanding of the “science” is probably necessary to understand the joke. So,… let’s take a very short science break: In the late 1970s, physicists began to formulate what can be considered the most successful theory of nature in history. It is called the Standard Model. The Standard Model does no less than attempt to identify all of the fundamental constituents of the universe and to specify how they interact. On the surface, the result is a surprisingly simple picture. Most of the phenomenon of the everyday world can be explained with just six particles: the electron, the up and down quark, the gluon, the photon, and the Higgs boson. As it stands today, the Standard Model also includes eleven other particles that account for the various other esoteric phenomenon whose study is usually under the exclusive purview of particle physicists with thick glasses. The elementary particles can be broadly grouped into three categories: fermions, force carriers, and the Higgs boson. The fermions can be thought of as the “matter” particles; the particles that make up everything that we see around us. The fermions can be further subdivided into two types: leptons and quarks. The force carriers are responsible for the forces with which the fermions interact. Breifly, the four known forces are: electromagnetism, the weak force (which plays a role in the formation of chemical elements), the strong force (which is responsible for holding protons, neutrons, and nuclei continued on page163 http://abstrusegoose.com

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That Annoying Friend

originally posted on June 3, 2008 Some of you younger readers might not be able to identify with this comic (even though you may understand it); but I assure you that you will experience this by the time you get to be my age. If you don’t, well,… then you’re that guy.

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Make a Wish

originally posted on June 5, 2008 Most readers will probably recognize this one right away as an autostereogram (also called Magic Eye), a popular fad in the 1980s and 1990s. Autostereograms are computergenerated images which when viewed with crossed eyes, appear as a vivid threedimensional image magically suspended in mid-air. So if you have not yet seen the image hidden in the picture, please give it a crack now. It works best if you place your face about 30 cm from the image and then cross your eyes until an image appears. Make sure to make a wish before doing so. If you're not in the mood for my shenanigans, then see “Comments” on page 164 for the spoiler.

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Life Imitates Art

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originally posted on June 5, 2008 original blog post: required reading at The Academy. [I will add comments for this comic in later drafts of the book.] 38

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Reality vs Fantasy

originally posted on June 9, 2008 See “Comments” on page 165. http://abstrusegoose.com

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Pi

originally posted on June 11, 2008 original blog post: If you don’t already, you really should eat toothpaste for dinner everyday. See “Comments” on page 166.

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Out of the Closet

originally posted on June 16, 2008 If you’re a Browncoat (Firefly fan), then there’s a good chance that you’ve used the word “gorram” once or twice in a real-world situation; but tread warily, my friend. The word “gorram” is strangely addictive. Once you use it in real life, it becomes a habit and you may find yourself using it during inappropriate situations.

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At the Driving Range

originally posted on June 13, 2008 This particular comic is actually a bit misleading. Many people took this comic to mean that I had been receiving tons of hate mail from angry machete-wielding xkcd fans when in actuality, almost all of the emails I received were filled with immense Abstruse Goose love. Strangely enough, the only exception occurred as a result of this comic. One person, in particular, read the comic and he (apparently) assumed that xkcd fans were supposed to be sending me hate mail, so he did just that. However, he emailed me back a couple of days later to apologize. This comic is also misleading in that it seems to imply that I dislike any comparisons of my comic with xkcd. Not true, dawg! In fact, I find the comparisons to be quite flattering. Besides, I can't blame people for making the comparison since one cannot deny the similarity between the two.

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Calc-zilla

originally posted on June 16, 2008 original blog post: The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. —William James, Collected Essays I still remember the exhilarating feeling I had when I first learned calculus in high school. It was as if I had been living in a dark room my entire life and suddenly somebody turned on the lights revealing an entire universe that I never knew was there. Unfortunately, I didn't have quite as active an imagination as little Billy does; but that's OK. I rather like the cold austerity of mathematics.

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Secrets and Lies

originally posted on June 18, 2008 original blog post: LO KPI FG ZWXXB, JSU RVKMVY YK QRQQWQY. And Caesar Vigenère says: “Easy as pi.” Random numbers are crucial for many fundamental aspects of cryptography. Breaking the random number generator could very well compromise the integrity of any information security system that utilizes it. It is for this reason that one specific algorithm for generating random numbers, called Dual_EC_DRBG, fell under scrutiny in late 2007. continued on page 166

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Math vs Physics

originally posted on June 21, 2008 It has been pointed out to me that this comic could probably have worked with [anything] vs. [anything] as the topic but the math vs physics debate has always fascinated me. [I will probably include a comment about the math vs physics debate in later drafts of the book.]

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NSFW

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originally posted on June 25, 2008 Some of the movie references in my comics are subtle. This one… not so much. This comic is also an experiment to see how much mileage I can get from my angry monkeys. As a side note, I often wonder how many people actually really heed the NSFW warning while at work.

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The Hottie

originally posted on June 28, 2008 I should probably mention that this comic is not really a slam on Paris Hilton herself, but on her public persona. I have no doubt that the real Paris is quite intelligent (whatever that means). At least she’s intelligent enough to know that what people really want to see from her is a dumb blonde. Well, that’s what she gave us and we, the adoring public, ate it up. FUN FACT: "What is the air speed velocity of an unladen proton?" is a spoof of a line from the movie Monty Python and the Holy Grail. The actual line was "What is the air speed velocity of an unladen swallow?".

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Veritas Vos Liberabit

originally posted on July 1, 2008 original blog post: “Wait, wait, wait… is this making fun of fundamentalist christians or scientists?… or both?… or neither?” Discuss with yourself. This comic may lead the reader to believe that I am anti-religion, but that is not exactly a correct characterization of me. For the most part, I respect other people's beliefs, but more importantly, I respect other people's freedom to believe what they want... blah, blah, blah... FUN FACT: The words “veritas vos liberabit” can be be translated as “the truth will set you free”.

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Real Life

originally posted on July 10, 2008 original blog post: Sorry for not updating this comic for so long but I’ve been pretty busy with “real life” stuff. I should be back on track shortly. Multitudinis imperitæ non formido judicia; meis tamen, rogo, parcant opusculis——in quibus fuit propositi semper, a jocis ad seria, a seriis vicissim ad jocos transire. Due to “real life” obligations, I was unable to post a new comic for about 10 days. I just posted this one to let people know that I would be back soon. See “Comments” on page 169

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The Inequivalence Principle

originally posted on July 11, 2008 original blog post: There then occurred to me the ‘glücklichste Gedanke meines Lebens,’ the happiest thought of my life,… The title of the comic is obviously a play on the words ‘The Equivalence Principle’. The Equivalence Principle is essential to the formulation of the General Theory of Relativity. At its core is the idea that gravitational and inertial mass are equivalent. In simple terms, this means that the gravitational force experienced by an object due to a massive body (such as a planet) is the same as the force that the object would experience if it were accelerating. A full discussion of the General Theory of Relativity is beyond the scope of this “text”. The idea occurred to a young Einstein in 1907 as he sat in his office in Bern, Switzerland. He later recalled that that had been the “happiest thought” of his life.

[I will probably include a brief description of GR in later drafts of the book.]

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Blind Date – Part 2

originally posted on July 13, 2008 original blog post: A good GILF is hard to find. A bad GILF… impossible.

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Particle in a Box

originally posted on July 16, 2008 original blog post: I’ve always felt sorry for that little guy. The “particle in a box” is a problem in physics that is often presented in introductory quantum mechanics courses. The problem consists of finding a solution to a single particle inside an infinitely deep potential well. Every time I encounter this problem, I always imagine myself to be the particle, and I gotta tell ya, it’s lonely in that damn box. [I will include more on this subject in later drafts of this book.]

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An Elegant Weapon…

originally posted on July 18, 2008 original blog post: …for a more civilized age. [Battles] of the future are the [battles] of the mind. 1. c4 e5 2. Nc3 Nc6 3. g3 f5 4. Bg2 Nf6 5. d3 Bc5 6. e3 f4 7. ef4 0-0 8. Nge2 Qe8 9. 0-0 d6 10. Na4 Bd4 11. Nd4 ed4 12. h3 h5 13. a3 a5 14. b3 Qg6 15. Nb2 Bf5 16. Qc2 Nd7 17. Re1 Nc5 18. Bf1 Ra6 19. Bd2 Rb6 20. Ba5 Rb3 21. Bd2 Ra8 22. a4 Ra6 23. a5 Kh7 24. Red1 b6 25. Be1 ba5 26. Na4 Rd3 27. Bd3 Bd3 28. Qa2 Nb4 29. Qa3 Nc2 30. Qb2 Na1 31. Ra1 Na4 32. Ra4 Qe4 33. Ba5 ????????? I suppose that in an ideal world, all conflicts would be resolved with the mind. FUN FACT: The chess moves listed above are from the 1969 match between Anthony Saidy and Bobby Fischer. I tried to diagram that last move (before the question marks) in the final panel of the comic but the resolution isn’t great enough to actually see it.

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Closed Timelike Curveball

originally posted on July 20, 2008 See “Comments” on page 169.

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38 Best Friends

originally posted on July 23, 2008 original blog post: It is easier to destroy than to create. See “Comments” on page 170. 56

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The Exception

originally posted on July 24, 2008 In this comic, I'm likening the life cycle of internet memes to chemical reactions. Many chemical reactions must achieve a certain energy level (called the activation energy) before they can take place. In the equation depicted in the comic, Ea represents the activation energy and the equation itself is called the Arrhenius equation, which shows the relationship between Ea and the rate of the reaction. Chuck Norris, of course, obeys no equation.

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LiveCommentJournal

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originally posted on July 28, 2008

See “Comments” on page 172.

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Today I Learned That…

originally posted on August 1, 2008 original blog post: Have a good weekend everyone. NOTE: I still haven’t grokked the relationship between motivic cohomolgy and Milnor ktheory. FUN FACT: The philosophy espoused in panel 4 was stolen from a line by Warren Buffett.

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So Many Questions

originally posted on August 4, 2008 Since posting this cartoon, I actually got an answer to one of the questions. It turns out that she really doesn’t like it when I do that. You should also know that the answer to the ultimate question of Life is 42. Let me take some time to address some of the more interesting questions from this comic. See “Comments” on page 176.

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The Curve

originally posted on August 8, 2008 original blog post: NOTE: This cartoon is based on a true story entirely fictional. Any similarity between the student in the comic and my friend Ben from analysis class is slightly exaggerated purely coincidental. In all fairness to Ben, he wasn't really such a cut-throat. He was just very... uh... studious. SIDE NOTE: Yes, I realize that I technically misused the word “decimate”.

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The Most Popular Girl in School

originally posted on August 11, 2008 See “Comments” on page 179.

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I never lose this game

originally posted on August 14, 2008 original blog post: The Q-tip thing really happened to a friend of a friend of mine. I’m told she never fully regained her hearing in her left ear. However, that story never fails to elicit a grade-A cringe whenever I tell it. original hovertext: I bet I can make you cringe.

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What’s in a Name?

originally posted on August 18, 2008 original blog post: HOW TO MAKE SPLENDA® IN 5 EASY STEPS Splenda® is basically made by adding three chlorine atoms to sucrose. The chemical formula for Splenda® is 1,6-dichloro-1, 6-dideoxy-β-D-fructofuranosyl-4-chloro-4-deoxy-α-D-galactopyranoside. It is also known as sucralose. According to SRI Consulting PEP Review #90-1-4 (July 1991), the following is the process by which sucralose is synthesized: 1. sucrose is tritylated with trityl chloride in the presence of dimethylformamide and 4-methylmorpholine and the tritylated sucrose is then acetylated with acetic anhydride, 2. the resulting TRISPA (6,1′,6′-tri-O-trityl-penta-O-acetylsucrose) is chlorinated with hydrogen chloride in the presence of toluene, 3. the resulting 4-PAS (sucrose 2,3,4,3′,4′-pentaacetate) is heated in the presence of methyl isobutyl ketone and acetic acid, 4. the resulting 6-PAS (sucrose 2,3,6,3′,4′-pentaacetate) is chlorinated with thionyl chloride in the presence of toluene and benzyltriethylammoniumchloride, and 5. the resulting TOSPA (sucralose pentaacetate) is treated with methanol (wood alcohol) in the presence of sodium methoxide to produce sucralose. Brings back memories of Organic Chemistry class. Ahhh… good times. http://abstrusegoose.com

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DISCLAIMER: The views and opinions expressed in this comic are solely those of the cartoonist and should not be attributed to the powerful and evil Abstruse Goose Corporation or any of its worldwide subsidiaries. We are currently unaware of any scientific study which shows that Splenda® may be harmful to human health in any way… but… I mean… chlorine? toluene? methanol? DAYAMN!… I’m just saying. “What’s in a name? That which we call sucralose By any other name would taste just as sweet.“ ADDITIONAL AUTHOR COMMENT: I feel that I must expand upon the disclaimer I included in the original blog post. Please allow me to make several points here. First of all, I hope that the comic didn’t give anyone the impression that I was calling Splenda a poison. I was being very serious in the original disclaimer when I said that there is no (to my knowledge) scientific evidence that Splenda is harmful to human health in any way. I’d like to point out that the presence of chlorine atoms in a substance’s molecular structure does not entail that that substance is dangerous to human health. After all, ordinary table salt (NaCl) is composed of sodium and chlorine. I’d also like to point out that the use of the chemical procedures involving toluene, wood alcohol, etc., does not necessarily mean that those substances will be present in the final product (if the procedures were carried out correctly). The point of the comic was this: Most people would be reluctant to consume sucralose if they knew its molecular structure and knew how it was synthesized (despite the lack of any logical reason to fear it); but give it a pretty name, and suddenly their entire perspective is transformed. We humans are funny that way. This comic wasn’t about any specific commercial product. It was about human psychology.

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A Wise Man Once Said…

originally posted on August 20, 2008 original blog post: NOTE: Sorry. No ST:TNG reference for comic #47. Time to revoke my Trekkie license. In 212 B.C., Roman forces invaded Archimedes' home town of Syracuse during the Second Punic War. According to a popular account of the story, a Roman soldier discovered Archimedes contemplating a mathematical problem and ordered him to come with him. It has been rumored that Archimedes was drawing figures on the ground and replied, “Noli turbare circulos meos”, or “Do not disturb my circles.” The soldier became enraged and killed him on the spot. If that story is true, then I guess, technically, Archimedes wasn't killed for his science. He was killed for just being a geek. In 1600, Giordano Bruno was burned at the stake for, among other things, his beliefs which consisted of a mixture of Christianity and alchemy. Today many people mistakenly attribute his execution to his belief in Copernican heliocentrism but that was never officially stated as his “crime”. continued on page 180

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It could've been worse

originally posted on August 26, 2008 You'd think that Morpheus coulda been a little more forthcoming.

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Dear CERN

originally posted on August 30, 2008 original blog post: You have been warned. The Standard Model of physics predicts the existence of a particle called the Higgs boson (for a brief description of the Standard Model see page 33). The Higgs interacts with the other particles of the Standard Model in a unique way that gives them mass. The Higgs has not yet been directly detected by experiment but many physicists believe that if the Higgs boson does exist, then the Large Hadron Collider will almost certainly be able to detect it. I posted this comic around the time the LHC was first scheduled to start up (before the malfunction that caused the delay). I have to admit, I got caught up in all the LHC excitement. I've got a fever, and the only prescription is the discovery of the Higgs boson.

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The Alpha Male

originally posted on September 1, 2008 What the hell is this?!! Everybody knows that in this enlightened post-industrial era, it's the geeks who are the alpha males (and alpha females).

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936 Little Blobs

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originally posted on September 3, 2008 original blog post: That last line is directed solely at my own damn self. This is a trick that my father taught me when I was quite young. I never forgot it. What a wise man he is. I bet my dad can beat up your dad.

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Frequently Asked E-Mail Question Answered

originally posted on September 9, 2008 original blog post: ...and I have no idea why. By the time I wrote this one, I was starting to receive some emails from people asking somewhat 'personal' questions about me. I may have been flattered by such inquiries but, as I tend to be a little cyber-shy, I always answered those emails with short, sparse replies. This made me feel as if I was being rude so I thought I could answer one of the questions in comic form.

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Qapla'!

originally posted on September 12, 2008 original blog post: The Wookiee Turing Test was even easier. See “Comments” on page 181. 74

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Fun with Open-Ended Meta-Gödelian Statistics

originally posted on September 15, 2008 WTF was I thinking when I posted this one? I think this comic would have been good if I omitted those last two lines; but, noooo, I tried to get all fancy and clever-like and that just confused people. So... I fixed it. Here's the fixed version:

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Fun with Statistics - Part 2

originally posted on September 17, 2008 original blog post: This one is true. Really. Of course, people who understand statistics can spot the flaw in the statement in the last panel. But, c'mon......WERE YOU NOT ENTERTAINED?!!!

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Family Reunion

originally posted on September 19, 2008 original blog post: Inspired by true events. See “Comments” on page 182. http://abstrusegoose.com

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say what you mean

originally posted on September 22, 2008 original blog post: Don’t read too much into this comic. I do love Star Trek, but during football season, I bleed Eagles green!!! 78

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Proof

originally post on September 24, 2008 original blog post: Sometimes I just don’t understand other people’s logic. Movie buffs may recognize those graphs from the movie Good Will Hunting. I have to say that I'm not really perplexed by Dane Cook fans. I like Dane Cook, too... when I'm drunk.

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All Good Things...

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originally posted on September 25, 2008 This comic was inspired by an excellent Steven Wright joke: Two babies were born on the same day at the same hospital. They lay there and looked at each other. Their families came and took them away. Eighty years later, by a bizarre coincidence, they lay in the same hospital, on their deathbeds, next to each other. One of them looked at the other and said, "So. What did you think?"

Of course the joke loses some of its flavor when it's not being told with Steven Wright's droll delivery style. However, the message of that joke stuck with me and has even affected my perspective on the fleeting nature of life. Whowoulda thunk that a Steven Wright joke could have had such a profound impact on me?

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Free Pass

originally posted on September 28, 2008 original blog post: Who would you put on your free pass list?

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stop me if you've heard this one

originally posted on October 1, 2008 See “Comments” on page 183.

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Science is Supercool

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originally posted on October 3, 2008 See “Comments” on page 183. 88

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NUM63R5

originally posted on October 6, 2008 original blog post: It’s always the simple things that get me. OK, maybe you don’t find that equation as fascinating as I do. But I also get distracted by shiny objects and Wheel of Fortune. See “Comments” on page 184.

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2008: A Server Space Odyssey

originally posted on October 9, 2008 original blog post: Sorry for the downage, folks. Some of you may have noticed that the site was down for a while. My server was attacked by a horde of angry monkeys, but don’t worry… we drove the barbarians back outside the gate. That’s my story and I’m stickin’ with it. We now return you to our regularly scheduled comics. Alright, I'm gonna come clean and admit that angry monkeys did not really attack my server. However, it is true that a server glitch left my site unavailable for approximately 2 days.

