The Gap

The Gap

A walk and sculpture for William Kingdon Clifford Phil Smith, Matthew Watkins and Tony Weaver (The Gap was performed by Matthew Watkins and Phil Smith at the British Association’s Festival of Science in Exeter in 2004. There is another account of the performance here. This stitched together text is the raw material that became smoothed and narrativelike only once it had begun to slide along the planes of car parks and slither down helical car park ramps. The sculpture created by Tony Weaver had been intended for the street outside the house of Clifford’s childhood in Longbrook Street, Exeter, but after fruitless attempts to gain planning permission, despite site visits, two applications and more than 100 emails, the sculpture was installed at the reception to the School of Physics at the University of Exeter in January 2009.)

01 – No. 82 Longbrook Street (We arrive, ignore the amassed audience, and simulate our discussion on the Z World dérive – all this on the pavement outside 82 Longbrook Street)

Matthew: (basic facts of Clifford’s life - born Starcross mid. 19th Century, lived here until he was 15, got chucked out of Cambridge for being an atheist, pre-empting of Einstein, eccentricity, Clifford algebras ).

Phil: (breaks out of this, direct to audience of walkers) That’s pretty much what happened here (?) months ago at the end of a day long exploratory walk – the third of three Z Worlds Walks, in fact. “Worlds” because they were walks to find little enclosed self-sufficient worlds, microcosms from which one might learn something about other places, maybe about every place. “Z” because on the first walk we caught the early morning Z bus, the first bus of the day, from the High Street as our catapult into exploring. My name is Phil Smith… Matthew: … and I’m Matthew Watkins… (two people who live in the same street and discovered, quite by accident, a common interest in a particular type of exploring cities: called psychogeography or mythogeography) Phil: Come up a little closer to the house. (The group moves up to the top of the small car park.) Phil: That day in 2003 Matthew and me had found plenty of Z worlds: a cool smooth crematorium lawn, derelict MOD land where rusting Z worlds were piled high in rusty warehouses, a patch of lichen on a wall of volcanic rock, a wilderness of rushes near Topsham, a tunnel of briars… this stop outside this initially rather unimpressive house – it was dark and I hadn’t really noticed the gothic tower yet… it didn’t really mean that much to me compared to all the wonderful worlds of shapes and textures we’d discovered earlier, but then, the next day, Matthew emailed me...

Matthew: the corkscrew dream, PhD, Chisholms, etc. (Photos of Clifford?) “A few nights ago, I dreamed I was walking through a city with my friend Amanda…We encountered a busload of kids, and one kid ran up to a lamppost, sort of jumped sideways in the air, grabbed the post, and spiralled downwards, his body remaining horizontal. When I woke up I remembered having read years ago in a little biographical piece on Clifford that he was an athletic child who invented a thing he called a ‘corkscrew’ (the thing the kid in the dream did). So Clifford as a child appeared in my dream, not the bearded professor - he lived in Longbrook Street as a child, and it was presumably in Exeter that he practised his ‘corkscrews’)." Phil: I was interested now. The dream seemed to place Clifford here and it changed the nature of this space – who was this child, rotating around lampposts, did he used to stare out of that gothic tower? What did he see and did it affect what he would do? Was he already thinking about the world? I knew almost nothing about him, and yet already he seemed a self-sufficient world in himself, this man who speculated that the universe was made of curved space, who’d made curved leaps around lampposts.

I went to the Central Library and I found a copy of his book The Common Sense Of The Exact Sciences, it was in the pre-50 Stack… (Take copy of the book from my rucksack.) I didn’t understand much to start with, I still don’t… but, I kept going because the book starts with shapes and spaces that I could recognise and the author writes as if he longs for everyone, even me, to understand.

On this walk, I really ought to be playing the role of the scatty, scantily clad assistant and Matthew should be Doctor Who, me asking the questions so the good Doctor can supply the answers that the plot requires, but … while Matthew is sometimes called “the Doctor” by his friends, I also used to be called “the Doc” but only by one person (Misha Glenny)… so I’m going to have a problem staying in character.