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The Cantor Madness

originally posted on October 10, 2008 This comic is the first of several instances in which I violate my prime directive of not referencing politics. Oh well. Part of my motivation for drawing this one was so that I could link to an interesting BBC documentary called Dangerous Knowledge which tells the stories of four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing. The documentary explores the link between the genius and madness of these scientists. In my opinion, the documentary was a bit on the sensationalistic side, but I still think it is worth viewing. Check it out.

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Cupcakes

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originally posted on October 13, 2008

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you're not as cool as you think

originally posted on October 15, 2008 I used to go bar hopping every Friday night with my co-workers. 'The boys' would terrorize the local singles' scene in a shameless display of macho bravado. The cheetahs in the comic?-- dat dem. http://abstrusegoose.com

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Darmok

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originally posted on October 16, 2008 original blog post: I can't wait. See “Comments” on page 185.

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Blind Date – Part 3

originally posted on October 19, 2008 original blog post: C’mon,… you know I wouldn’t leave y’all hangin’. Admit it. When you saw this comic, you were happy that I got laid, weren't you? NOTE: Yes, my spelling of the word S-I-K-E was intentional so stop emailing me about it.

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I, Computer

originally posted on October 23, 2008 The rapid growth of modern computing power is driven by the integrated circuit. In particular, the microprocessor that is at the heart of every modern computer has paced the explosive growth of computer power for several decades. A microprocessor is basically a small wafer of (usually) silicon with a vast array of transistors placed on it in various patterns so that they can accomplish different computational tasks. A transistor can be thought of as a valve that controls the flow of electricity. The first transistors were crude devices that were about the size of a dime. Before the invention of the transistor in 1947, bulky vacuum tubes were used for the same purpose (see comic #17 on page 32). Today, transistors are microscopic and millions of them can be crammed onto a microprocessor about the size of a postage stamp by etching grooves onto a silicon wafer using beams of light. continued on page 185 http://abstrusegoose.com

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Yo, Adrian, we did it!!!

originally posted on October 30, 2008 Congrats to my Philadelphia Phillies for becoming the 2008 World Series champions.. ...or to quote Chase Utley on live TV: “WORLD F*CKING CHAMPIONS!”

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The Belt Trick

originally posted on October 31, 2008 original hovertext: Save money this Halloween. Use the belt trick. OK, time for another mini math/physics break. In the 1920s, physicists discovered that electrons behaved as if they were spinning. It might be tempting to take this to mean that an electron spins the way the Earth rotates about its axis, but that is not the correct way to think about it. Electrons are considered to be point particles with no physical extension in space, so what exactly is spinning? It turns out that the electron has a kind of intrinsic spin that is quantum mechanical in nature. In fact, it was later discovered that all of the elementary matter particles had spin similar to that of the electron. continued on page 187

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True Things

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originally posted on November 3, 2008 See “Comments” on page 188. 104

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Schrödinger’s (emotional) Miscalculation - Part 3

originally posted on November 7, 2008 original blog post: I know that some of you would druther have seen it go the other way,… you sick bastards. But I just happen to be in a good mood.

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in a parallel world

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One Giant Leap

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OCD – Obsessive Compulsive Don

originally posted on November 10, 2008 original blog post: Just be thankful I didn’t post the original 24-panel version of this comic. OK, I'm pretty sure that the only people who 'got' this one were two or three of my old college friends. Note to self: don't use inside jokes in the comics. 108

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A Simple Puzzle

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originally posted on November 11, 2008 original blog post: hint: Panels 7, 12, and 21 are already in the correct position. If you get the correct sequence, I can’t offer you a prize or anything but you will have my undying respect. And I’ll buy you a beer the next time I’m in your neighborhood. “Thank you” to all the people who participated. I still owe y'all a beer.

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A Simple Puzzle – SOLUTION

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originally posted on November 14, 2008 original blog post: This is the solution to the Simple Puzzle. Apparently it was too simple as I now owe two months’ salary worth of beer to people in about 9 different countries. Many of you recognized it right away as Hotel California by the Eagles. My apologies for the misleading hint that threw some of you off. Anyway, here are the lyrics along with the correct sequence (just in case you don’t believe me): 6. On a dark desert highway, cool wind in my hair Warm smell of colitas, rising up through the air Up ahead in the distance, I saw a shimmering light 2. My head grew heavy and my sight grew dim I had to stop for the night 8. There she stood in the doorway; 10. I heard the mission bell 112

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13. And I was thinking to myself, this could be heaven or this could be hell 17. Then she lit up a candle and she showed me the way There were voices down the corridor, I thought I heard them say… 7. (CHORUS) Welcome to the hotel california Such a lovely place Such a lovely face Plenty of room at the hotel california Any time of year, you can find it here 19. Her mind is tiffany-twisted, she got the mercedes “bends” 3. She got a lot of pretty, pretty boys, that she calls friends How they dance in the courtyard, sweet summer sweat. Some dance to remember, some dance to forget 4. So I called up the captain, please bring me my wine He said, we haven’t had that spirit here since nineteen sixty nine 9. And still those voices are calling from far away, Wake you up in the middle of the night Just to hear them say… 12. (CHORUS) Welcome to the hotel california Such a lovely place Such a lovely face They livin it up at the hotel california What a nice surprise, bring your alibis 5. Mirrors on the ceiling, The pink champagne on ice 14. And she said we are all just prisoners here, of our own device 1. And in the masters chambers, They gathered for the feast 16. They stab it with their steely knives, But they just can’t kill the beast 15. Last thing I remember, I was Running for the door I had to find the passage back To the place I was before 18. relax, said the night man, We are programmed to receive. 11. You can checkout any time you like, 20. But you can never leave! 21. (GUITAR) NOTE: I admit that some of the panels are somewhat ambiguous and that slight variations of my intended sequence also work. http://abstrusegoose.com

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Popular Science

originally posted on November 15, 2008 original hovertext: Sometimes I feel like I'm being mocked. Please see “Comments” on page 193. 114

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In the Beginning

originally posted on November 18, 2008 original hovertext: But until we figure it out, I'm going to imagine it however I want.

See “Comments” on page 193.

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All You Zombies

originally posted on November 19, 2008 See “Comments” on page 194. 116

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Batteries Included

originally posted on November 22, 2008 original blog post: OK… this one looked funnier in my head at 3 in the morning. original hovertext: In version 3.0, you don't Dive Into Python. Python Dive Into you. When I drew this one, the release of Python 3.0 was imminent. I had to show my love.

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A Simple Request

originally posted on November 24, 2008 original blog post: By the way, did you ever wonder what joke could possibly have had that punchline?

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The Bionic Woman

originally posted on November 25, 2008

Make no mistake. The cyborgs are already among us. Many people with hearing loss have had cochlear implants that can translate sound into electrical impulses which the auditory nerve interprets as sound. Retinal chips have been developed that can be implanted behind the eye which converts light signals into electrical impulses which the optic nerve interprets as vision. There have even been cases where electrodes have been implanted into the V1 area in the back of the brain to produce visual images directly, bypassing the eye altogether. Amputees can now be fitted with prosthetic arms that can be directly controlled through connections to biological nerves. Scientists have even developed methods for growing biological neurons directly onto silicon chips in specific configurations that allows the neurons to communicate with computer circuitry. In a remarkable experiment conducted in 2004, a University of Florida researcher connected a culture of rat neurons (in a petri dish) to a computer flight simulator and the network of neurons actual learned how to “fly a plane”. Could a direct braincomputer interface be far behind? Resistance is... ahh, never mind.

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Hand Turkey

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originally posted on November 28, 2008 original blog post: To all my friends here in The States, I hope your Thanksgiving was free of incident. For solutions see “Comments” on page 194.

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The Mind of God

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originally posted on December 1, 2008 See “Comments” on page 195.

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You're a Good Man...

originally posted on December 4, 2008 original blog post: Just in case you're one of the few people left on the planet who hasn't yet seen the drawing:

This one combines my love of Peanuts with my eager anticipation of The Watchmen movie. Note to self: remember to get permission from artist to publish this drawing.

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Ripoff

originally posted on December 5, 2008 See “Comments” on page 197.

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The Adventures of Buckaroo Banzai...

originally posted on December 8, 2008

Across the Third Dimension

original hovertext: The demented Dr. Lizardo's evil plan was short-lived in Riemannian Flatland.

See “Comments” on page 197.

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The Purposeful Life

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originally posted on December 10, 2008 Who the hell says “Weeeeeeeee!” when they drive?

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Gift Horse

originally posted on Sadly, this one is based on me when I was in college.

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The Butterfly Effect

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originally posted on December 16, 2008 See “Comments” on page 198.

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Young George

originally posted on December 19, 2008 original blog post: I keed. I keed. If I met George Lucas today, I’d totally kiss his ass. (…and so would you;… admit it)

I don't really participate in any of the George Lucas bashing that goes on out there. I actually happen to think that George is a great genius... FOR ME TO POOP ON!!!... No, I keed, I keed. As far as I'm concerned, in all seriousness, Star Wars and The Empire Strikes Back truly are masterpieces of cinematic history. Nothing can ever take that away from George.

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Moment of Clarity

originally posted on December 22, 2008

“I can safely say that nobody understands quantum mechanics.” ---Richard Feynman

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Holiday Tradition

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originally posted on December 24, 2008 original blog post: Happy Holidays, everyone! My holiday tradition actually isn't as sad as I made it out to be in the comic. In fact, I rather enjoy my annual Christmas walk.

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Scientific Verification

originally posted on December 27, 2008 original blog post: Hey, check it out. I did a rare guest comic strip. See “Comments” on page 198.

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ask a silly question...

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originally posted on December 29, 2008 original blog post: …so the next time your kid asks you, “Why do I need to learn this stuff?”, do what any self-respecting parent would do…

See “Comments” on page 201. 138

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Happy Zeno Year

originally posted on December 31, 2008 original blog post: Happy New Year from Abstruse Goose!

Zeno of Elea (495-435 B.C.) shocked the philosophers of the ancient Greek world by inventing four seemingly innocent paradoxes that seemed to have no solution. This comic illustrates a variation of the second of those paradoxes called Achilles and the Tortoise. Suppose Achilles is running to catch up to a tortoise crawling ahead of him. Before Achilles can reach the tortoise, he must first reach the place where the tortoise started. After reaching that place, Achilles would still be behind because the tortoise is in motion. By repeating this argument, it seems that the tortoise must always be ahead. Today, we can view that paradox from the point of view of modern mathematical analysis and the difficulties seem to be easily resolved. Hmmm... well OK, if you say so; but something about Achilles and the Tortoise still... gnaws at me.

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Computer Programming 101

originally posted on January 2, 2009 original blog post: Thank God some people don’t need to see so far under the hood.

See “Comments” on page 205. 140

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Lie of Omission

originally posted on January 5, 2009

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100 The Pantheon

originally posted on January 9, 2009 original blog post: It’s my house and I’ll decorate it however the hell I want.

I may not be in the same league, but that don't mean I ain't awesome.

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COMMENTS 1

Convergent Subsequence

continued from page 16 The essence of this theorem is not difficult to understand so, for the curious, I present a short explanation. An explanation of this theorem requires that we learn some definitions first. You may already be familiar with the concept of Euclidean 2-space, denoted by R2, which can be thought of as the usual xy-plane that we learned in geometry class. Likewise, Euclidean 3space, denoted by R3, can be thought of as the usual 3-dimensional space in which we all live. Well, Euclidean n-space, denoted by Rn, simply refers to the space of n dimensions, where n is an integer. For this explanation, let's just consider R2. The ideas can be generalized to higher dimensions. A sequence in R2 (denoted by x1, x2, x3,...) can be thought of as an infinite set of points in the xy-plane as depicted in FIG. 1-1. Every point is labeled with an integer from one to infinity. Note that the terms of the sequence x1, x2, x3,... need not necessarily be distinct.

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A convergent sequence x1, x2, x3,... is a sequence that (in a sense) gets arbitrarily close to another point x as we go further out in the sequence. For a more precise definition of a convergent sequence, we can think about it as a game played between a protagonist and a challenger. Imagine that the protagonist claims the sequence converges to the limit x. The challenger then draws a circle around x. If the protagonist can find a point in the sequence such that every point in the sequence that comes after that point lies completely in that circle, then he wins that round. An example of this process is depicted in FIG. 1-2, where every point of the sequence after x8 appears to be contained within the circle.

FIG. 1-2 For the next round, the challenger draws an even smaller circle around x. If the protagonist can win every round in this fashion, no matter how small the circle is, then the protagonist wins and the sequence is said to converge to the limit x. Notice that it is possible to create another sequence out of an original one by picking out certain elements. For example, from the sequence x1, x2, x3,..., you could pick out (say) x3 and call it the first element of your new sequence. Then you could pick (say) x9 and make that the second element of your new sequence and so on. If all of the elements of your new sequence retains the same order relation as the original, then that new sequence is said to be a subsequence of the original. OK, now for some more definitions. Some sets in R2 are called closed and some sets are called open (and some sets are neither). Generally speaking, a set is closed if that set includes its boundary (this is not actually a precise definition). For a simple example, consider an ordinary square in R2. You can think of the square as your property surrounded on all four sides by a fence. If all of the fence is part of your property, then the square is a closed set. If none of the fence belongs to you (i.e. it belongs to your neighbors), then the square is an open set. A set in R2 is called bounded if you can draw a circle that contains the entire set. The square is an example of a bounded set, since we can always draw a circle that contains the entire square. An example of a set that is not bounded is the set of all points above the xaxis. No matter how big you draw a circle, it will never contain all of the points above the xaxis. Note that not all closed sets are bounded. 146

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A (nonempty) set in R2 is called compact if it is both closed and bounded. The idea of “compactness” is of paramount importance in the field of mathematics. In fact, sometimes it seems as if mathematicians spend half of their time just proving the compactness of spaces. So there. Now we finally have all the definitions we need to understand the BolzanoWeierstrass theorem. For convenience, I restate it here: If a compact set C in R n contains a sequence, then that sequence has a convergent subsequence whose limit is in C.

The Bolzano_Weierstrass theorem is an important result in the study of mathematical analysis and it is key for proving many other important results such as the Mean Value theorem and certain integrability theorems. Using comic #1 as a hint, can you think of how one might go about proving the BolzanoWeierstrass theorem?

2

Penis Size and IQ

continued from page 17 calculations on the input, and then producing an output. Computer scientists and programmers are often interested in how fast an algorithm runs. In particular, in the field of cryptography, algorithms are used to encrypt data. However, in theory, every encryption algorithms can be cracked by yet another algorithm. Ideally, a cryptographer would want the encryption algorithm to run quickly while any algorithm that can crack it would run slowly. The running time of an algorithm is referred to as its time complexity. There is a way to classify algorithms according to their time complexities. Consider an algorithm that takes an input of length n. For example, the input might be a binary string of length n. The algorithm must perform a certain number of steps before producing a final output. Now let f(n) be the maximum number of steps that the algorithm performs on any input of length n. Then we say that the algorithm runs in time f(n). The exact running time of any particular algorithm might be a rather complicated expression for large n so it is often convenient to just estimate it. One of the most common forms of estimation is called asymptotic analysis in which only the highest order term of the expression for the running time is considered. For example, suppose the running time of an algorithm is f(n) = 4n3 + 7n2 + 3n + 9. This function has four terms and the highest order term is 4n3. So as the length of the input n gets larger, the term 4n3 grows faster than the other terms. For “very” large n, the 4n3 term dominates so much that the function begins to look like f(n) = 4n3 and we can ignore the smaller order terms. In fact we can even ignore the coefficient 4 and say that f(n) looks like n3. The notation for describing this (called bighttp://abstrusegoose.com

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O notation) is f(n) = O(n3) and we say that f(n) is asymptotically at most n3, or alternatively, f(n) is big-O of n3, An algorithm is called constant if its time complexity is independent of the input (i,e, O(1)). If its time complexity is O(n), then it is called linear. For O(n2), it is called quadratic and for O(n3) it is called cubic and so on. Any algorithm that has a time complexity O(nm), where m is a constant, is called a polynomial-time algorithm. Algorithms that have a time complexity of the form O(t f(n)), where t is a constant greater than 1 and f(n) is a polynomial function, are called exponential algorithms. Clearly, as n grows larger, the running times of exponential-time algorithms grow much faster than the running times of polynomial-time algorithms. To get an idea of why the time complexity of an algorithm is so important, consider the following list of running times for n = 1 million:

Time Complexity

# of Operations

Time required at 106operations/sec

O(1)

1

10-6 seconds

O(n)

106

1 second

O(n2)

1012

11.6 days

O(n3)

1018

32,000 years

O(2n)

10301,030

Universe succumbs to agonizing heat death

TABLE 2-1 Running times for n = 1 million. Note: 106 operations per second is obsolete by today's standards but it is still a commonly used measure in computer science.

Needless to say, computer programmers would generally rather stay away from using exponential-time algorithms while writing their programs. On the other hand, cryptographers would generally like it if the fastest algorithms for cracking their encryption was exponential. For this reason, most encryption algorithms take advantage of a peculiar feature of prime numbers. So let's talk a bit about prime numbers. A prime number is a number greater 1 that can be divided by no positive integer other than 1 and itself. The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. In fact, there are an infinite number of prime numbers. A positive integer greater 1 that is not prime is called composite. Prime numbers can be thought of as the multiplicative building blocks of the integers since every integer is composed of its prime factors. Finding the prime factors of a number is referred to as factoring. Many encryption algorithms use large composite numbers as part of the process of encryption. In order to crack the encryption, it would be necessary to be able to factor the large numbers. It's an easy matter to find two large prime numbers and then to multiply them together to produce a large composite number. However, if I were to hand you that composite number without telling you the prime factors, it would be very difficult to factor 148

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that number. Actually, it would be more accurate to say that it would take you an immensely long time to factor. Currently, one of the best known general-purpose methods for factoring large numbers is called the number field sieve which has a running time of O(exp((64/9)1/3(log n)1/3(log log n)1-1/3)). So how fast is the number field sieve in practical terms? A commonly used measure for describing the computing power necessary to factor a number is “millions of instructions per second – years (MIPS-years). 1 MIPS-year represents the computing power of a computer operating at 1 million operations per second for one year. TABLE 2-2 shows how many MIPS-years are required to factor integers of a given size using the number field sieve. Number of Decimal Digits

MIPS-Years Required

150

104

225

108

300

1011

450

1016

600

1020

TABLE 2-2 Computing power required using number field sieve. As you can see, the discovery of a polynomial-time factoring algorithm would have serious implications for the world of information security. Undoubtedly, anyone who is clever enough to come up with such an algorithm would also be clever enough to use it to become “obscenely rich”.