What might keep me in there is that I know that Matthew has a way into Clifford’s algebraic world, and I don’t… I believe it’s there in some way, I can very, very hazily describe its most basic qualities, but I can’t really ‘see’ in its language… and yet I have this sense that if I could learn more of that world, it will tell me things about other worlds, maybe all other worlds… So this walk today… is about gaps – between what I know and what I might know, between this world (touches the ground) and Clifford’s algebra, between Matthew and me, and gaps between one thing and another that maybe open up everything. Matthew: To help us begin our journey, we’re going to go somewhere where we can begin to see Clifford’s mathematics and geometry take shape - a two dimensional space he might have been rather surprised by. (Matthew leads us towards the road. At the roadside.) Phil: We may be talking about matter in 4 and sometimes more dimensions, but that won’t affect just how much it will hurt if you get run over, so be very careful crossing the roads, stick together and let’s try and cross as a group. [82 - carpark] Matthew: points out π (pi) among the graffiti) 02 - just inside Howell Road car park between Longbrook Street and Exeter Central Station

Matthew: I begin by forewarning you that we're not going to launch straight into Clifford's original ideas - too deep. We'll have to build up to it by working our way through some of the ideas that preceded Clifford. Some of it is fairly basic stuff, some of it he would have probably been thinking during his teenage years here. We're going to get into his own stuff later, but now I shall attempt to guide you through some ideas inspired by the style of presentation which appears in Clifford's popular science classic Common Sense Of the Exact Sciences – using this car park!

Phil: Clifford might have been surprised to find the car park here. But the prison would have been familiar to him. Maybe he even saw executions there. (Showing the audience the slip in the book.) The last person before me to borrow The Common Sense from the library was a prisoner.

(Phil hands the book to Matthew) Matthew: the railway line/maybe saw the first steam train to arrive (Matthew begins to chalk a straight line perpendicular to a wall of the car park and another, similarly scaled, line parallel to the first.) Phil: And then, of course, up there, the courts would have been familiar to him as the place where his father, a JP, would sit, interpreting the laws of the judicial system, within the walls of the old castle. Was he thinking of that space… when he wrote his nonsense piece, ‘The Giant’s Shoes’ for a collection of Fairy Tales called The Little People? (Maybe read this from the book of Clifford’s Essays from the Institution.) “Once upon a time there was a large giant who lived in a small castle… From his earliest youth up, his legs had been of a surreptitiously small size, unsuited to the rest of his body; so he sat upon the south-west wall of the castle with his legs inside… (Phil sits on the car park to demonstrate, is there a square to do it in?) … and his right foot came out of the east gate, and his left foot out of the north gate, while his gloomy but spacious

coat-tails covered up the south and the west gates; and in this way the castle was defended against all comers…” Unfortunately, the giant came to this arrangement prior to putting his shoes on – something, subsequently, he was unable to do for lack of space. “In one of (the shoes) his wife lived when she was at home; on other occasions she lived in the other shoe. She was a sensible, practical kind of woman, with two wooden legs and a clothes-horse, but in other respects not rich. The wooden legs were kept pointed at the ends, in order that if the giant were dissatisfied with his breakfast he might pick up any stray people that were within reach, using his wife as a fork.” Even in a fairy tale his giant is a set of orthogonal axes, his legs and back at right angles. But it’s not an arid game of geometrical references – the story of The Giant’s Shoes is alive with the promise of shapes that transform themselves, it being in their very nature to change – a wife into a fork, a church steeple into a dagger, a meal of hay into a good thatch of hair. Matthew: (material to cover: • positive whole numbers as steps • numbers generally as steps (or vectors) • negative numbers as backward steps • addition and subtraction in terms of steps • commutativity of addition

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the number line orientation of number line multiplication as geometric operation on number line commutativity of multiplication on line number line as “1-dim space with algebraic structure” the implications of stepping off the line)

Mention that Clifford was a 'space man', used space to explain number, typically. His chapter on number pretty much starts with steps (Matthew passes the Common Sense book to Phil) The sequence of ideas to be explained is then: number as quantity, as quantity of anything, as quantity of steps, this allowing us to associate numbers with locations along a line. Number as an 'instruction' - we'll call that a vector. Adding demonstrated with steps. Commutativity of addition. What about subtraction? Hitting brick wall, move to other line, discover negative integers. Negative numbers as instructions, as vectors. Multiplying as an instruction, a magnification rather than a shift. "Multiplying by 3 as a stretching of the line where 0 stays put, 1 -> 3, -1 -> -3, etc.". We now have a 'number line', or a '1dimensional space' with a simple algebraic structure, which very simply means we have rules for adding and multiplying. So the usual system of numbers with + and x (if we include negative numbers) is thought of by mathematicians as "a 1-dimensional space with an 'algebraic structure'." Clifford would have described it, as