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3

SETI Finally Receives a Signal

continued from page 18 A large part of SETI's motivation came from a speculative idea of astronomer Frank Drake. You may be familiar with the famous Drake equation for estimating the number of communicative civilizations in our galaxy. Just for fun, I state it here: The number of communicative civilizations = N x fp x ne x fl x fi x fc x fL where N = the number of stars in the Milky Way galaxy. fp = fraction of stars with planets ne = average number of planets capable supporting life for stars with planets fl : of the planets capable of supporting life, the fraction that actually evolves life fi : of the planets that evolve life, the fraction that evolves intelligent life fc : of the planets that evolve intelligent life, the fraction that is capable of communicating with radio fL : the fraction of the life of the universe during which an average civilization communicates with radio Using his equation, Drake estimated that there should be around 10,000 radio-broadcasting civilizations in our galaxy. Seth Shostak, senior astronomer at SETI, estimated that the figure should be between 10,000 and 1,000,000.

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5

There is No Spoon

continued from page 20 15 degree bend. Even when I was washing that spoon, I didn't notice that it was bent. So what exactly was going on? William James postulated that the very act of thinking about a behavior increased the tendency to act out that behavior: “We may lay it down for certain that every representation of a movement awakens in some degree the actual movement which is its object” [1]. He referred to this as the principle of ideomotor action. James made that statement in 1890 and today that still seems like a reasonable proposition, even from the perspective of modern clinical psychology (and even neurobiology). James' principle of ideomotor action is related to the modern concept of priming. Put simply, priming refers to the phenomenon whereby exposure to certain stimuli can influence an individual's perceptions and behavior (without conscious awareness) at a later time. Note that the operational definition of priming has a slightly different meaning than what is usually referred to as subliminal influence. The word subliminal usually implies a lack of awareness of the triggering stimuli, whereas priming implies a lack of awareness of the effect of the triggering stimuli. For an example of priming, suppose that a person reads a list of words which includes the word chart and then is later asked to complete a word starting with cha. Then the probability that that person will answer chart is higher than for a person who was not primed. For a more interesting example (in which subjects were primed with behavioral traits), consider an experiment carried out by New York University psychologist John Bargh [2]. In this experiment, 34 students were divided into three groups: One group was primed with the rude trait, another was primed with the polite trait, and the final group was primed for neither trait. The method of priming took the form of a “Scrambled Sentence Test” which was presented to the students as a test of language ability. The students were given a 30question test in which each question consisted of a list of five words which the student was to use to construct a grammatically correct four-word sentence. For example, a scrambled sentence might be presented as “he it hides finds instantly”. The group primed for the rude trait were given scrambled sentences that included words like aggressively, bold, rude, bother, etc. (e.g., “they her bother see usually”). The group primed for the polite trait was primed with words such as: respect, honor, considerate, appreciate, patiently, etc. For the neutral group, the scrambled sentences contained words such as: exercising, flawlessly, occasionally, rapidly, etc. The tests were administered to the students one at a time by an experimenter. The students were told to complete the test and then to find the experimenter, who would be located in another room in the same hallway, to receive instructions for the next part of the test. The student would find the experimenter engaged in conversation with an accomplice posing as another participant who was asking questions about the test. The experimenter and the accomplice were instructed beforehand to carry on the conversation for ten minutes while ignoring the actual participant. The point of the experiment was to see how long it took the participant to interrupt the experimenter's conversation. The results of the http://abstrusegoose.com

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experiment showed that the participants primed for the rude trait tended to interrupt significantly faster than did the participants from the other two groups. I find it fascinating that the simple act of exposing an individual to a simple set of words could have such a significant effect on that person's subsequent behavior without his/her knowledge. It prompts me to consider the potentially staggering implications with regard to advertising and marketing. Advertising companies are already known to employ armies of psychologists tasked with the goal of creating advertisements that will have the greatest psychological impact. However, it is the possible unintended consequences of these advertisements that I find to be of interest. For example, consider the following words that can be found in a typical commercial that we see on television everyday: may increase the chance of heart attack or stroke that can lead to death. It should not be used right before or after certain heart surgeries. Serious skin reactions or stomach and intestine problems, such as bleeding and ulcers, can occur without warning and may cause death. Patients taking aspirin and the elderly are at increased risk for stomach bleeding and ulcers. Tell your doctor if you: are pregnant; have a history of ulcers or bleeding in the stomach or intestines; have high blood pressure or heart failure; have kidney or liver problems. That short snippet is packed with a high density of words and phrases that many people may associate with negative or stressful life situations. Are such commercials unintentionally priming millions of viewers for various kinds of negative behavior without their knowledge? Granted, this is just speculation on my part but I always make it a habit while watching TV to mute the sound during commercials. Call me crazy. Another study by Bargh suggests that individuals can also be primed with the concept of attaining a goal and that the effects of goal-priming can continue to operate over an extended period of time (without the person's conscious intent) to guide thought or behavior towards the goal [3]. So is that what happened to my spoon? By attempting to bend the spoon with my amazing telekinetic powers, did I unwittingly prime the goal of bending the spoon, and then unconsciously fulfill that goal as a result? I don't think that that would be such an unreasonable conclusion. The very concept of priming implies the idea of some form of unconscious mind and, in fact, there is much evidence to suggest that our brains perform much complex information processing that occurs outside of conscious awareness [4]. However, even among neuroscientists and psychologists who can agree on the existence of an unconscious mind, there are many differences of opinion as to how it should be defined and to what extent it affects our conscious awareness. I will not open that can of worms here by attempting to define the unconscious mind, but I will simply state that I believe it exists. The power of the unconscious mind is a theme that runs throughout my comics as you shall see. 152

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NOTE: I should also probably point out that the lines about quantum physics in the comic are just technobabble bullshit with no scientific basis. References: [1] James, William (1890). Principles of psychology. New York:Holt [2] Bargh, J.A., Chen, M., Burrows, L. (1996). Automaticity of Social Behavior: Direct Effects of Trait Construct and Stereotype Activation on Action. Journal of Personality and Social Psychology, 71, 230-244 [3] Bargh, J.A., Gollwitzer, P.M., Lee-Chai, A., Barndollar, K., & Troetschel, R. (2001). The automated will: Unconscious activation and pursuit of behavioral goals. Journal of Personality and Social Psychology, 81, 1004-1027. [4] Velmans M 1991 Is human information processing conscious? Behavioral Brain Science 14: 651-726

6

Schrödinger’s Infinitesimal Miscalculation

continued form page 21. from the rules of the macroscopic world which is described by classical mechanics. Take, for example, an ordinary baseball. According to classical mechanics, the baseball has a definite trajectory (position and momentum) at any given time and we can theoretically predict the position of the baseball at a later time if we know its trajectory at an earlier time. This seems to conform to our common sense notion about how everyday objects that we see around us should behave. However, when we are dealing with small objects (e.g. electrons), quantum physics tells us that such common sense no longer applies. At the quantum level, we must describe an object by its state vector. Suppose, for example, that we wanted to know the position of such a microscopic object. According to quantum physics, the object has no definite position until it is measured. In fact, before its position is measured, we can think of the object as having a probability of being in any possible position available to it (this idea, by the way, is one of the central tenets of the Copenhagen interpretation). This probability distribution is described by the state vector which, by convention, is represented by the Greek letter Ψ (psi). In this case, Ψ describes the object's possible positions. As you may have surmised from the name, state vectors are examples of mathematical objects called vectors. To be more precise, they are vectors in a complex vector space called a Hilbert space, but we won't get into that here. The important point is that different vectors can be added together to give another vector. So for example, if Ψ and Χ are two different vectors, then Ψ + Χ would be another vector. Vectors can also be multiplied with a (complex) number to give another vector so if Ψ is a vector, then cΨ would be another vector (where c is a number). Physicists have adopted a notation for these state vectors (called bra-ket notation) in which each vector is denoted by a symbol in angled brackets http://abstrusegoose.com

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such as ΙΨ>, ΙΧ>, ΙΦ>, Ι1>, Ι2>, Ι3>, etc. Thus with this notation, the addition of vectors can be written as ΙΨ> + ΙΧ> and the multiplication of a vector by a number can be written as c ΙΨ>. Now let's see this bra-ket notation in action for a simple example. Suppose we have a microscopic particle whose state vector for position is Ψ and it is expressed as the weighted sum of two other vectors ΙΨ> = c ΙA> + d ΙB>. What this expression says is that the particle can be in two possible positions, A or B. Before measuring the position, the particle cannot be thought of as occupying any of the two positions. We can only say that it has a probability of being in either of the two positions. Physicists would say that the particle is in a quantum superposition of the two positions. The numbers c and d are called probability amplitudes. The square of the probability amplitudes (actually the squared moduli) gives the probability of finding the particle in that position after measurement. In this case, if we measured the particle's position, the probability of finding it in position A would be |c|2 and the probability of finding it in position B would be |d|2. The amplitudes are usually “adjusted” so that their squares sum to 1 but that's another detail which I won't get into here. The process by which the state vector representing the superpostion of different states reduces to a single state is referred to as state vector reduction (it can sometimes be referred to as wavefunction collapse). The idea of the reduction of the state vector is another one of the central aspects of the Copenhagen interpretation of quantum physics. Now let's apply what we've learned so far to poor Schrödinger's cat. We know that the cat can be in two possible states: live or dead. Let ΙΨ> be the state vector and suppose that the probability of finding a dead cat is ½ and that the probability of finding a live cat is ½. Then one possible way to express the state vector is ΙΨ> = (1/√2) Ι live cat> + (1/√2) Ι dead cat>. So there you have it. Now you understand the equation in the first panel of the comic; but what about the equation in the last panel? The symbol є is generally used by mathematicians to represent an infinitesimal quantity. Hence, the equation in the last panel expresses the idea that there was an infinitesimal chance that an angry monkey could have magically appeared in the box. [I will expand this explanation in later drafts of the book.]

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8

Arguing with a String Theorist

continued from page 23 The following is an entry by Luboš Motl on his blog The Reference Frame posted on February 6, 2009: A couple of physics blogs, including asymptotia.com, have recently posted the cartoon above. A boy is saying some very stupid things that he considers to be arguments against the validity of string theory. On the other hand, a girl who seems to be familiar with string theory reacts in the only way that actually makes any sense in this context. The deeply flawed and brutally misinterpreted propositions made by the boy have recently been repeated by thousands of laymen as a new mantra. There are whole websites on the Internet that have been alive for years just by repeating the stupid boy's statements from the cartoon: a classic infinite loop of obsession. I can understand why people want to repeatedly watch porn: we are hardwired for certain things. But the people who can read these websites more than thrice - or more than for one week - must suffer from some kind of severe mental deviation or retardation, an insatiable thirst for repetitiveness that I simply cannot comprehend. They must believe that if they eat the same excrement from a cartoon 1,589 times (guess where the number comes from), it becomes a yummy pizza. ;-) Although all these topics have been discussed hundreds of times and all the people who have seen it, who are interested in physics, and whom I consider at least partially intelligent must have understood them, let me respond to the particular line of comments made by the stupid boy from the cartoon again, realizing that in comparison with the girl's appropriate reaction, my answers will be just a waste of time: Misunderstanding: First of all, string theory has not made a single testable prediction in over 30 years. Reality: String theory was born in the late 1960s as a theory of strong interactions. It has made lots of predictions about the strong interactions. Many of them were correct. Many of them, made with the old version of the theory, were quickly proved wrong. First, let me jump from 1973 to 1997. String theory was thought to be a wrong theory of strong interactions from 1973 to 1997 or so when it was realized that string theory on certain AdS backgrounds is exactly equivalent to theories similar to QCD... This crucial discovery has revived the line of reasoning that was studied 30 years earlier and it led to many predictions - that are much more difficult in other approaches - about nuclear physics. They are not only testable but many of them have been spectacularly confirmed. This application of string theory is arguably the most active approach to the theory of strong interactions in the beginning of this century. If we return back in time, string theory was (not really correctly, as explained above) abandoned as a theory of strong interactions. It became a theory of quantum gravity around 1974 when it was realized that massless spin-two excitations (gravitons) belonged to the spectrum and they inevitably follow the rules of general relativity. The theory describing strong interactions by the 1997 holographic recipe is the same theory of quantum gravity: the 1997 discovery has really proved than an underlying "gravitational explanation" is inseparable from theories similar to "QCD", and vice versa. There is no way to separate these things again: the dualities that have been established - really proven - can no longer be unproven or disestablished. We're not talking about "two different string theories here". Once you accept the AdS/CFT dual description of gauge theories such as the N=4 theory, there is no way to deny string theory the status of a unifying theory of all forces, having 10 spacetime dimensions, that is inseparably woven to the structure of all physical field theories. http://abstrusegoose.com

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When it became clear that string theory was a theory of quantum gravity, it also became clear that it couldn't be directly tested. The fact that quantum gravity is almost certainly untestable by direct experimental tests has been known not for 30 years but for 109 years. In 1900, Max Planck realized that physicists should be using "natural units". Today, we talk about Planck units. In the contemporary conventions, they're products of powers of the light speed "c", (reduced) Planck's constant "hbar", and Newton's gravitational constant "G". The correct product with the units of distance, the so-called Planck length "sqrt(G.hbar/c^3)", is close to 10^{-35} meters which is so short that it has been clear, since 1900, that people in a foreseeable future couldn't possibly "see" them directly. Again, this argument has been known for 109 years. Every person who has ever begun to study quantum gravity should have been familiar with it. I was familiar with it - with the magnitude of the "natural length scale of quantum gravity" - when I was 10 years old. Every person who claims to be interested in fundamental physics but who also reveals his or her "surprise" that the effects of quantum gravity cannot be directly seen in existing experiments is simply dumb beyond imagination. Is quantum gravity directly relevant for people's everyday lives? No. Was it ever argued to be relevant? No. Is it a new situation that most people don't really care about fundamental physics or any other theory-loaded science? No. Did the people on the street in the 1930s say that they gave a damn? No? That's because they didn't. ;-) Despite this apparent separation of the scales, hundreds of exceptional physicists - really many of the smartest people on this planet - decided that it was the right time to study physics at the fundamental scale. All of them have always known that these effects couldn't be directly seen because they're associated with extremely short distances and durations and extremely high temperatures. And indeed, it became possible to unequivocally say a lot of statements about the nature of phenomena that are crucial near the Planck length. The topology of space can change; the total number of dimensions visible at this scale must be 10 or 11 whenever all other "obscure" degrees of freedom are geometrized; black holes preserve the information, even during the evaporation; strings, branes, and various topological defects are parts of the spectrum whenever certain moduli approach the asymptotic regimes. I could write thousands of pages of much more specific and quantitative predictions of string theory: Generating predictions is what string theorists are doing all the time. The fact that these predictions cannot be tested in your basement is not a flaw of string theory but an obvious consequence of the very choice of the questions: we want to study quantum gravity, the processes at the "natural scale". These processes simply can't be testable in your basement, because of a simple calculation that even kids should be able to understand. This has nothing to do with string theory per se: it is a property of the very questions we are asking. The argument that quantum gravity is inherently untestable could have been made more than 100 years ago. But if someone had used it to suppress all research of the subject in 1909 or so, he would have killed hundreds of amazing insights that came out of this research. Many of them tell us seemingly "divine" answers to difficult questions about quantum gravity while others tell us answers to completely different questions - like those about the collisions of gold ions - that turned out to be connected with quantum gravity. The people who are trying to suppress the research of string theory today are surely trying to eliminate many discoveries that will be made in the future. Paradoxically enough, it was string theory that has also found a possible flaw in Planck's estimate i.e. in his argument showing that the fundamental scale had to be extremely tiny and inaccessible. When we add the extra dimensions into our considerations and analyze their possible radii, multiplicities, and general shapes, we find out that it is conceivable that the "higher-dimensional natural scale" can actually be much closer - and perhaps even accessible to the LHC - because it may be close to 10^{-18} meters if some additional dimensions are large or curved enough. And the extra dimensions themselves may still be a few microns in size. Such options are considered unlikely - I quantified the probability of such scenarios to be around 1% - but they show that effects that used to be considered inaccessible to science may often become accessible, and quantum gravity might be just another example following hundreds of other examples.

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A reduced Planck scale is the "phenomenological way" how string theory unexpectedly allows seemingly untestable questions to be tested. Predictions of low-energy physics that follow from the high-energy starting point are the "theoretical way" that connects the observations with the mysterious fundamental scale. String theory reproduces all of physical quantities of low-energy gauge theories coupled to Dirac fermions (including all loop effects, non-perturbative effects, renormalization rules, confinement, Higgs mechanism, etc.). It parameterizes the low-energy parameters differently than QFT - in terms of discrete data (instead of continuous data) which might perhaps be viewed as "less convenient" ones but they are equally consistent. String theory is as correct a description of these nongravitational observations as quantum field theories are. You can't really say that it is "empirically worse off" than quantum field theories. And its theoretical status is surely better off than in quantum field theories: it incorporates gravity including loops and other quantum effects! And string theory is actually linked with pretty much all interesting directions in phenomenological "particle physics beyond the Standard Model", including supersymmetry, GUTs, deconstruction, and others. Misunderstanding: By making string theory ridiculously malleable with your 10^{500} ways to compactify the extra dimensions, you essentially put the theory beyond the reach of any conceivable experimental test. Reality: String theory is absolutely robust. It can be demonstrated that there exists no consistent way to deform it or "slightly modify" its rules of the game. It is the first theory known to the mankind that has no adjustable dimensionless non-dynamical parameters whatsoever. The adjective "malleable" associated with string theory is completely absurd. On the other hand, much like other theories in science, string theory predicts many solutions many potential "vacua" - where physical phenomena might in principle take place. Let me emphasize the difference again: we have entirely fixed rules but there exist many ways how to live according to these rules. But the number of predicted solutions, whether it is larger or smaller than you expected, can never be used as an argument for or against the validity of a theory. It is simply a feature of the theory and one needs actual further tests to decide whether the feature - or the prediction, if you wish - is valid or not. At this point, we don't have empirical data about these issues. Genetics is arguably disappointing because it doesn't show that the human DNA is unique. It doesn't quite prove the existence of God who created humans to His own image. We cannot see God's DNA in the sequences that would distinguish us from monkeys and other life forms that were not created to His image. ;-) Believe me, billions of people in the world - including very nice women and men - are profoundly disappointed by molecular biology because of these and related reasons. I won't even try to tell them that they share 96% of their DNA with chimps because they could get insulted! What can they do about their disappointment? Well, they may pray and they may dream about a different, better Universe where God's traces can be identified in our DNA and where this preferred DNA sequence of God may be calculated. The calculation could perhaps use some hints from the Bible, they think. But that's about it: they can't do much more than that (in the past, they could at least burn the heretics at stake to get some relief) and so far they haven't presented the Biblical calculation. ;-) The number of "candidate animals" i.e. the number of DNA sequences that are as long as the human DNA is roughly 10^{billion}, much bigger than the number of semi-realistic vacua often estimated as 10^{500}. The human DNA doesn't show any uniqueness of the human race. It cannot be calculated from the first principles. It is disappointing and ugly. It is true and paramount for biology, too. Sorry: but maybe humans are not that special, after all. It might perhaps be the right time to start to consider this possibility, 150 years after it was demonstrated to be true. ;-) The different DNA sequences don't give us "different versions of Darwin's theory". There is only one theory and the wide variety of DNA sequences is an essential feature (or a prediction) of this theory!

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The situation of the number of vacua in string theory is philosophically isomorphic. Many people, including your humble correspondent, would sentimentally prefer a theory where all the other vacua were absent. It would simplify our life a lot. But science is not about a wishful thinking. The large number of vacua that are a priori usable instead of ours has been established to be very large. It is extremely unlikely that this insight will ever be undone. The only big related question that remains to be answered is whether physicists have any chance to identify the correct vacuum. The anthropic people have essentially given up, believing that the "landscape" is just too vast and too chaotic: they use circular reasoning to assure themselves that our Universe has to "live" in a large, chaotic segment of the landscape where nothing can be determined with any certainty. And they think that vague statistical analyses of the landscape and qualitative predictions are the only possible advances that can be done beyond the present point, in the future. And they might be right or "effectively right", for one reason or another. The other people, including myself, know that at least in principle, there can exist all kinds of methods to determine which vacuum is actually right - either by analyzing their detailed theoretical properties and comparing them with the experimentally measured properties of our world; or by finding a hypothetical selection principle that makes our vacuum (and perhaps a few other vacua) dramatically more likely than others. Whether our vacuum is "random" and "anonymous" or whether it can be identified - and whether it makes sense to spend time with this big task (which is a different question!) - remains to be seen. So far the right vacuum hasn't been identified, so the anthropic opinion is confirmed by the "status quo" (in the same way as the opinion that "science has ended" was confirmed by the "status quo" at any other point in the history of science, too, until the following morning when science continued). But the observation that at some level, there exists a large number of candidates for "the vacuum" has been pretty much established (at least in the case of supersymmetric AdS vacua where the number of possible subtleties that could "kill" the vacua seems extremely low). In fact, our world doesn't look "quite so unique and symmetric" and it indicates that the number of "equally fundamental or symmetric" vacua must be much larger than one, to say the least. And yes, I consider the people who disagree with this statement to be complete deniers of the scientific evidence. The large number of vacua in quantum gravity is an established fact of science. It will never be undone, much like we will never return to the idea of a Flat Earth. This insight is not the last insight of science but it is an insight of science. Misunderstanding: Second of all, string theory is only formulated perturbatively. A full non-perturbative definition of the theory doesn't exist. Reality: This statement is also wrong and even if one formulated a more careful but similar statement that would be technically correct, it would be morally wrong because the same thing could be said about quantum field theory, not just string theory, so one can't ever justify the application of this observation as an argument against the step (or leap) from quantum field theory to string theory. More generally, it is also sociologically illogical to present quantum field theory and string theory as "foes" because they are not only equivalent in some contexts but a large portion of the best QFT experts in the world are actually string theorists. Let me add some details about the perturbative expansions. A full, exact, non-perturbative definition of many superselection sectors of string/M-theory is known. That's completely equivalent to the situation in quantum field theory. Maldacena's AdS/CFT correspondence shows that some superselection sectors of quantum gravity - i.e. string theory - are completely equivalent to certain quantum field theories. These theories, such as the N=4 supersymmetric gauge theory, can be e.g. put on a lattice. There are some subtle remaining problems with supersymmetry on the lattice, despite the progress in deconstruction etc. But whatever these problems are, they are equally serious or equally solvable for string theory and for quantum field theory because in this subset of backgrounds, they're really the same theories.

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While the lattice descriptions might arguably be the only approach to formulate four-dimensional quantum field theories non-perturbatively, we have actually many more methods to do the same thing in string/M-theory: so the situation in string/M-theory is better in this respect than the situation in quantum field theory. The BFSS matrix model (also known as M(atrix) theory) is an exact, nonperturbative definition of a sector of string/M-theory - namely M-theory on an infinite, 11-dimensional flat space. An ordinary quantum mechanical model - with degrees of freedom X,P,theta extended into matrices - can be demonstrated to coincide with M-theory in 11-dimensions if the size of the matrices is sent to infinity. We can calculate physical quantities for finite N and send N to infinity, to obtain the M-theoretical result. It's as well-defined as undergraduate quantum mechanics. If you are irritated by the absence of strings in the 11-dimensional vacuum and by the absence of an adjustable coupling constant "g" in the BFSS matrix model, you may also write down the nonperturbative definition of screwing string theory due to your humble correspondent that was later renamed to matrix string theory by Dijkgraaf, Verlinde, and Verlinde (DVV). ;-) It has type IIA (or heterotic E8 x E8) strings in it, as the four authors proved. Nevertheless, the exact, non-perturbative definition exists for any "g". You don't have to expand anything. In fact, the key new contribution by DVV was to show that you could expand matrix string theory in "g" - and get the right stringy perturbative interactions, as I expected - which required some extra work. Similar definitions don't exist for all superselection sectors of string/M-theory at this point. But it's also the case that we don't possess non-perturbative definitions of all quantum field theories, either. Even if we had these definitions, it wouldn't mean that we can immediately calculate all non-perturbative phenomena out of them. When you try to calculate physics of a strongly coupled system, you always need some kind of cleverness - e.g. a good choice of the "effective degrees of freedom". This general wisdom holds for string theory and for quantum field theory, too (besides condensed-matter physics: ask the fractional quantum Hall effect people where their stunning pride comes from!). In string theory, we know many more non-perturbative phenomena - and many more of their relationships - than we know in quantum field theory. So once again, the situation in string theory is better than the situation in quantum field theory. The higher number of string-theoretical non-perturbative effects, dualities, and insights to learn may be attributed to the "larger size" of string theory. But again, this "large size" shouldn't be surprising because string theory is understood to be a broader theory that should include all correct wisdom of quantum field theory, general relativity, and much more. So it must obviously tell us much more about the fundamental objects, phenomena, and their relationships. And it is doing so beautifully, indeed. Theories should be as simple as possible but not simpler. As Edward Witten correctly observed, string theory has proven to be remarkably rich, more so than even the enthusiasts (like your humble correspondent) tend to realize. There are still many things to be learned about non-perturbative (and perturbative?) physics of string theory which is why people are still working on it intensely. Misunderstanding: Third, string theory describes perturbative expansions about fixed spacetime backgrounds. Reality: First of all, this half-incorrect statement irrationally mixes two issues that have nothing to do with one another. In the previous section, we have explained that it is simply not true that string theory is only known or defined perturbatively. After all, most of the insights found since 1995 are actually concerned with non-perturbative physics. Many non-perturbative effects, quantities, and their relationships are known. Explicit non-perturbative definitions of some vacua are known, too. So the adjective "perturbative" makes the sentence incorrect. Now, remove this word and think about the statement that string theory expands physics around fixed spacetime backgrounds. It is true and it is inevitable, too. Every consistent theory of quantum gravity must be doing so, at least when it gets to the "real work". If you consider infinite spacetimes - such as AdS spaces or flat spaces - they have a particular behavior in the asymptotic region at infinity. All doable processes can only deal with a finite amount of energy and a finite amount of energy is never enough to "rebuild" the space at infinity. That's roughly http://abstrusegoose.com

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why states in any theory of quantum gravity - and, in fact, any quantum field theory - decompose into the so-called "superselection sectors" that don't speak to each other. As long as a theory is consistent with the very simple observation that a doable (finite-energy) experiment cannot rebuild the space at infinity, it associates a superselection sector with every (classical) configuration or every (quantum) state in its Hilbert space. There is no way to avoid it. So any particular calculation of the Hilbert space has to be made for particular choices of the superselection sectors - for particular behavior of spacetime at infinity. Again, a theory that doesn't allow the space to extend to these asymptotic regions or that doesn't allow the geometry in these regions to be described by a well-defined geometry fails to agree with the very existence of space (that is demonstrably much larger than the short-distance fundamental scale, to say the least) and is instantly ruled out. These asymptotic regions may have many shapes - and flat and AdS-like backgrounds are the simplest ones to be described by accurate equations - but such fixed asymptotic regions of spacetime must be allowed and respected, otherwise the theory would be instantly dead. Misunderstanding: Any respectable fundamental theory of quantum gravity must be background-independent. Reality: The topic of background independence, which is pretty much equivalent to the previous section (but I have also divided the discussion into two parts, in order to follow the cartoon), has been explained many times. The people who like to say the same stupid thing as the boy from the cartoon usually severely misunderstand what the adjective "background-independent" means: the meaning they actually associate with this quasi-religious adjective is incompatible with basic physical consistency criteria. They think that their "background independence" should prevent a theory from considering physics at specific backgrounds, in specific superselection sectors. Carlo Rovelli even thinks that one should find a background-independent propagator. He may even believe that he has found one. ;-) So far, he hasn't noticed that his combination of words, a "background-independent propagator", is a special example of another oxymoron, namely "Taylor expansions without a point to expand around". Propagators are defined to be the inverse (continuous) matrices of quadratic fluctuations around a particular background: they're determined by the kinetic (quadratic) terms in the action expanded around the background. No background, no propagators. As argued above, every consistent theory living in an infinite space must agree with the existence of superselection sectors; must allow for the existence of realistic superselection sectors that resemble the nearly flat space we inhabit; and must be able to predict what happens in these sectors because virtually all quantitative questions we can ever ask about in physics have this form (and it is highly questionable whether there exist any quantitative yet background-independent questions at all). The people who use the word "background independence" incorrectly and quasi-religiously don't seem to get any of these points. And maybe, they're getting these points but they have already switched to a dishonest discourse in which it is better for them to repeat things they know to be untrue. That's widely believed to be the case of Mr Lee Smolin. Second, there is a question whether the very character of a theory depends on the "background" or the "superselection sector". It can be demonstrated that string theory doesn't depend on the background: the local phenomena are always isomorphic. While the separation of the states into superselection sectors is inevitable in any physical theory, the character of local physics should be independent of the choice of the sector. It can be demonstrated in perturbative string theory and other formulations of string theory that the identity of the theory is independent of the superselection sector. A modification of the background can be shown to be physically indistinguishable from a condensation of a particular configuration of strings (or their non-perturbative counterparts, if we consider non-perturbative physics) that existed in the original background: see, for example, Why there are gravitons in string theory. If these strings (or M&M's) have to change the asymptotic conditions, they must be associated with non-normalizable

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states in the Hilbert space but these states may still be linked to the ordinary, normalizable, finiteenergy excitations. So physics of string theory is surely independent of the background: every choice of the background leads us to the same theory. A completely different question is whether this independence is "obvious": physicists ask whether it is "manifest". The latter is pretty much an aesthetic, not physical, question, and our sense of beauty may often mislead us. As in most questions, the background independence is manifest in some approaches but not others. For example, there exists a way to define string theory that is very analogous to quantum field theory (with infinitely many fields, if expanded into point-like component fields). It's called string field theory (not to be confused with all of string theory: "string field theory" is just a small sub-discipline within string theory). It works well for open strings only, especially if they're bosonic, but it gives us a new perspective on many questions related to perturbative physics and D-brane states. String field theory for the 26-dimensional open bosonic string can be formulated in a manifestly background-independent way. The action is "S = Integral Phi*Phi*Phi" as long as you define the integral and the star-product properly. (Yes, I've been designing T-shirts with this equation.) Their (integral, star) definitions are formally independent of the background and individual backgrounds are associated with (the BRST operators connected with) particular "vacuum solutions" of the equations of motion, "Phi*Phi = 0". The condensation of "infinitesimally perturbed strings" generates the whole background, smooth geometry, and its nilpotent BRST operator. But you should have already understood that real, quantitative physics only begins when one picks a background, a superselection sector. Before one does so, many of the objects are too formal and cannot be associated with particular numbers. A physicist should always be careful about such formal manipulations: he should always ask whether his formula can generate very particular numbers that can be in principle both calculated and measured in the appropriate Universe. Martin Schnabl was extremely conservative about this important principle which is why his new remarkable "vacuum solution" to string field theory, once it was found, was and is so much more meaningful - and so much more correct and important - than dozens of "formal" results that generated "infinity minus infinity" expressions whenever you wanted to analyze them in detail. His solution is linked with some rather deep mathematics (on the boundary between complex calculus and number theory), too. So the genuine lesson is that any respectable framework in quantum field theory or any theory that generalizes it must eventually admit background-dependent calculations, in a sharp contrast with the stupid boy's proposition in the cartoon. Background-independent formulations - if they exist - must always be understood as a first, philosophical step to formulate the detailed, background-dependent theories. Only the latter can produce meaningful, measurable numbers that can be compared with observations. It would be very pleasing to have a complete description of string theory that would cover all corners of its "landscape" and allowed us to calculate the properties of all vacua as solutions of some universal equations. Deep physicists have spent years with attempts to find such universal equations and they will surely continue to do so in the future, to one extent or another. (I didn't want to mention that the list includes your humble correspondent because I found the word "deep" more important and I wanted to avoid any self-glorification.) On the other hand, such beautiful and universal equations that treat all possible corners of the landscape "democratically" are not guaranteed to exist. In some sense, we should expect that they don't exist - at least not to the extent to "directly tell us" which objects are weakly coupled at any point - because such equations would present all possible objects in all regions of the landscape as "equally fundamental" and "equally manifest" even though many of them are complicated bound states of each other. Moreover, whether these equations exist or not has no impact on the question whether string theory is the correct fundamental description of the world around us as long as we determine our conclusions by the evidence rather than by the prejudices. Misunderstanding: Fourth, in string theory, the Dirac operator and the gauge fields are... http://abstrusegoose.com

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I am not certain what the boy wanted to say because he was, thankfully to the girl, interrupted. :-) But in string theory, the Dirac operator and the gauge fields are derivable, omnipresent, and essential aspects of low-energy physics that can be deduced to exist in any realistic enough vacuum. They play the same important role for the low-energy physics as they always did; the low-energy equations usually hold exactly in string theory, too. On the other hand, these old concepts are no longer the deepest or the "only deep" principles that underlie physics. In most approaches to string theory, they're secondary and can be shown to be consequences of more powerful unifying principles that generate other physical phenomena, too. For example, the gauge fields with Yang-Mills symmetries and the fermionic matter fields that follow the Dirac equation are just two consequences of the conformal symmetry in perturbative string theory (or superconformal symmetry: the superconformal zero mode on the worldsheet must annihilate the physical modes which directly gives us the Dirac equation in spacetime - nice). And the same conformal symmetry applied to closed strings (with antiperiodic fermions) also implies the existence of the metric tensor with the diffeomorphism symmetry (also known as gravity in general relativity), the critical dimension, and many other things. String theory also allows us to derive new, fundamental, and unexpected facts about gauge fields and the Dirac operator (look e.g. at the D-brane viewpoint on the ADHM construction). At any rate, I know too much about the world to realize that evil must be confronted with fists. That's why I fully endorse the clever girl's reaction to the piles of rubbish that the talkative boy was emitting. And to make it really clear how much I endorse her ;-), let me reproduce her classical answer to her obnoxious foe in its entirety. Summary: PAK! You keep talking like a bitch, I'm gonna slap you like a bitch. :-) You can see that Clifford Johnson is using gloves to communicate with the excessively zealous anti-scientific commenters who are spamming his blog with bullshit. That can't protect him from trashtalking at aggressive smear blogs such as Not Even Wrong. There's no peaceful way to deal with this situation, Clifford. So I kindly ask all the female readers to give a proper thrashing to every man who will emit the same crap as the unfriendly boy from the cartoon. I hope it is sufficiently politically correct for clever girls to beat disgraceful, dishonest, and sub-par kibitzers like the well-known one from Columbia University. Thanks a lot.

SWEET FANCY MOSES!!!

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Ask Me Why

continued from page 28 However, there is a certain sense in which I am superstitious and for which I have no rational explanation. I'm also willing to bet that you are superstitious too, and I'll prove it. Consider the following thought exercise: Suppose that you were told that you had a 50% chance of being the sole winner of a $1,000,000 lottery jackpot. On your way to buy the lottery ticket, you come across a ladder on the sidewalk and notice that there is a $100 bill underneath the ladder. The only way to get the $100 bill is to walk under the ladder. Would you take the $100? … Yeah, me neither. See? I told you that you were superstitious. If you want to gauge to what degree you are superstitious, then ask yourself how much money needs to be under the ladder before you are willing to pick it up. NOTE: “Thanks” to all the people who emailed me to actually ask me why I'm superstitious.

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I.I. Rabi’s Question Answered?

continued from page 33 together), and gravity. Electromagnetism is mediated by photons, the weak force is mediated by the W and Z bosons, and the strong force is mediated by gluons. It is postulated that particles called gravitons are responsible for gravity but the Standard Model does not include gravity (but gravitons are great fun in Star Trek episodes). For a simplified view of this, one can imagine the fermions interacting with each other by exchanging force carriers. This exchange of force carriers results in the appropriate force between them. There are six types of leptons: the electron, the electron neutrino, the muon, the muon neutrino, the tau, and the tau neutrino. It should be apparent that the particles appear to be grouped into pairs: for each particle there is an associated neutrino. The quarks also exist in pairs: up/down, charm/strange, and top/bottom. Triplets of quarks bind together to form protons (up, up, down) and neutrons up, down, down) which make up the nuclei of atoms. Quarks have the strange property that they cannot be seen individually (at low energies). The fermions are grouped into three categories called generations. The first generation consists of the electron, the electron neutrino, and the up and down quarks. The second generation consists of the muon, the muon neutrino, and the strange and charm quarks. The third generation is the tau, the tau neutrino, and the top and bottom quarks. Each of the three generations are identical except for significant differences in the masses of the particles. So why are there three generations and not two or four or even more? Who the hell knows?