we just did, using geometric analogies. His main contribution to mathematics, Clifford Algebras, are sometimes called "geometric algebras", and reflect something of his deep understanding of the nature of space, what it is and how it works. Something about orientation. JSRC - Hamilton had already talked about vectors before Clifford. Phil: I’m still with this. Perhaps because I can relate this to my walking. Taking one step after another. And yet – apart from those times when one might deliberately attempt to walk a straight line in order to find what obstacles might lie across its path - most of the time I’m stepping off the line, always trying to sidle and crab away from the straight and narrow… (Phil steps off the line.) … so, geometrically, where am I now? Matthew: OK - so we're off the line. It's not really a LINE in the strict geometric sense (being physically impossible), but it's suggestive of a line. Similarly, the car park is suggestive of a plane, though far from a pure geometric plane. But we'll step off the line into a plane. Stepping off the line is meaningless if we’re still thinking of numbers as passive quantities, but if we think of them as active ‘instructions’ (Clifford’s directed steps) then there’s a natural extension. 03 - deeper inside the Howell Road car park

(material to cover: • points in plane as vectors, so, rather like negative numbers – a new kind of number • the existence of complex numbers before WKC • adding vectors in a plane • commutativity of vector addition • multiplying vectors in a plane (why and how) • commutativity of complex multiplication • plane as 2-d number plane or 2-d space with algebraic structure • 2-d-ness of a plane, axes • orientation in 2-d • 3-d spaces and extension of vector addition into 3-d, axes and hint of orientation) (We walk further into the car park, near to the far lamppost, in view of Debenhams. Matthew gets set up somehow - Phil tells helicopter/kite/ sheep anecdote. Phil: When we came into the car park one evening to work some of this out, the police helicopter got interested in us and hovered over us for a few minutes – Clifford would have been very interested by that – he made a number of preliminary experiments with the intention of designing a flying machine – using very large kites and very long pieces of string – which had to be laboriously laid out in a straight line before take off… (Phil draws a long line with a piece of chalk)

…the experiments came to an end, when after laying out the string Clifford and his helpers went off to lunch and a flock of sheep, including its shepherd, got entangled in the string… (Phil draws a huge tangle knot in the string.) … and that seems to have been the end of that. Matthew: Now, by the 18th century, mathematicians were already starting to deal with a 2-dimensional number system or a "plane" of numbers. Stepping off the line into a plane, what is that number? Bit like negative numbers. You can extend them in a meaningful way. We're going to see how a 2-d number system works. Sequence: Extending the idea of vectors to correspond to points in plane. Vector addition. Acting out vector addition in 2-d. The commutativity of vector addition. Now if we're going to try to deal with geometrical issues, like "area", we need to be able to multiply. So what about multiplying. If we're going to talk about a 2-d number system with an algebraic structure, we need to be able to multiply any two points (vectors in plane). Use lengths of knotted string and 3 volunteers. Predetermined lengths, they can choose angle, use compass to determine where they are, mentally add angles, etc. Get a third person. There's an element of magnification and an element of rotation. Note that it's commutative. Multiplying in this way might seem arbitrary, but there's a very deep and beautiful branch of maths called complex

analysis (used by physicists to describe the deep structure of the world). Notice how the complex numbers include the usual numbers (multiplication stays the same on horizontal axis). Point out the number i, the most famous complex number, which multiplies our volunteer in a particular way. i x i = -1 This system of complex numbers already well-established by the time WC started learning maths. He explained it in CSOE. Keep reminding them that this is a modern version of Clifford explaining the ideas that led up to his ideas. "Don't panic" -> just as we can interpret points in 1-d as numbers, adding and multiplying them, we can interpret points in 2-d as "numbers", adding and multiplying them. Explain why the car park is "2-dimensional". Two numbers needed - either (horizontal, vertical) or (distance, angle). You could get into left, right orientation (whether we measure angles clockwise or anticlockwise. Car park as 2-d also means only 2 lines at right angles...Put umbrella upright to illustrate where the 3rd would have to go. Phil: Like where the ceiling and walls meet in the corner of a room. Or the giant’s legs and backbone…

(Phil sits on the car park, legs apart, to illustrate)

Matthew: Or…. Pointing out Debenhams. What 3-d space is. 3 lines at right angles. Points described with 3 numbers. The fact you can extend vector addition to 3d (imagine flying instructions, or lifting our arrows off the carpark and waving them about in the air. Phil: I look up at that building, and I hear Matthew talking – and part of me feels comfortable, because I know that the basement of Debenhams was created to serve as a hospital in the event of nuclear war and so it has a meaning for me, but also I feel like I’m on the edge up

there… on the edge of the gap where standing doesn’t quite meet falling – inside a geometrical version of my vertigo: my attraction to I don’t know what, maybe to what I fear, a compelling suspicion that there’s something concealed, incomprehensible and sinister… and yet reading the The Common Sense, listening to Matthew here in the car park I’m beginning to see places and spaces through a ghostly grid, to vectors… get a sense of buildings in motion, geometrically alive… but I know that coming up is stuff that’s going to scare and bewilder me … a gap is going to open up between what this universe is, and what my mind can do... and I’m going to fall, geometrically, off Debenhams. Come on!