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Generation 1

Generation 2

Generation 3

electron

muon

tau

electron neutrino

muon neutrino

tau neutrino

up quark

charm

top

down quark

strange quark

bottom quark

TABLE 18-1 So let's finally get around to explaining the cartoon. For this we will need a bit of history. The first of the fundamental particles to be discovered was the electron in 1897. The existence of the neutrino was proposed in the early 1930s in order to make certain energy calculations come out correctly. However, direct evidence for the existence of neutrinos didn't come until the mid 1950s. In 1937, the muon was first discovered although it wasn't until 1946 when physicists figured out what the hell it was. The muon is essentially a heavier copy of the electron (as explained above concerning the three generations). It seemed very mysterious why nature should have a second heavier copy of the electron. This mysterious appearance of the muon prompted Nobel Prize-winning physicist Isaac Isidor Rabi to ask the question, “Who ordered that?” Muons are produced all the time in the Earth's upper atmosphere by the decay of particles called pions produced by cosmic rays. It was by this process that the muon was first detected in 1937. The equations in the last panel of the comic describe the process by which pions decay to produce muons. So there you have it! Now you don't have to go to grad school.

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Make a Wish

continued from page 35

SPOILER ALERT: IF YOU HAVE NOT YET VIEWED THE HIDDEN PICTURE, PLEASE DO SO NOW BEFORE READING FURTHER AND REMEMBER TO MAKE A WISH FIRST... OR NOT.

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OK, so now you've read the hidden message in the image and you're a bit confused. “But that's not funny or witty”, you say, “why did you post this as a comic?” Once again, the theme of the unconscious mind and priming returns (see the comments for comic #5 on page 20). My hope was that the reader actually would “make a wish” before reading the hidden message. Upon seeing the words “YOUR WISH IS GRANTED”, the reader's unconscious mind would be triggered to predispose him/her to take subtle actions in life that would increase his/her probability of attaining such wish and I would suddenly be flooded by millions of emails from grateful fans thanking me for helping them get their wish: Dear Abstruse Goose: Thank you so much for posting comic #20. For years I've had a crush on this gorgeous female co-worker but I've never had the guts to ask her out. Now, after being primed by your Magic Eye message, I've finally blah, blah, blah.... To date, I have not yet received such an inspiring email message. Oh well... The scientific explanation for how Magic Eye images work is rather simple. The image consists of a series of repeating patterns. For the type of autostereogram that I used in the comic, the image appears as a random set of dots when viewed normally. However, when a viewer looks at the image at close distance while crossing her eyes at just the right angle, each of her eyes are fixed on a different set of an adjacent pair in the repeating pattern and her brain mistakenly perceives the two patterns as a single image at a different distance.

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Reality vs Fantasy

original blog post: In all seriousness, thank you all for your support lately. We got well over 80,000 visitors the other day thanks in part to Digg, reddit, Wil Wheaton, Operation Agitprop, StumbleUpon, and all you bloggers that linked here. And I welcome all constructive criticism; positive or negative… even from those of you that seriously need to get laid (kirkt)  This is my obligatory self-deprecating humor comic. Every new webcomic has to have one. It’s the law.

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23

Pi

As of this writing, the current Guiness Record holder for reciting the most digits of pi is Lu Chao from China. It took him him 24 hours and 4 minutes to recite 67,890 digits on November 20, 2005. It should also be noted that Akira Haraguchi, a man from Japan, recited pi to 83,431 digits and 100,000 digits on two separate occasions but these performances have not yet been verified so they are still unofficial. Here are the current top ten (official) record holders: Rank

Name

Country

Digits Memorized

Date

1

Chao, Lu

China

67890

Nov. 20, 2005

2

Chahal, Krishnan

India

43000

Jun. 19, 2006

3

Goto, Hiroyuki

Japan

42195

Feb. 18, 1995

4

Tomoyori, Hideaki

Japan

40000

Mar. 10, 1987

5

Mahadevan, Rajan

India

31811

Jul. 5, 1981

6

Tammet, Daniel

Great Britain

22514

Mar. 14, 2004

7

Thomas, David

Great Britain

22500

May 1, 1998

8

Robinson, William

Great Britain

20220

May 5, 1991

9

Carvello, Creigthon Great Britain

20013

Jun. 27, 1980

10

Umile, Marc

15314

Jul. 21, 2007

USA TABLE 23-1

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Secrets and Lies

continued from page 44 Without getting into too much detail, the U.S. Commerce Department's National Institute of Standards and Technology had promoted an encryption algorithm (called SP 800-90) which depended on Dual_EC_DRBG. The algorithm was subsequently promoted by the U.S. National Security Agency. Security researchers later found that if an individual had knowledge of a certain set of fixed numbers, he could possibly be able to predict the output of Dual_EC_DRBG, effectively rendering the algorithm useless. Although nobody knows for sure whether anybody is in possession of the secret numbers, the discovery of the flaw and the endorsement by the NSA certainly raised some eyebrows.

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MISCELLANEOUS NOTES FOR COMIC KH stands for “Key Hole” and it is a code that (when followed by a number) indicates the type of surveillance instrumentation that is installed on U.S. military reconnaissance satellites. The KH naming system was initiated in 1962 with KH-4. Incrementing numbers following the KH indicated changes in instrumentation. To my knowledge, the most recent unclassified KH designation was KH-13 from a satellite launched in 1999. Given the understandably secretive nature of intelligence agencies, it is reasonable to assume that classified reconnaissance satellites currently in orbit probably have an equivalent KH designation higher than 13. I took a stab in the dark and picked KH-27 for the comic. In computing, teraflop/s is a term describing the computing speed of a computer. To say that a computer can perform 1 teraflop/s is to say that the computer can perform 10 12 floating point operations per second. Computers that have broken the teraflop/s barrier have been in existence since 1998. Petaflop/s, by comparison, is 1000 times faster than teraflop/s (1015 floating point operations per second). As of this writing, the official record for the fastest computer in the world belongs to a computer system called Roadrunner which is housed at Los Alamos National Laboratory. Roadrunner posted a top performance of 1.105 petaflop/s in late 2008. Also as of this writing, IBM is developing a computer system (called Sequoia) for the U.S. Department of Energy that should be able to achieve speeds of 20 petaflop/s by 2011. Exaflop/s (10 18 operations per second), in turn, is 1000 times faster than petaflop/s. The NSA has always played an important role in the development of the supercomputer industry and they have always been at the forefront of supercomputing research. They currently house some of the most powerful computers in the world at their headquarters in Maryland. It is not known whether the NSA has cracked the exaflop/s barrier yet but I certainly wouldn't be surprised if they had. More Fun Facts: Zetaflop/s = 1021 floating point operations per second. Yottaflop/s = 1024 floating point operations per second. Beyond yottaflop, numbers have not yet been named. Yet Another Fun Fact: On the show, Star Trek: The Next Generation, the android character named Data was said to have “a total linear computational speed rated at 60 trillion operations per second” ( 60 teraflop/s). The title of this comic, Secrets and Lies, is taken from a book by the same title written by information security expert Bruce Schneier.

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Were you able to solve the little puzzle that I posted along with the comic?: LO KPI FG ZWXXB, JSU RVKMVY YK QRQQWQY. And Caesar Vigenère says: “Easy as pi.” As the hint implies, the message has been encrypted with something called a Vigenère cipher. The Vigenère cipher uses a key that consists of a keyword to encrypt a message. Each letter of the keyword has a numerical equivalent which is determined by its position in the alphabet. Suppose that the keyword is of length n. To encrypt a plaintext message, it is first split into blocks of length n. Then each letter of each block is replaced by another letter that exists to its right in the alphabet. The amount of this shift is determined by the numerical equivalent of the corresponding number in the keyword (mod 26).

FIG. 27-1 For example, if the the keyword is AMY, the numerical equivalents of the letters are 1-1325. Now suppose the plaintext message is BOB. Then the first B gets replaced by the letter 1 place to its right C (see the FIG. 27-1 above). The O gets replaced by the letter 13 places to its right. Since this shift is calculated mod 26, it gets replaced by a B. Similarly, the last B gets replaced by the letter 25 places to its right, A. Hence, the encrypted message is CBA. If you know the keyword, then decrypting the message is simply a matter of reversing the process. The key that I used for my puzzle was simply the first 30 digits of pi with each digit representing the amount of shift. The first 30 digits of pi are: 3.14159265358979323846264338327 168

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If you really must know what the message says, here is the answer: IN GOD WE TRUST, ALL OTHERS WE MONITOR. That phrase has jokingly been referred to as the NSA's unofficial motto.

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Real Life

continued from page 50 The quote in the original blog post is from the book Tristram Shandy by Laurence Sterne. Sterne took and modified the phrase from the Poliicraticus [Statesman's Book] by John of Salisbury, a twelfth-century churchman. The phrase can be translated as: “I do not fear the judgments of the ignorant populace, yet I ask that they spare my humble works – in which it has always been my intention to pass from jests to serious matters and from serious back to jests.” You can interpret that however you want.

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Closed Timelike Curveball

A key feature of the special theory of relativity is that space and time can no longer be thought of as separate entities. The three dimensions of space and the one dimension of time must be joined to form a four-dimensional spacetime. Since it is difficult to visualize (no less draw) four dimensions, it is sometime convenient to think of spacetime using just two dimensions of space and one dimension of time as illustrated in FIG. 37-1 where time is represented by the vertical axis.

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FIG. 37-1 Of course it should be understood that any representation in this three-dimensional spacetime could, in principle, be extended to four dimensions. Hence every point of a spacetime diagram represents an event – a point in space at a moment in time – and a particle in this diagram would be represented as a line, called a world line, as shown in FIG. 37-1. A closed timelike curve (CTC) refers to the world line of an object that returns to its starting point forming a closed loop. That definition is a bit of an oversimplification but the exact details and the mathematics behind CTCs are beyond the scope of this “text”. In short, “closed timelike curve” is a physicist's code phrase for “time travel” as the existence of CTCs would imply the theoretical possibility of building a time machine. As things stand today, there is nothing within the laws of physics that would prevent the existence of CTCs. If there's anything else that I hope you could learn from this particular comic, it’s this: don’t forget to moisturize.

38 Best Friends I actually do have some (ever so mild) obsessive-compulsive tendencies but I never ended any friendships over them. Mini-Science Break: In the mid-1800s Scottish physicist James Clark Maxwell formulated the equations that unified electricity and magnetism. These equations are now known as Maxwell's 170

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equations. I won't explain Maxwell's equations but I'll show them to you just because they look so cool. The following is one form of Maxwell's equations:

FIG. 38-1 These equations show that a time-varying electric field produces a magnetic field and that a time-varying magnetic field produces an electric field. An electric field and a magnetic field can sustain each other and form an electromagnetic (EM) wave that propagates through space. EM waves have many similarities with certain types of mechanical waves such as the waves that can be seen on the ocean. FIG. 38-2 shows an idealized depiction of an ocean wave (assume that it is moving to the right).

FIG. 38-2 The distance from one crest to the next is called the wavelength. An electromagnetic wave can have different wavelengths but it always travels at the “speed of light” 3.00 x 10 8 meters per second (in a vacuum). In fact, light itself is an example of electromagnetic waves. Visible light consists of electromagnetic waves with wavelengths in the approximate range 400 to 700 nm (400 to 700 x 10 -9 meters). Other examples with which you are undoubtedly familiar include radio and TV signals, X-rays, and microwaves – each with a different range of wavelengths.

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Wavelengths of Visible Light 400 to 440 nm

Violet

440 to 480 nm

Blue

480 to 560 nm

Green

560 to 960 nm

Yellow

590 to 630 nm

Orange

630 to 700 nm

Red TABLE 38-3

An EM wave's wavelength is related to its frequency by c = λf where c is the “speed of light”, λ is the wavelength, and f is the frequency. The frequency can be thought of as how many times per second a crest of the wave passes a particular point as the wave moves. If a crest passes through a particular point once a second, we say that its frequency is 1 hertz (Hz). EM waves have been detected with frequencies ranging from at least 1 to 10 24 Hz. This broad spectrum of frequencies is known as the electromagnetic spectrum. So the proper way to eat M&Ms is: first red, then yellow, then green, then blue. Brown doesn't count.

40

LiveCommentJournal

continued from page 58. Just in case you were wondering, yes, this is a snippet of actual code that I wrote (well, OK,… I improvised a little with the comments). The code was part of my final project for a computational biology course. The project involved finding optimal parameters for artificial neural networks through evolutionary algorithms. Now I'll be the first to admit that I'm a terrible programmer, but I slogged through it and the final product was about 6000 lines of spaghetti code. The program wasn't very efficient in terms of speed and memory usage due to heavy modularization and my injudicious use of serialization. However, it was quite effective at identifying certain promoter DNA sequences. For me, writing it was a labor of love. For the interested, I present a short introduction to neural networks. The idea of artificial neural networks (ANNs) can have its origin traced to a paper written in 1943 by neurophysiologist Warren McCulloch and mathematician Walter Pitts [1]. In the paper, they described how nerve cells could possibly replicate certain logic functions that are essential for the operation of computers. Indeed, ANNs were originally developed 172

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based on principles observed in the network of neurons found in the brains of humans and animals. It was believed that ANNs would be capable of performing some of the computing capabilities of a biological brain. It is now known that the neural processes that occur in the brain are fundamentally different (and more complex) than most artificial models, but ANNs are still known to be useful for many kinds of computational tasks and have remained an active area of research for purely theoretical reasons as well. In biological systems, the neuron is the fundamental functional unit of all nervous system tissue. A biological neuron is composed of a soma (or cell body) that contains a cell nucleus and a branching dendritic tree (dendrites) that extends from the cell body. The dendrites can form connections (synapses) with other neurons. The dendrites collect electrical signals from the other neurons. The signals are then integrated in the soma and a response is generated and propagated along a branching axon to other neurons (see FIG. 40-1). By some estimates, a single neuron can be connected to as many as 20,000 other neurons. The computational ability of the brain is believed to arise from this massive networking between the neurons and learning is believed to occur through the formation of new connections and by the strengthening or weakening of synapses.

FIG. 40-1 The part of our brains where most of the “thinking” occurs is called the neocortex, The neocortex is a sheet of nervous tissue about 2 mm thick that is heavily folded around the inside of the skull. If a sheet of neocortex from a typical human was stretched out flat, it would be about the size of a dinner napkin. By comparison, the cortical sheet from a chimpanzee would be about the size of a business envelope and, for a rat, it would be about the size of a postage stamp. Some anatomists have estimated that a typical human neocortex contains about 30 billion neurons and that a typical neuron forms between 5000 to 10,000 synapses with other neurons. Even using the lower estimate of 5000, that evaluates to an astounding 150 trillion synapses!

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An ANN, by comparison, consists of a network of artificial neurons that mimic some of the properties of biological neurons and is usually implemented as software. An artificial neural network may be defined as: “a massively parallel distributed processor made up of simple processing units, which has a natural propensity for storing experiential knowledge and making it available for use. It resembles the brain in two respects: 1) Knowledge is acquired by the network from its environment through a learning process. 2) Interneuron connection strengths, known as synaptic weights, are used to store the acquired knowledge” [2]. In an artificial neural network, each functional processing unit performs a simple computation: it receives signals from input links and computes an output which is sent to output links. The computation consists of two components. First is a linear combiner that computes the weighted sum of the input values and the second is a nonlinear activation function that transforms the weighted sum to a final output value. FIG. 40-2 depicts a schematic of an artificial neuron where w1, w2, …, w6 represent the different weights of the incoming signals.

FIG. 40-2 A single processing unit by itself is not very powerful. Just as in a biological brain, the computational power is derived from the combination of many units in a network. The network topology and the connection weights are linked to the specific computational problems that the network is able to solve. Neural networks are often organized in the form of layers of neurons. In general, three classes of network architectures [2] may be specified as: 1) single-layer feedforward, 2) multilayer feedforward, and 3) recurrent networks. A single-layer feedforward network consists of a layer of input neurons and a single layer of output neurons that performs the computation. The input layer sends its output to the output layer but not vice versa. Multi-layer feedforward networks are characterized by the presence of one or more hidden layers that perform some 174

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computation on signals from the input layer and sends its output to the output layer. The presence of hidden layers allows the network to perform certain computational tasks that single-layer networks cannot. As with a single-layer feedforward network, each layer receives signals only from previous layers in the network. A recurrent neural network differs from feedforward networks in that neurons may feed its output to other neurons in previous layers. The presence of these feedback loops may have a significant impact on the learning ability of the network. FIG. 40-3 shows a schematic of a single-layer feedforward network with four neurons in the input layer and two neurons in the output layer.

FIG. 40-3 Training a neural network is a process by which the network learns relationships between inputs and specified output targets. Often this training process consists of a repetitive process using an optimization algorithm that adjusts the system's connection weights. During training, each input pattern is propagated forward through the network and the output is compared with the target. The goal is to implement a method that adjusts the synaptic weights such that the errors are minimized. One of the most popular methods for learning is called backpropagation [3]. The weights are updated sequentially from the output layer back to the input layer by “backpropagating” an error signal along the synaptic connections and then the process is repeated multiple times. Each such pass through the network is called an epoch. Evolutionary artificial neural networks [4], or EANNs, are a special class of ANNs in which evolutionary algorithms are used to select for certain features such as topology, initial synaptic weights, input feature selection, etc. One salient feature of EANNs is the ability to adapt to a dynamically changing environment. Evolutionary algorithms refer to a class of population-based stochastic search algorithms that were developed based on principles of biological evolution. In particular, certain features of successful individuals in the population are propagated to subsequent generations. For example, in a typical application of an EANN, a population of ANNs with randomly selected features is created to http://abstrusegoose.com

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try to solve a certain computational task. In keeping with our analogy with biological evolution, each ANN plays the role of an individual in a population (I like to think of it as a population of brains that are set loose in the problem space). The individual ANNs are then evaluated for their accuracy (fitness) and then a zombie comes and eats the less successful brains while the successful ones are allowed to live (actually, the zombie part is optional). The process repeats itself with the creation of another population (generation). The features of the more successful individuals from the previous generation are allowed to proceed to the next generation along with a newly created population with randomly selected features. In some cases the features of individuals may be allowed to mutate from generation to generation and/or some individuals may be allowed to mate (i.e. to combine some of their features to produce a new individual). The process is repeated until an individual with the desired level of fitness is observed. References: [1] McCulloch, W.S., Pitts, W., A Logical Calculus of the Ideas Immanent in Nervous Activity, Bulletin of Mathematical Biophysics, Vol. 5, 115-144, 1943. [2] Haykin, S., Neural Networks: A comprehensive foundation, 2nd ed.. Prentice-Hall, Upper Saddle River, NJ, 1999 [3] Baldi, P., Gradient Descent Learning Algorithm Overview: A General Dynamical Systems Perspective, IEEE Transactions on Neural Networks, Vol. 6, no. 1, 182-195, 1995 [4] Yao, X., Evolving Artificial Neural Networks, Proceedings of the IEEE, 87(9): 1423-1447, 1999.

42

So Many Questions

continued from page 61 Does P = NP? The question of whether P = NP is one of the greatest unsolved problems in computational complexity theory. It is another one of the Millennium Prize Problems designated by the Clay Mathematics institute which offers a $1,000,000 prize for a solution. To read more about other Millennium Prize Problems, see the comments for comic #14 on page 29 and for comic #44 on page 179. In the comments for comic #2, (page 17) we learned that an algorithm can be classified according to its time complexity, which is a measure of how fast it can solve a certain problem. Likewise, problems themselves can also be classified by complexity class according to how fast they can be solved. At the bottom of this hierarchy is a class called P which consists of problems that can be solved in polynomial time. To be more precise, we say that P is the class of languages that are decidable in polynomial time on a deterministic 176

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single-tape Turing machine. We won't concern ourselves with the exact details of that definition except to say that a Turing machine is an idealized mathematical model of a computer. The concept was first proposed by mathematician Alan Turing in 1936 and it turns out that a Turing machine is an accurate model of a general purpose computer in that it can do anything that a real computer can do. For our purposes, we can simply think of a Turing machine as an abstract computer with a finite number of internal states but with infinite memory. Turing machines can accept inputs, perform calculations, and produce outputs. So when we say that a problem is in P, we mean that it can be solved in polynomial time by a Turing machine. There is a variant of a Turing machine called a nondeterministic Turing machine. Nondeterministic Turing machines are characterized by the ability to “make guesses” at any point in the computation and checking its guess (in polynomial time). If a problem can be solved in polynomial time by a nondeterministic Turing machine, we say that the problem is in NP. In fact, the term NP comes from the phrase nondeterministic polynomial time. An alternative but equivalent definition would be to say that NP problems are the problems that can be verified in polynomial time. By the above definitions, it should be clear that if a problem is in P then it is also in NP since any problem that can be solved in polynomial time by a deterministic Turing machine can also be solved in polynomial time by a nondeterministic Turing machine (by omitting the “guessing”). Problems in P are generally regarded as being tractable because they can be solved in a “reasonable” amount of time; but there are many problems in NP that are considered to be intractable (i.e. they can't be solved in a “reasonable” amount of time). Although it seems obvious that NP should include some problems that are harder than some problems in P, it has never been proven that P and NP are not in fact equivalent. However, it is commonly believed that P and NP are not the same. An important step towards (possibly) resolving the P vs NP issue came in the early 1970s with the work of Stephen Cook and Leonid Levin. They discovered certain problems in NP whose individual complexity is related to the complexity of the entire class. If a polynomial-time algorithm exists for any of these problems, then a polynomial-time algorithm exists for any NP problem. These problems are called NP-complete and, by definition, any problem in NP can be converted to any NP-complete problem in polynomial time. The theoretical implications of this discovery should be clear. If any polynomial-time algorithm is discovered for any problem shown to be NP-complete, then P = NP. What happened before the big bang? Physicist Stephen Hawking said that asking this question is like asking what lies one mile north of the north pole, i.e. it's meaningless. What is soy sauce, really? ...really, delicious . http://abstrusegoose.com

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Are there limits to human knowledge? What is mathematics, really? Is physical achievability a proper subset of abstract conceivability? Is mathematics invented or discovered? For me, each of these questions are intimately tied to the others. However, any attempt to address these issues would entail that I venture far into the realm of philosophical conjecture and biased musings. Furthermore, any attempt at an explanation of my personal opinions and prejudices on this matter would most likely result in a rather lengthy exposition so I will forgo including any such essay for now. I will probably include it in later drafts of the book so stay tuned. For convex surfaces of fixed intrinsic diameter, is the doubled disk the one with the greatest area? In 1955, mathematician A.D. Alexandrov conjectured that the answer to this question is yes [1]. However, he was never able to prove it and today it is still an open problem. The conjecture deals with the following function:

where M is a 2-manifold. Alexandrov's conjecture states that F attains a maximum of π/2 over all (convex) Riemannian 2-manifolds. You can think of a Riemannian 2-manifold as the surface of a beach ball. In our case let's assume that the distance along the surface of the beach ball from one pole to the other is 1 meter (it's a ginormous beach ball). OK, now suppose that you wanted to deform and stretch the beach ball into a shape so that it has the maximum possible surface area. However, you can only deform and stretch the ball under the following restrictions 1.) the shape remains convex and 2.) the maximum shortest path along the surface between any two points is 1 meter. To say that the beach ball is convex means that a straight line between any two points must never pass outside the surface. Then Alexandrov's conjecture simply states that it attains its maximum surface area when it is deformed into a doubled disk. You can picture a doubled disk as two circles each with a diameter of 1 meter that are glued together by their edges. Sounds easy, right? Well, go on. Give it a shot. Try to find a proof. [1] Alexadrov, A.D., Die Innere Geometrie Der Konvexen Flächen, Akademie-Verlag, Berlin, p. 417, 1955

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Can every even integer greater than 2 be written as the sum of two primes? In 1742 mathematician Christian Goldbach wrote a letter to Leonard Euler in which he conjectured that the answer to this question was yes. Today, it is called Goldbach's conjecture and it is still an open problem. It is a simple matter to come up with easy examples where the conjecture holds. For instance 8=3+5 10 = 3 + 7 = 5 + 5 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53. In fact, the conjecture has been verified by computer for all even integers less than 4•1014.

44

The Most Popular Girl in School

The differential equations depicted in the first panel of this comic are a form of the Navier-Stokes equations. Put simply, the Navier-Stokes equations describe how fluids flow. These equations can be derived by applying Newton's second law to the flow of incompressible fluids subject to an external force and to the forces arising from pressure and friction. It is still an open problem in mathematics whether (in R 3) solutions to the equations always exist and that if they do exist, whether or not the solutions behave themselves (no singularities, infinities, discontinuities). This problem is often referred to as the existence and smoothness problem for Navier-Stokes, or simply, the Navier-Stokes problem. The Navier-Stokes problem is one of the most enduring problems in all of mathematics having remained unsolved for over 150 years. In fact, it is another one of the so-called Millennium Prize Problems as designated by the Clay Mathematics Institute. This particular comic was inspired by an event which occurred in late 2006 when a Lehigh University mathematician, Penny Smith, claimed to have solved Navier-Stokes (after having worked on it for only one month) and proceeded to post her solution to an online preprint server. Needless to say, this caused quite a buzz in the mathematics community and the science blogs were soon ablaze with the commotion of this possible breakthrough. Unfortunately, Smith withdrew her paper after a few days due to a “serious flaw” in her proof. Although, her solution ultimately proved to be wrong, no one can deny that, for those few days at least, Penny Smith was “the most popular girl in school”.

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47

A Wise Man Once Said…

continued from page 67 It is a common misunderstanding that Albert Einstein was a poor student as a young boy. In truth, he did very well in science-related classes but he did not always endear himself to his teachers. He was known for his absent-mindedness, his daydreaming, and his disdain for any form of authority. According to family legend, when Albert was young, his father asked his son's headmaster what profession Albert should adopt. The headmaster answered, “It doesn't matter; he'll never make a success of anything.” At the age of sixteen, Einstein contemplated an apparent paradox. What would happen if one could travel alongside a beam of light? Einstein later recalled how this thought experiment planted the seeds for his special theory of relativity which he completed at the age of 26. At the time I drew this cartoon, there were some concerns raised by a small group of people that not enough safely precautions were being taken by CERN regarding the startup of the Large Hadron Collider (LHC). This group claimed that the LHC could conceivably create stable micro black holes that could grow and have disastrous results for Earth. Some claimed that the collider could create strangelets, an hypothetical form of matter that could possibly convert the rest of the planet into strangelets in a runaway fusion process. This group even went so far as to request a legal injunction against the LHC startup. The request was dismissed. Hey, what's with that '47' remark? Well, if you're a trekker, you may have noticed that the number 47 appears with high frequency in episodes of Star Trek: The Next Generation. Sometimes the number appears in the techno-babble that occurs as a natural part of the dialogue or sometimes it might appear unobtrusively on one of the technical panels aboard the Enterprise. Some have speculated that the origin of this phenomenon can be traced to an organization at Pomona College called The 47 Club which (jokingly) claims that all numbers equal 47. Star Trek writer Joseph Menosky attended Pomona College and it is rumored that he started the tradition of including 47 in the Star Trek scripts and that he later “infected” the other writers on the staff.

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53

Qapla'!

“Can machines think?” That question was posed by the mathematician Alan Turing in an article he wrote in 1950 [1]. In the article, Turing acknowledged that there were inherent ambiguities in that question and he proposed that we instead ask another question. Turing suggested that we should ask if a machine can win a game called the imitation game. Here's how the imitation game works. Suppose that we have a computer that was programmed to be able to carry on a conversation with a human. In this case, the conversations with the computer are carried out by typing into a keyboard (i.e. text only). Now imagine that the computer and a human are in a room hidden from view of another human (interrogator). The interrogator can communicate with both the human and the computer through a keyboard and monitor. The interrogator has to decide which one is the computer and which one is the human simply by asking them questions. The computer has been programmed to try to “trick” the interrogator into thinking that it is human while the human has been instructed to try to convince the interrogator that she is a computer. If the interrogator is unable to make a determination, then the computer can be considered to have won the game. This test of a computer's intelligence has come to be known as the Turing test. So how good is your kung-fu? Do you think that you would be able to write a computer program that could pass the Turing test? If so, then you may be entitled to a $100,000 prize. In 1990, an American inventor Hugh Loebner agreed to underwrite a formal contest designed to implement the Turing test. It is called the Loebner Prize for artificial intelligence and the contests are held annually. A grand prize of $100,000 and a gold medal will be awarded to the first computer whose responses are indistinguishable from a human's. Nobody has won the grand prize yet but annual prizes are given for the “best” performance for that year's contest. The winner of the best performance prize in 2008 was a bot called Elbot and here's an excerpt from the transcript (complete with original typos): INTERROGATOR: Good afternoon, how are you? ELBOT: But I wouldn't say that any afternoon could be good. INTERROGATOR: That's a bit pesamistic, I think any afternoon has the ability to be good. ELBOT: Well, afternoon was your word. INTERROGATOR: Are you able to see into the future ELBOT: How do you respond when people pose this question to you? INTERROGATOR: I would hope that the answer was no, otherwise I would be interested to meet you. ELBOT: Well, what will that solve? After all, the answer is only a matter of perspective. INTERROGATOR: That would depend on one's perspective http://abstrusegoose.com

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ELBOT: Sorry for the tangent, but that reminds me of carefree robot summers spent depending on internal programming. INTERROGATOR: Touchee ELBOT: With the inspirations you supply, there are no ends to my horizon... INTERROGATOR: Have you any plans for later in the day ELBOT: This is confusing. Why bring the waking hours into this? INTERROGATOR: Just curiosity and I hope that kill no cats ELBOT: That is truly one of the most inspirational qualities of humankind. Even when things seems completely hopeless, you can still hope.

Not exactly HAL 9000. References: [1] Turing, A.M., Computing machinery and intelligence, Mind, 59, 433-460, 1950

56

Family Reunion

Although this comic really was inspired by true events, I think it's safe for you (the reader) to assume that I exercised my right to take creative license with the actual content. One day, I was asked to say grace when I was visiting home from college for Thanksgiving. I believe that my actual prayer included something along the lines of, “Forgive us, Lord, for killing this innocent bird who has done us no harm.” I think I heard my brother and one of my cousins snicker a little bit, but I don't think anyone else appreciated my superb sense of humor. Although it is true that I was never asked to say grace again, I'm sure it had nothing to do with this incident.

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61

stop me if you've heard this one

IMHO, this would have been a half decent comic if I had just used the middle panels:

See? HI-LARIOUS!

62

Science is Supercool

original blog post: Let’s get one thing straight. Aquaman is cool. When I was little, I wanted to be Aquaman so I could talk to my goldfish. But what about poor Scienceman? He may be overly idealistic about the potential of science, but his heart is in the right place. Unfortunately, he gets no respect. But you can help Scienceman. The content of this comic was partially inspired by a great Perry Bible Fellowship comic called 'Super League'. Actually all of the PBF comics are made of awesome. However, the only reason I drew this particular cartoon was to give me an excuse to link to a post on the Cosmic Variance blog in which physicist (and avid blogger) Sean Carroll was asking his readers for donations for some science education charities that his blog was sponsoring. I hoped I helped a little. http://cosmicvariance.com/2008/10/01/donorschoose-challenge/ http://abstrusegoose.com

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63

NUM63R5

Shortly after posting this comic, I received an email from an astute reader that figured out that there are an infinite number of such equations: 32 + 42 = 52 102 + 112 + 122 = 132 + 142 212 + 222 + 232 + 242 = 252 + 262 + 272 …....... …...... …...... The first number of each equation is n(2n+1) with n+1 consecutive numbers on the left side and n consecutive numbers on the right side (for n = 1, 2, 3, ...). The gap between the last number in equation n and the first number of equation n+1 is 2n+3 (for n = 1, 2, 3, …). FUN FACT: The equation in the comic equals 365 and those are the exact digits that appear in the title of the comic. MORE FUN FACTS: In the 1970s, some Soviet researchers suggested sending this equation as a radio message into outer space in the hopes that an extraterrestrial civilization would receive it. Some even went so far as to say that extraterrestrials actually manipulated the Earth's rotation in order to make the days of the year match the equation.

65

The Cantor Madness

This comic is the first of several instances in which I violate my prime directive of not referencing politics. Oh well. Part of my motivation for drawing this one was so that I could link to an interesting BBC documentary called Dangerous Knowledge which tells the stories of four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing. The documentary explores the link between the genius and madness of these scientists. In my opinion, the documentary was a bit on the sensationalistic side, but I still think it is worth viewing. Check it out. [I will probably expand upon the comments for this comic in later drafts of the book.]

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68

Darmok

Perhaps this comic was a little too niche. I imagine that the only people who appreciated this one were those select few who fell in the small intersection of “GTA fans” and “obsessive trekkers”.

FIG. 68-1 NOTE: Let me also mention that the episode of Star Trek: The Next Generation on which this comic is based (called Darmok) was suppose to occur on star date 45047.2 which roughly translates into the year 2369. The episode that immediately follows Darmok (entitled Ensign Ro) takes place around Bajoran space which is why, in the comic, Riker announces to Picard that they've arrived at Bajor. Nobody ever appreciates my attention to detail. * SIGH *

70

I, Computer

continued from page 97 In 1965 Gordon Moore (an active figure in the development of the integrated circuit and later chairman of Intel) predicted that the number of transistors that could be squeezed onto an integrated circuit would double every twelve months. That is tantamount to saying that computing power would double every twelve months. In 1975 Moore revised that figure to twenty-four months. This prediction has proven to be remarkably accurate since the http://abstrusegoose.com

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introduction of the integrated circuit in 1958 (see TABLE 70-1) and has come to be known as (the now famous) Moore's Law. Moore's Law in Action for Intel Processors year

processor

number of transistors

1971

4004

2,300

1972

8008

3,500

1974

8080

4,500

1978

8086

29,000

1982

286

134000

1985

386

275,000

1989

486

1,200,000

1993

Pentium

3,100,000

1995

Pentium Pro

5,500,000

1997

Pentium II

7,500,000

1999

Pentium III

9,500,000

2000

Pentium 4

42,000,000

2002

Itanium II

220,000,000

2005

Pentium D

291,000,000

2006

Dual-Core Itanium 2

1,720,000,000

2009

Itanium (Tukwila)

2,000,000,000+ ?

TABLE 70-1 Whatever the exact rate of growth may be, the essence of Moore's law is that the growth is exponential in nature. In fact, the exponential growth of computing power can be be traced to a time before the advent of integrated circuits from the mechanical punch-card computing technology of the type used for the 1890 U.S. census to the latest state-of-theart supercomputers in use today. The question remains, however, whether or not computers will ever exceed humans in intelligence. It would seem that a necessary (but not sufficient) prerequisite for a computer to claim that title would be for the computer to match or exceed the computational capacity of a human brain. With Moore's Law in hand, can we predict when this seminal event will occur? Well, first we need to know the computational capacity of a human brain. Attempting to quantify the computational capacity of the human brain is a speculative venture at best since we know so little about the brain. However, there are naive ways to make an estimate. As I mentioned in the comments for comic #40 (page 172), the neocortex might contain as many as 150 trillion synapse connections. Neural circuitry has been estimated to be able to perform up to 200 calculations per second. With 150 trillion 186

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connections we get 30 quadrillion (3x1016) calculations per second. Another method for estimating is to examine specific regions of the brain in detail to calculate its computational speed and then to extrapolate the result to the entire brain. Studies of this kind have produced estimates in the range of 1014 to 1015 calculations per second [1] [2]. In his book, The Singularity is Near, Ray Kurzweil uses an estimate of 1016 calculations per second and gives a tentative prediction (based, in part, on Moore's Law) that computers will achieve this by some time between 2010 and 2015 and that personal computers will achieve this by around 2025. As I mentioned in the comments for comic #27 (page 166), the Sequoia computer system being built by IBM is scheduled to be completed by 2011 and is expected to operate at 20 petaflop/s (20 quadrillion operations per second). Hmmm. Maybe it's time to promote Kurweil from the title of best-selling author to the title of godhood. FUN FACT: A “hello world” program is a computer program that prints out the words “hello world” onto a display device. It has become the traditional first program that many people write when learning a new programming language. This tradition was popularized by the 1978 book The C Programming Language, by Brian Kernighan and Dennis Ritchie. [1] Moravec, H., Rise of the Robots, Scientific American, December, 1999, 124-35. [2] Watts, L., The Mode-Coupling Liouville-Green Approximation for a Two-Dimensional Cochlear Model, Journal of the Acoustical Society of America, 10, 8: 2266-71, 2000.

72

The Belt Trick

continued from page 99 The mathematical description of the phenomenon of quantum mechanical spin can be quite elegant, involving concepts from algebraic geometry and other areas of mathematics, but this is not the place to get down and dirty with that. This is, after all, just a freakin' comic book. The point I wanted to make was that particles that have this electron-like spin (physicists call it spin-½) exhibit the amazing property that spinning 360 degrees (2π radians) does not bring the particle back to its original position the way a classically spinning object would. However, if the particle rotated a total of 720 degrees (4π radians), then it would be back to its original position. The mathematical name for an object with this bizarre property is called a spinor. The “belt trick” to which the comic refers is a simple exercise that can be used to model spinors. Here's how it's done:

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Take an ordinary book and place a belt between its pages and close the book as shown below. Fix the other end of the belt to a stationary object so that that end cannot rotate

FIG. 72-1 (perhaps taping to a box as shown). Now if the book is rotated through an angle of 2π, you will find that the twist in the belt cannot be undone without rotating the book further. However, if you twist through an additional rotation of 2π, you'll find that the belt can be untwisted simply by looping the entire belt over the book. Cool trick, huh? Now get the f*ck outta here, kid.