03 (a) On the way to Bishop’s Move – junction of Howell Road and Longbrook Terrace? Matthew: It’s appropriate that we’re here on Howell Road…. Howell being King Howell, King of the Celts… because we’ve only one more mathematical step to go before we’re ready to consider Clifford’s work and that step was made by a Celt… the Irish mathematician… Hamilton… Mention the basics about Hamilton's quaternions (1843 – 2 years before birth of William Kingdon Clifford in Starcross) - how we're just now a short walk from Clifford's ideas.

Hamilton and his quaternions – Probably most significant Irish contribution to mathematics, and first example of an algebra that wasn’t commutative. I'm sure Robert Anton Wilson wrote something about how only an Irishman could have come up with this. This is actually praising the Irish for their ability to think laterally. Perhaps it's a certain Celtic mindset. You could mention that the Celts primary contribution to maths was geometric involved those wonderful bits of knotwork - and we'll come back to that. We now walk along narrow pavement and cross a road. 04 - Bishops Move

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material to cover: • ecological perspective 2-d spaces in 3-d, rotation, etc First hint of ‘bivectors’ Clifford algebras in computer algorithms (make clear we haven’t described C. algebras yet) Reminder about vector addition Orientation in 2-d space Anti-commutativity

Matthew: Side wall of building has a perspective chessboard. This is an example of projective geometry. It's a 2 dimensional grid hanging in 3-d space viewed from an oblique angle, rendered as a 2-dimensional image made of quadrilaterals. This shows how 3-dimensions can be represented by a trick in 2-d. Projective geometry involved. The mind seems to instinctively grasp this - it's like it's wired in. It is able to "fill in the gaps" in your perception. These days a graphics designer making a logo like this might use a sophisticated bit of software which would use mathematics to calculate how a grid would look in perspective at a selected angle. This is exactly the sort of thing Clifford algebras would deal with very beautifully and with maximal efficiency. Although most mathematicians are barely familiar with Clifford algebras, and in the 1970's only a tiny handful, now people involved in robotics, optical imaging, space vehicle design are using Clifford Algebras. This has all happened since JSRC'86. Do I want to do my "game" quote here or further along just before we do the game? This isn't a quote from Clifford, but it's from Newman's introduction to CSOE, so we could quote it without necessarily deceiving anyone and claiming it to be WKC outright. "Geometry considered as a pure science of ideal space is an exercise in logic comparable to a game played with formal rules. As in any game, there are pieces or counters (elements: lines, points, etc.), their

properties fixed by definition (postulates), their operations prescribed by rules (logical inference). There is no point in asking what the game means; it is essential only that it be consistent and played according to the rules. If the game points some moral or there is discernible in its patterns some similarity to the patterns of physics, politic, or psychoanalysis, the coincidence is interesting, but not necessarily important. The pure logic of games has little to do with the erratic wanderings of nature." He believed geometry in the physical world was an experimental science, rather than the pure science of the abstract geometry (a consistent game) csoe xxi (Newman). "Geometry considered as applied mathematics is quite another case. The postulates and elements based upon experience purport to describe the space around us and the extensional properties of matter. Pure geometry can no more be wrong than the game of dominoes; like dominoes, also, it cannot be right. Geometry as applied mathematics on the other hand, can be right or wrong in describing measurable relations." I get a real chessboard out with some stickers on it, demonstrate orientation and anticommutativity at the same time. Get the mirror out, etc. The choice here: do we mention bivectors? I can't see any good reason to. They've just seen vectors in a plane multiplied to produce a vector, and Hamilton's quaternions

are sort of the same. This could all be very confusing...May have to hide behind a smokescreen of artistic vagueness. You're going to go on to tell them it's a bivector or trivector?? Some people notice these things! Lines in space had long been described in terms of the vectors pointing along them, but Clifford's innovation was to come up with a system where you could multiply two vectors to get a "bivector", which describes a plane with an orientation - Perhaps produce a chequered board you can use to describe how clockwise on one side is anticlockwise on the other, and that there are two ways you can orient a plane. e1e2 = -e2e1 Also a BISHOPS MOVE on the board could be thought of as a sum of horizontal and vertical vectors. e1 + e2 or e1 e2 The idea of "oriented space". A geometric figure can have a "negative area" in Clifford's description of space. This turns out to be a very useful and elucidating concept once you get the hang of it. [This is probably a bit wacky, but we could just stretch the psychogeography, point out that we are on the road to Crediton, and the Bishop of Crediton (where the Cathedral was) MOVED to Exeter along this axis. Perhaps prefigure any mention of religion by pointing out how powerful the bishop of Exeter would have still been in the 1850's (I assume he was), and how there would have been a major