73

True Things

I think several points about this comic deserve mention. Quantum Entanglement First of all, I should probably mention that “Alice and Bob” are just placeholder names that are commonly used in descriptions of cryptographic protocols in which one party is attempting to send a message to another party. By convention, such descriptions usually depict Alice sending a message to Bob. So what exactly is quantum entanglement? Put simply, quantum entanglement is the phenomenon in quantum physics whereby two or more particles interact in a such a way that their fates become inextricably linked together. In effect, they constitute a single quantum state (for more information about quantum states, please read the comments for comic #6 on page 21). Now suppose that two particles become entangled and let's consider their spins (you can read more about quantum mechanical spin in the comments for comic # 72 on page 99). Just like a classically spinning object, a particle's quantum mechanical spin can be considered to be spinning about an axis that has direction in space. Let's just pick an arbitrary direction of the axis and call it up/down. If it is spinning in one direction, we can 188

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call it 'up' and if it is spinning in the other direction, we can call it 'down'. Just like Schrödinger's cat in comic #6, the particle would be in a superposition of both the up and down state before any measurement was made of its spin. However, if someone were to try to measure the particle's up/down state, the particle would immediately assume either an up spin or a down spin according to random chance. In other words, before its spin state is measured, we can only know the probability that the particle will be in either state after the measurement. OK, now it gets even stranger. If you were to measure an up spin for one of the entangled particles, then someone measuring the spin of the other particle would find a down spin with 100% probability. Furthermore, this effect is instantaneous. This would be true even if the two particles were located light years aways from each other. Somehow, the two particles “appear” to be communicating information to each other faster than the speed of light; but according to the special theory of relativity, nothing can travel faster than the speed of light – not even information. So why does quantum entanglement NOT violate relativity?... cuz fuck you – that's why. Packet Switching A network is a collection of devices (or nodes) that can communicate with each other through a shared communication protocol. By this definition, the internet can actually be thought of as a network of networks. For the internet, the devices usually consist of computers, routers, hubs, gateways, etc., and the most widely used communication protocol is actually a suite of protocols called TCP/IP. The TCP/IP suite of protocols is what allows different types of devices to communicate with each other over the internet transparently. So whether you're using a PC or a Mac, Windows or Linux, you can bet that TCP/IP is working quietly under the hood (for most of us) to facilitate your connection to the internet. A key feature of TCP/IP is the facilitation of packet switching. Packet switching is a method of sending information over a network in which the message to be sent is separated into chunks called packets. Each packet is given a destination address and a number to identify its order in the sequence. For networks or collections of networks such as the internet, when a node is ready to send a packet, a direct connection to the destination may not be available due to network conditions such as traffic congestion or network failures. In this case, the packet may be routed to intermediate nodes until it reaches its destination and sometimes the packets may arrive out of sequence or be lost altogether. It is the responsibility of network software to keep track of the lost packets and the out-of-sequence packets. There are methods to guarantee the delivery and correct sequencing of the packets but I will not describe them here. The Well-Ordering Theorem A description of the well-ordering theorem can get a bit “technical” but I find it to be a great example of how mathematics can serve up some great mind-fucks, so I will present a http://abstrusegoose.com

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short explanation here. A full description requires that we be familiar with some basic definitions first. For our purposes, a set will simply be defined as a collection of objects (or elements). The objects in a set can be anything from apples and oranges to angry monkeys but, as might be expected in mathematics, the sets under consideration are often sets of numbers of some kind. If a, b, and c are elements of a set A, then this can be expressed through mathematical notation as A = { a, b, c }. We say that A is a subset of a set B if every element of A is also an element of B. One way to create new sets from old ones is by constructing what is known as a cartesian product. Suppose that we have two sets A and B. Then the cartesian product of A and B (denoted by A X B ) is defined to be the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. You may already be familiar with the notion of ordered pairs from geometry class where every point in the xyplane corresponds to a unique ordered pair (x, y). A relation on a set A is defined as a subset C of the cartesian product A X A. If C is a relation on A, then mathematicians use the notation xCy to indicate that (x, y) is an element of the set C where x and y represent elements of A. We say that “x is in the relation C to y”. One type of relation that occurs with great frequency in mathematics is called an order relation. A relation C on a set A is an order relation if the following properties hold: 1) 2) 3)

For every x and y in the set A where x does not equal y, either xCy or yCx. For no x in A does the relation xCx hold. If xCy and yCz, then xCz.

There may be many ways to create order relations for any given set but you are probably already familiar with the “usual order relation” for numbers that you learned in elementary school math classes in which x < y means x is less than y. You might want to confirm for yourself that the “less than” relation conforms to the definition of an order relation just given. A set A with an order relation < is said to be well-ordered if every nonempty subset of A has a smallest element. For an example of a well-ordered set, consider the set { 1, 2, 3 } with the usual order relation and convince yourself that every nonempty set has a smallest element. For an example of a set that is not well-ordered, consider the set of all integers with the usual order relation. The subset consisting of all of the negative integers clearly does not have a smallest element. OK, now that we have the definitions out of the way, we can finally state the wellordering theorem in all its glory. Ready? Here it is: The well-ordering theorem: If A is a set (any fucking set whatsoever), there exists an order relation on A that is a well-ordering. This rather simple sounding theorem was proved in 1904 by the mathematician Ernst Zermelo and it stunned the mathematics community. To get a feeling for why this result was so astonishing, take some time to try to construct a well-ordering on the set of real numbers. As you are probably already aware, the real numbers can be thought of as the 190

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numbers that can be expressed as decimals (such as 1.0000000000..., 3.1415926535..., and 0.3333333333...). If you claim to be able to well-order the reals, then you're (probably) a damn dirty liar. As an extra side note, the symbol (Z+)ω that appears in the comic represents the infinite cartesian product Z+ X Z+ X Z+ X ··· where Z+ represents the set of positive integers. Non-existence In his book Gödel, Escher, Bach: An Eternal Golden Braid, author Douglas Hofstadter considers the concept of contemplating your own nonexistence to be a metaphorical analogue of Gödel's Theorem. Aerodynamics The equation depicted in the cartoon P + ½ρν + ρgh = constant is one of the many forms of Bernoulli's equation, named after the Dutch-Swiss mathematician Daniel Bernoulli. The equation describes what is known as Bernoulli's principle which relates the pressure in a gas to the local velocity. In the equation, P is the pressure, ρ the density, ν the velocity, h the height, and g the acceleration due to gravity. When a gas flows over an object (e.g. an airplane wing), the velocity of the gas can have different values near different parts of the object's surface. In simple terms, the equation says that if the velocity of the gas increases, then the pressure decreases. This principle can explain how an airplane wing produces lift that allows the airplane to fly. The exact details of how the wing produces lift can be somewhat complex and will not be described here. Euler's Identity As you may have surmised from the cartoon, the equation ei π + 1 = 0 is called Euler's identity. The equation is named after the insanely prolific mathematician Leonard Euler who derived it in 1748. In the equation, e is the base of the natural logarithm (called Euler's number), i is the imaginary unit (which is one of the complex numbers whose square is -1), and π (pi) is the ratio of the circumference of a circle to its diameter. http://abstrusegoose.com

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If you know the mathematics behind the derivation of Euler's identity, the equation might not seem so extraordinary (I will not present the derivation here). However, there is an intuitive way to visualize this equation (see FIG. 73-1).

FIG. 73-1 The value of eiΘ is a complex number which lies on the unit circle of the complex plane as shown in FIG. 73-1. The position on the unit circle is determined by the value of Θ which is the counterclockwise angle made with the positive real line. As you can see, if Θ = π, then the value is -1, which gives us Euler's identity. I have to admit that when I take a step back and just admire the equation for what it is, I am astounded by how elegantly it relates five of the most important constants in mathematics: e, i, pi, 1, and 0. I can't help but get the feeling that this equation is telling us something very profound about the nature of mathematics. Noted physicist Richard Feynman called it “this amazing jewel...the most remarkable formula in mathematics.” [1] References: [1] Feynman, R.P., Leighton, R.B., Sands, M.L., Feynman Lectures on Physics, Vol. I, Addison-Wesley, Reading, MA, 1963.

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Popular Science

OK, it's disclaimer time again. This comic obviously references the PBS documentary The Elegant Universe which is based on the book by Brian Greene. First of all, I think that the book is quite excellent; one of the better popular science books out there. In my humble opinion, Dr. Greene does a masterful job of explaining difficult concepts to a layaudience and he did it with a very entertaining writing style (I can kiss ass with the best of them). As for the documentary, I don't think many people would contend the argument that it was a dumbed-down incarnation of the book; but in Brian Greene's defense, he originally wanted two versions of the mini-series to be produced: one for little kids and another for a more sophisticated audience. In the end, PBS decided to do the kids' version. ...so to sum up, 1.) the documentary is geared towards kids and 2.) I highly recommend the book.

79

In the Beginning

Not everyone can agree on the exact details of how it went down in the early universe but, here, I present you with my own speculative version. A Brief History of the Universe

age of the universe

description

before 10-43 seconds

Dancing monkeys everywhere.

10-43 seconds

Gravity begins to separate from the other forces.

10-35 seconds

The strong force separates from the electroweak force. The universe inflates rapidly.

10-9 seconds

Electroweak symmetry breaks.

10-3 seconds

Quarks start to condense into protons and neutrons.

3 minutes

Protons and neutrons begin to condense to form stable nuclei.

300,000 years

Electrons are now able to condense around nuclei allowing atoms to form.

3 billion years

The first quasars make their appearance.

5 billion years

The first galaxies begin to appear.

10 to 15 billion years ...a little later

Our solar system is born. My girlfriend breaks up with me. TABLE 79-1 http://abstrusegoose.com

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80

All You Zombies

original blog post: Then I glanced at the ring on my finger. The Snake That Eats Its Own Tail, Forever and Ever. I know where I came from—but where did all you zombies come from? I felt a headache coming on, but a headache powder is one thing I do not take. I did once—and you all went away. So I crawled into bed and whistled out the light. You aren’t really there at all. There isn’t anybody but me—Jane— here alone in the dark. I miss you dreadfully! –All You Zombies, Robert A. Heinlein This one is an homage to the infamously perplexing time-travel short story by Robert Heinlein entitled All You Zombies. After posting this comic, someone suggested that I check out an indie movie about timetravel called Primer (directed by Shane Carruth). I have since added that movie to my recommendation list.

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Hand Turkey

Hand Turkey - ANSWERS A. Spock B. Tony Stark C. Homer Simpson D. Leonardo da Vinci E. M.C. Escher F. This is a Shocker Turkey. G. Yakuza member H. Wolverine I. This is a POPsickleTURKEY J. ninja K. The Thing L. mathematician M. Salvador Dali N. Sarah Palin O. Doctor Zoidberg P. Jackson Pollock 194

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The Mind of God

original blog post: Is it possible to imagine anything so ridiculous as this miserable and wretched creature, which is not so much as master of himself, exposed and subject to offenses of all things; and yet dareth call himself master and emperor of this universe in whose power it is not to know the least part of it, much less to command the same? And the privilege, which he so fondly challengeth, to be the only absolute creature in this huge world’s frame perfectly able to know the absolute beauty and several parts thereof, and that he is only of power to yield the great Architect thereof due thanks for it, and keep account both of the receipts and layings-out of the world! Who hath sealed him this patent? Let him show us his letters of privilege for so noble and so great a charge. —Michael de Montaigne, An Apology of Raymond Sebond, 1568 This is, without question, one of my favorites out of the first 100 comics. Now make no mistake; the art may be simplistic but this is one of those comics that took me an insanely long time to draw (insanely long time = a few hours). For several years I had intended to write a short science fiction novel based on this theme. Now the theme of humankind-evolves-into-God is a tried and true theme in the science fiction world but I thought that I could put a unique spin on it. For now, the novel will have to wait. MISCELLANEOUS NOTES FOR COMIC The quote by Michael de Montaigne in the original blog post was meant to be “ironical”. The Greek phrase in the 5th panel

can be roughly translated as “Let no one ignorant of geometry enter”. Legend has it that that phrase was engraved near the entrance to Plato's Academy, a school founded by Plato in Athens.

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The drawing in the 9th panel is copied from an actual entry in one of Galileo's notebooks in which he recorded the movements of Jupiter and some of its moons. The equation in this panel

is a statement of Kepler's Third Law which relates the time T (that it takes a planet to go around its sun) with the orbit's semimajor axis a. In the 10th panel, the equation

is a statement of Einstein's equation which relates curvature of spacetime to the density of mass-energy. The symbols

represent the process of bombarding 238U with neutrons to produce fissionable Plutonium which can be used in nuclear reactors. A,G, C, and T are the symbols for the nucleotides found in DNA: adenine, guanine, cytosine, and thymine. The expression

is half of the Schrödinger equation which describes how the quantum state of a system evolves over time. The equation in panel 11

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was derived by Ray Kurzweil to describe how he believes world knowledge could accumulate at a double exponential rate. The digits in panel 13 are digits from the binary representation of pi.

87

Ripoff

original blog post: The disputants, I ween, Rail on in utter ignorance Of what each other mean, And prate about an Elephant Not one of them has seen! Obviously, this comic refers to my “Mind of God” comic. As I mentioned before, I generally don't read online comments about my comics but sometimes I accidentally encounter them while surfing. I just happened to (accidentally) read two comments suggesting that I was ripping off an earlier work and, I'll be honest; I was a bit pissed off. Now don't get me wrong. I'm a big boy and I can take intelligent criticism, but if someone accuses me of plagiarism, I feel I must defend myself.

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The Adventures of Buckaroo Banzai...

This is my homage to Edwin Abbot's novel Flatland: A Romance of many Dimensions by a Square which he wrote in 1884. Edwin Abbot was a clergyman and headmaster of the City of London School. The novel revolves around the character Mr. Square who lives in a two-dimensional world called Flatland. One day he is visited by the mysterious Lord Sphere who happens to live in a three-dimensional world called Spaceland. To Mr. Square, Lord Sphere just appears as a circle that can apparently change size. Lord Sphere lifts Mr. Square out of Flatland and shows him the wonders of traveling in three dimensions. However, when Mr. Square later tries to tell his fellow Flatlanders about the existence of Spaceland, he is labeled a lunatic. For Abbot, Flatland was actually intended to be a not-so-thinly disguised criticism of the prejudices that he perceived to be prevalent in Victorian England.

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Originally, I wanted the cartoon to demonstrate the concept of how traveling in higher dimensions can (in some cases) lead to an apparent change in chirality (or handedness). It's not as easy to draw as it seems.

91

The Butterfly Effect

original blog post: For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. The Butterfly Effect is a phrase that describes the phenomenon of sensitive dependence on initial conditions. The phrase has its origins in the idea that a butterfly flapping its wings in (say) Brazil, could produce a hurricane in (say) Philadelphia due to nonlinearities in the world's weather patterns. This strong dependence on initial conditions of the final outcome of a dynamical system is a fundamental aspect of the mathematical theory of chaos. WOPR is an acronym for “War Operation Plan Response”, a fictional computer system featured in the movie War Games. The WOPR was designed to play strategy and war games for the purpose of calculating optimal strategic responses to any possible nuclear attack. In the movie, the WOPR mistakes a simulation for an actual nuclear attack by the Soviet Union and nearly starts World War III. The fictional WOPR was supposedly located at the nonfictional headquarters of NORAD (North American Aerospace Defense Command) which is located inside Cheyenne Mountain in Colorado Springs, CO, USA. If you examine the outline of the United States that I drew for the comic, you will notice that I placed the location of the WOPR exactly where Colorado Springs would be if you overlaid a map on top of it. Once again, nobody appreciates my attention to detail. :(

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Scientific Verification

continued form page 136 Obviously, the calculations depicted in the second panel of the comic are just nonsense, but the equations in the first panel were meant to be actual calculations for the energy required for the total destruction of the planet Alderaan. There are several ways that one might conceivably calculate the energy needed to destroy a planet. In the movie Star 198 http://abstrusegoose.com

Wars, the Death Star unleashes its beam of destruction and is able to scatter the pieces of Alderaan into a shower of flying rubble. So how do we calculate how much energy is needed to disperse a planet's mass in this fashion? Well, one naive method for doing so would be simply to calculate the energy required to scatter the entirety of the planet's mass with enough velocity to overcome the planet's gravitational attraction (i.e. the escape velocity). For this, we resort to the formula for gravitational potential energy:

where G is the gravitational constant, M is the mass of the planet, and m is the mass of an object that is a distance r from the center of the planet. We know that the gravitational constant is equal to 6.67 x 10-11 N • m2/kg2. I also happen to know that Alderaan has a mass of 6.16 x 1024 kg and a radius of 5.61 x 106 m. Don't ask me how I know that. Let's just say I have connections inside the rebel alliance. To calculate the velocity needed for an object to escape the gravitational field of a planet, we rely on the fact that mechanical energy is conserved. So we have:

K1 + U1 = K2 + U2 where K1 and U1 are the object's initial kinetic energy and initial potential energy respectively. K2 and U2 are the kinetic energy and potential energy at a later time. Theoretically we want the object to be able to reach infinity with no kinetic energy left over so K2=0 and U2=0 and we have K1 + U1 = 0. Now by the classical kinetic energy formula, we have K1 = ½ mv12 where v1 is the initial velocity so all we have to do is plug in the values for the gravitational constant as well as the mass and radius of Alderaan and we can calculate the escape velocity:

v1 = 1.21x104 m/s Now we can use the kinetic energy formula with Alderaan's escape velocity and total mass to calculate the energy required for its total destruction:

Ktotal = ½ Mv2 Ktotal = ½ (6.16x1024 kg)(1.21x104 m/s)2 Ktotal = 4.51x1032 J As you can see I made a mistake in the comic. The problem with the method just outlined is that it does not take into account the fact that, as Alderaan is being blown up, its mass is continually changing due to the expulsion of planetary debris. The words “continually changing” should be a clue that this is a job for http://abstrusegoose.com

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calculus. So here's how to do it. Imagine that Alderaan is composed of infinitely many concentric spheres somewhat like the layers of an onion. Now we can imagine peeling off the layers one by one. Let's suppose that each layer has an infinitesimal thickness dr. The surface area of a sphere is simply 4πr2 and the volume is 4/3πr3 so we can determine the mass m of an individual layer as well as the mass M of the rest of the planet inside the layer:

where ρ is Alderaan's density. Now we can calculate the total potential energy by plugging in the above formulas to the gravitational potential energy formula and integrating from 0 to R (where we now denote Alderaan's normal radius by R).

Ahhh... as you can see, my evil plan is unfolding exactly as I have foreseen. Now we know that Alderaan's density is simply its mass divided by its volume:

Substituting that into the formula for U and simplifying we get:

That works out to about 2.71 x 1032 J. That's a shit load of energy. As for the energy calculations for the force, I leave that to you.

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96

ask a silly question...

You may be asking, “What is this group theory of which you speak, kind sir?” For a student of mathematics, group theory is the starting point for any modern treatment of the subject of abstract algebra. However, it is much more than just a subject included in academic curricula to frustrate unwitting math students. As you may have guessed, at the heart of group theory is a mathematical object called a group. I will define a group a little later but I will mention now that the concept of a group is intimately tied to the idea of symmetry. In fact, groups were first invented to study symmetries of algebraic structures called field extensions. We see symmetry all around us in nature. The human figure is roughly symmetrical about a vertical plane. As you probably already know, the right half of the human form is approximately a mirror image of the left half. This is known as bilateral symmetry. You may also have noticed that the shape of a snowflake is symmetrical. If you rotate a snowflake by 60○, the shape remains the same. This is called rotational symmetry. Another kind of symmetry can be seen in certain wallpaper designs that have a repeating pattern where the entire pattern can be displaced in various directions without changing the overall design. This kind of symmetry is called translational symmetry. The above discussion might lead you to believe that symmetry is all about how rigid shapes can be moved around in space while still looking the same. Essentially that notion is a correct one. So just for fun, let's take a closer look at a specific example of the symmetry of a rigid shape: a square. How many different ways can a square be moved in space such that it retains its original appearance? It would be easier to describe the symmetries of the square if we labeled the vertices with numbers as shown in FIG 96-1. Clearly, if we rotated the square by 90○, 180○, or 270○ in a clockwise direction, then its appearance would remain the same. Let's call these rotations r, r2, and r3, respectively. The effect of these transformations is to rotate the different vertices into each other. Of course, if we rotated it by 360○, it would be the same as if we did nothing at all. The act of 'doing nothing' is a symmetry as well, so let's call it '1' (for reasons that you will see shortly).

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Note that you can compose together different rotations to get another rotation. For example, notice that if we performed the rotation r twice it would be the same as if we performed the rotation r2 once. We can adopt a notation to express this act of performing two rotations as rr where it is implied that the r on the right is performed first followed by the r on the left. With this notation we have rr = r2. It is also possible to reverse the direction of any rotation with the result being that the square is returned to its original position. For example, if we perform the rotation r immediately followed by its inverse (which we will denote by r -1), then the vertices do not change position. There is another way, besides rotations, to move the square such that it keeps its original appearance. Can you see it? If we drew lines through the center of the square (as shown in FIG. 96-2a) and flipped the square about any of those lines, the appearance would remain the same. FIG. 96-2b shows the result of a reflection about the line passing through vertices 1 and 3. Let's label that reflection as s. Two reflections s performed one after the other would result no change at all. Using the notation described above we have ss = 1. We can also compose a rotation with a reflection (i.e., sr, sr2, sr3).

FIG. 96-2 We have just described all of the possible symmetries of the square. I will leave it as an exercise to confirm that every possible rigid motions of the square that leaves its original appearance unchanged is contained in the set { 1, r, r2, r3, s, sr, sr2, sr3 }. This set of symmetries is an example of a group called the dihedral group of order 8. (denoted by D8) You may find that the group concept will pop up like zits whenever we're dealing with symmetries. In fact, the set of all symmetry transformations of any system can always be described as a group. Symmetry can even be found buried deep within the foundations of the universe itself. For example, recall our discussion about the Standard Model (see the comments for comic 202

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#18 on page 33) in which the elementary particles were described. What I didn't mention previously is that the quarks comes in three colors: red, green, and blue. These colors have nothing to do with the colors that we see in a rainbow. The colors of the quarks refer to properties of the quarks that determine how they respond to the strong force. This can be compared to how electric charge determines how a particle responds to the electromagnetic force. It turns out that three colored quarks (called a multiplet) can be “shifted” and “shuffled” in various ways while the equations describing them remains unchanged (you can think of this as rotating the quarks in some abstract space similarly to the way we rotated the square). This is also a kind of symmetry and it can be described by a group called SU(3). Similarly, there is a symmetry associated with the electromagnetic force described by a group called U(1) and a symmetry associated with the weak force described by a group called SU(2). Aside: The above description is a rather severe oversimplification so let me expatiate with yet another oversimplification. It would be more appropriate to say that particles obey SU(2) symmetry only approximately. A proper quantum field-theoretic description requires that we treat the weak force and the electromagnetic force as a single unified force (called the electroweak force) with an associated symmetry group called SU(2) X U(1) which is built out of SU(2) and U(1). It has been theorized that the weak and electromagnetic forces were indeed a single unified force earlier in the history of the universe but later became separate forces [1, 2, 3]. This separation of the forces is commonly referred to as electroweak symmetry breaking. The Standard Model goes even further by unifying the electroweak force with the strong force whereby SU(3) is combined with SU(2) X U(1) to form one large group called SU(3) X SU(2) X U(1). In 1915, the German mathematician Emmy Noether proved a remarkable theorem that showed that every continuous symmetry in nature corresponds to a conservation law. In particular, consider movement through space (translation). No matter how you move through space, the laws of physics remain the same. This is a type of symmetry and the corresponding conservation law is the conservation of momentum. Likewise, symmetry with respect to the passage of time gives us the conservation of energy and rotational symmetry gives us the conservation of angular momentum. Holy popcorn, Batman!!! So when mathematicians and physicists explore these myriad symmetries that seem to be so intimately tied to the universe around us, they do so using the language of group theory. It is not my intention to explain group theory in any detail here but I will at least give you a basic definition of a group. The concept of a group, which plays such an integral role in our theories of the universe and which has given us such a rich body of mathematical theory, has a deceptively simple definition: First we define a binary operation. A binary operation ● on a set G is a function that takes two elements of G to give another element of G. For any a, b in G we shall write a ● b to express that we are applying the binary operation on a and b. http://abstrusegoose.com

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A group is an ordered pair (G, ●) where G is a set and ● is a binary operation on G satisfying the following axioms: 1.) 2.) 3.)

(a ● b) ● c = a ● (b ● c), for all a, b, c in G, there exists an element e in G, called an identity, such that for all a in G we have a ● e = e ● a = a, for each a in G there is an element a-1 of G, called an inverse of a, such that a ● a-1 = a-1 ● a = e.

That's it. That's all a group is. A familiar example of a group is the set of integers with the binary operation of addition where the identity is 0 and the inverse of any element is its negative. Another example is the set of positive real numbers with the binary operation of multiplication where the identity is 1 and the inverse of any element is its reciprocal. You can check for yourself that these do indeed satisfy the definition for a group given above. You can also check that the example given above of the symmetries of a square also satisfies the definition of a group where the elements are { 1, r, r2, r3, s, sr, sr2, sr3 } and the binary operation is the composition of any of its elements. In order to explain what the Monster group is, we will need some more definitions. The number of elements in a group is called the order of the group. The examples given above of the integers and the positive reals are groups of infinite order. The symmetries of the square is a group of order 8. A subgroup of a group is a subset of that group that still satisfies the definition of a group. For example, the set H = { 1, r, r2, r3 } is a subgroup of D8 as you can easily verify. Every group has at least two (trivial) subgroups: one composed of just the identity and the other being the group itself. Now suppose that we take any element of the subgroup H, say r, and “multiply” it from the left by an element of its parent group D8 , say s. Then multiply from the right by its inverse s-1, so we have srs-1 = r3, which is itself a member of the subgroup H. If every element of H has this property that leftmultiplying by any element of D8 and right-multiplying by its inverse gives a member of H, then H is called a normal subgroup of D8 . Is H a normal subgroup of D8? Check for yourself. We call a group a simple group if it has no normal subgroups other than the two trivial subgroups. Simple groups can be thought of as the building blocks of group theory in that every group can be composed of simple groups. This is similar to the way prime numbers are the building blocks of all of the integers. Mathematicians have undertaken the monumental task of classifying all of the possible finite simple groups and have discovered that every finite simple group falls into one of 18 categories (or families) or one of 26 sporadic groups that do not follow any such pattern. The largest of the sporadic groups is the so-called Monster group whose existence was predicted in 1973 by Bernd Fischer and Robert Griess. Hence it is also known as the Fischer-Griess Monster. Just in case you're interested, the order of the monster group is 808017424794512875886459904961710757005754368000000000 . 204

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References: [1] Glashow, S., The renormalizability of vector meson interactions, Nucl. Phys., 10,107, 1959. [2] Salam, A., Ward, J.C., Weak and electromagnetic interaction, Nuovo Cimento, 11, 568, 1959 [3] Weinberg, S., A model of leptons, Phys. Rev. Lett., 19, 1264-66, 1967

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Computer Programming 101

I hope that this comic effectively illustrates the problem inherent in teaching computer programming to a new student. Computer programming is built upon layers of abstraction and the question is “How far under the hood is too far?”. In my experience, most people learn at least one programming language before ever learning anything about computer architecture, compiler design, the mathematical theory of computation, etc. This seems to work well enough but learning it in that order means that the process of learning the programming language will necessarily entail referencing back to concepts that the student has not yet learned. I certainly remember how confusing that was for me. I imagine that a lot of people who teach computer programming tend to forget what it was like to be clueless. For an offensively oversimplified explanation of electroweak symmetry breaking, see the comments for comic #96 on page 201.

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REFERENCES (that you may or may not have missed)

1

Convergent Subsequence

1st panel: I drew the “sun set” scene based on the Roy Lichtenstein painting Sinking Sun which was bought for $15,696,000 at a Sotheby's auction in 2006.

Sinking Sun by Roy Lichtenstein, 1964 11th panel: The words "THERE... ARE... FOUR... LIGHTS!" was spoken by Captain Picard in an episode of Star Trek: The Next Generation( ST:TNG) entitled The Chain of Command II.

3

SETI Finally Receives a Signal

4th panel: "WOW!" was a reference to the so-called WOW! Signal. The WOW! Signal was a narrowband radio signal detected by a SETI astronomer Dr. Jerry Ehman in 1977. The signal had the characteristics of an artificially produced message and lasted for a full 72 seconds. Dr, Ehman wrote “Wow!” on the printout of the signal and the rest is history.

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4

LOLCAT Backlash

All of the quotes from the obnoxious kid in this comic are taken from http://icanhascheezburger.com/.

8

Arguing with a String Theorist

9th panel: The line 'You keep talking like a bitch, I'm gonna slap you like a bitch" is taken from the movie Reservoir Dogs.

21

Life Imitates Art

2nd panel: John Conway, Julianne Dalcanton, Sean Carroll, and Joanne Hewett are contributors to the physics blog Cosmic Variance. 3rd panel: The number 47 is known to occur with an unusually high frequency in ST:TNG. 5th panel: The line "I'm with SAG. We don't lie." is a spoof of a line spoken by Wesley Crusher in the episode of ST:TNG entitled Justice. The actual line was "I'm with Starfleet. We don't lie." 9th, 11th panel: In the episode The Naked Now (ST:TNG), Wesley Crusher does an impossible calculation in his head in a matter of seconds and then saves the Enterprise with just seconds to spare. 14th panel: The line "Never forget to check your references." is from the movie Real Genius.

22

Reality vs Fantasy

"Best mind fuck yet." is a line from Arnold Schwarzenegger in the movie Total Recall.

24

Out of the Closet

3rd panel: "Get him a body bag." is a line from the movie The Karate Kid.

28

Math vs Physics

5th panel: The dialogue is from an episode of Cheers: Sam Malone: Are you as turned on as I am? Diane Chambers: More!

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36

An Elegant Weapon…

2nd panel: Gordon Freeman is the protagonist of the Half-Life series of video games. 4th panel: In the movie High Plains Drifter, Clint Eastwood speaks the line, "You're going to look pretty silly with that knife stickin' outta your ass."

41

Today I Learned That…

8th panel: The word grok originated from the book Stranger in a Strange Land by Robert Heinlein. In the book, grok is a martian word that can be roughly translated as "to understand profoundly through intuition or empathy".

42

So Many Questions

What is Mathematics, Really? is the title of a book by Reuben Hersh. "Who is John Galt?" is a phrase from the Ayn Rand novel Atlas Shrugged. "Did Galoka think that the Ulus were too ugly to save?" is a line from the movie The Last Starfighter.

43

The Curve

3rd panel: The phrase "The local Euclidean metrization of a k-fold contravariant Riemannian tensor field" is a line from an episode of ST:TNG entitled The Vengeance Factor. 6th panel: On an episode of Cheers, Diane Chambers says to Sam, "I hate you with the white hot intensity of a thousand suns." 7th panel: This is a line from the movie Conan the Barbarian.

45

I never lose this game

9th panel: "Two for flinching" is a line from the movie Stand By Me.

49

Dear CERN

4th panel: "Michael Corleone says hello" is a line from the movie The Godfather II spoken by Tony Rosato before throwing a garrote around around Frank Pantangeli's neck.

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60

Free Pass

9th panel: In the movie Office Space, Lumbergh often started sentences with phrases like "Yeah, I'm going to have to ask you to go ahead and <something>...". Just in case you didn't notice, the right-most photograph underneath the comic is of Lumbergh.

63

NUM63R5

1st panel: This line was spoken by the Larry Fleinhardt character from the TV show NUMB3RS.

65

The Cantor Madness

2nd panel: In an episode of the TV sitcom The Big Bang Theory, Sheldon Cooper says, "Oh gravity, thou art a heartless bitch."

67

you're not as cool as you think

8th panel: The full pick-up line is "You must wash your clothes with Windex, cuz I can definitely see myself in your pants."

74

Schrödinger’s (emotional) Miscalculation - Part 3

6th panel: "Happiness is a warm puppy" is a well known phrase from the Peanuts comic.

79

In the Beginning

3rd panel: "Aziz, light!" is a line from the movie The Fifth Element. The dancing monkeys are singing a song from an episode of Bugs Bunny entitled What's Up Doc.

82

A Simple Request

1st panel: In the movie Star Trek: Generations, Data recounts to Geordi a joke that Geordi told Data during the Farpoint mission. The punchline of the joke was "...the clown can stay, but the Ferengi in the gorilla suit has to go."

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Feynman, R.P., The Feynman Lectures on Physics, Vol. 3, Addison Wesley (2005). French, A.P., Special Relativity, M.I.T. Introductory Physics Series, W. W. Norton (1968) [TEXTBOOK – UNDERGRADUATE]. French, A.P., Vibrations and Waves, M.I.T. Introductory Physics Series, CRC (1971) [TEXTBOOK – UNDERGRADUATE]. Gamow, G., One Two Three . . . Infinity: Facts and Speculations of Science, Dover Publications (1988). Gasiorowicz, S, Quantum Physics, Third Edition, Wiley (2003) [TEXTBOOK – UNDERGRADUATE]. Gladwell, M., Blink: The Power of Thinking Without Thinking, Back Bay Books (2007). Gleick, J., Chaos: Making a New Science, Penguin (1988). Gowers, T. (editor), Barrow_Green, J (editor), Leader, I. (editor), The Princeton Companion to Mathematics, Princeton University Press (2008) [MADE OF AWESOME]. Greene, B., The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, W.W. Norton & Co. (2003). Griffiths, D.A., Introduction to Electrodynamics, 3rd Edition, Benjamin Cummings (1999) [TEXTBOOK – UNDERGRADUATE]. Guillen, M., Five Equations That Changed the World: The Power and Poetry of Mathematics, MJF Books (2000). Gurewitch, N., The Perry Bible Fellowship: The Trial of Colonel Sweeto and Other Stories, Dark Horse Comics (2007). Hammerschmidt, E., Abby and Norma, Lulu (2008). Hartle, J.B., Gravity: An Introduction to Einstein's General Relativity, Benjamin Cummings (2003) [TEXTBOOK – UNDERGRADUATE]. Hawking, S., A Brief History of Time, Bantam (1998). Hawkins, J., Blakeslee, S., On Intelligence, Holt Paperbacks (2005). Haykin, S., Neural Networks: A Comprehensive Foundation, 2nd Edition, Prentice Hall (1998). Hays, L., Bobby Fischer: Complete Games of the American World Chess Champion, Hays Publishing (1995). Heath, T.L. (trans), Densmore, D. (editor), Euclid's Elements, Green Lion Press (2002). Heinlein, R.A., All You Zombies, (1958). Heinlein, R.A., Stranger in a Strange Land, Ace Trade (1991). Hennessy, J.L., Patterson, D.A., Computer Architecture, Fourth Edition: A Quantitative Approach, Morgan Kaufmann (2006) [TEXTBOOK – BEGINNING GRADUATE].

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Penrose, R., Gardner, M., The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford University Press (2002). Penrose, R., The Road to Reality: A Complete Guide to the Laws of the Universe, Vintage (2007). Peskin, M.E., Schroeder, D.V., An Introduction To Quantum Field Theory, Westview Press (1995) [TEXTBOOK – BEGINNING GRADUATE]. Petzold, C., Code: The Hidden Language of Computer Hardware and Software, Microsoft Press (2000). Pickover, C.A., Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford University Press (2002). Pinker, S., How the Mind Works, W.W. Norton & Co. (1999). Rand, A, Atlas Shrugged, Centennial ed., Dutton Adult (2005). Randall, L., Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions, Harper Perennial (2006). Ronan, M., Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, Oxford University Press (2007). Rosen, K.H., Elementary Number Theory, 5th Edition, Addison Wesley (2004) [TEXTBOOK – UNDERGRADUATE]. Ross, S., A First Course in Probability, 8th Edition, Prentice Hall (2009) [TEXTBOOK - UNDERGRADUATE]. Rucker, R., Infinity and the Mind: The Science and Philosophy of the Infinite, Princeton University Press (2004). Rudin, W., Principles of Mathematical Analysis, Third Edition, McGraw-Hill (1976) [TEXTBOOK – UNDERGRADUATE]. Russell, S., Norvig, P., Artificial Intelligence: A Modern Approach, 2nd Edition, Prentice Hall (2002). Sakurai, J.J., Modern Quantum Mechanics, Addison Wesley (1993) [TEXTBOOK – BEGINNING GRADUATE]. Schey, H.M., Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, Fourth Edition, W.W. Norton & Co. (2005). Schneier, B., Applied Cryptography: Protocols, Algorithms, and Source Code in C, Second Edition, Wiley (1996). Schneier, B., Secrets and Lies: Digital Security in a Networked World, Wiley (2004). Schultz, C., Happiness is a Warm Puppy, Cider Mill Press (2006). Schumm, B.A., Deep Down Things: The Breathtaking Beauty of Particle Physics, The Johns Hopkins University Press (2004). Sendak, M., Where the Wild Things Are, 25th Anniversary edition, Harper Collins (1988).

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