religious component to WC's life in Exeter. Perhaps find out who was the Bishop was in the 1850's and an anecdote? He seems to be the one who 'moved' when the cholera hit – Bishop Philpotts.] – Matthew, if we’ve time it’s good if you can finish with this anecdote – as you, the mathematician conjure up a bit of psychogeography, just before I, the artist (!), encourage everyone to see place in terms of shapes and framing …(Phil) Produce a chessboard, and remind people that the Bishop moves diagonally, the sum of horizontal and vertical vectors which the rook ('Castle') moves along. Sums of vectors versus products of vectors Phil: When I first heard Matthew mention this thing about our minds filling in the gaps geometrically, I remembered how since Matthew had first mentioned the name William Clifford to me, outside no 82 down there, I’d started to come across his name in different places: and one of those places was in the bibliography of The Perception Of The Visual World by James J. Gibson – his Common Sense of the Exact Sciences is in there – because just as Clifford builds up his geometrical science from simple steps, lines and shapes, so Gibson builds up his theories of perception from the ecology of the world around us… from the array of light that creates complex matrices as it reflects from the surfaces around us.. and the very nature of those surfaces… from the grander vistas.. like these walls going up Poltimore Square towards the car park… to the texture and pigment of the bricks… and the process of perception itself is not unconnected with these vistas and textures and pigments – perception can fill

in the gaps because that perception has been formed in response to exactly these combinations of light arrays and vistas and textures and pigments … like Clifford, Gibson sees perception as based on difference… Clifford writes: “consciousness begins with difference” (??) … Gibson that there is no memory bank against which we compare what we see now with previous experiences, but rather what we experience are the transitions from one moment’s perception to the next and those differences are what tell us what we see… meaning is in difference itself… also, like Clifford, Gibson is a great one for refusing to take anything for granted… he’s a fellow peeler-away of assumptions – As I’m talking to you I’m looking up Longbrook Street to where the old Theatre Royal once stood, and I’m thinking of Clifford addressing an audience in a theatre… and as he does he describes how he sees the inside of the theatre and how he sees the members of the audience… him there, them there – there’s a sort of rotational quality to theatre, isn’t there? – a plus and a minus, that’s easily reversed – Clifford talked about how when he looked at the faces of the audience raised towards him, he had a sense of the backs of their heads – even though he could see nothing of them… but not as if he could see around the back of their heads… but rather as if he could see the backs of their heads from the inside… (doing corkscrew motion with hand) …our way of looking may not be as straightforward as we think… at least not William Clifford’s… imagining the

rotational quality of a head, but from the inside… he corkscrews inside their skulls! (again, doing the corkscrew motion with hand) When you arrived at 82 Longbrook Street this afternoon/evening you probably saw the place like I did – as part of a boundless space, its walls running down into foundations, reaching up towards the sky, when you turned your back on it you felt it was still there, when you looked at it… taking account of you wearing the right glasses if you need them … it all looked in focus, part of the same world as the trees and the road and the houses opposite… yes? But if I’d asked you, down there, to fixate on one detail, you might have seen it differently… look at this spot on the Bishop’s Move sign… now, keeping you focus on that spot, move your head slightly… Gibson based all his work on seeing the world as we’re in motion… which is what we are almost all the time… now if you move your head you should see some differences (stay focussed on the spot!)… you should see that your sight is not boundless, but it is framed by the side of your nose, your eyebrows and maybe your fringe of hair, you may notice that though the centre of your vision is in focus towards the edges it blurs - and if you move your head again - you’ll see that the part of the vista leaving your sight tends to bend as it leaves and the part coming into sight is bent and straightens up as it enters the centre of your vision… This is what J. J. Gibson calls seeing the “visual field” as opposed to how you perceived 82 Longbrook Street when you arrived, which he would call the “visual world” –

The more I learn about Clifford, indeed about algebra and geometry in general – despite all my trouble in trying to grasp them - the more natural they seem and the more geometrical and algebraic nature seems… and I begin to see – more and more of the time - things as a series of “fields”…

05 – Stairwell window Matthew: point out Debenhams again and hand out cubes and pencils and instructions (Phil sneaks up inside the stairwell putting on his Matthew mask. He then draws a right hand pair of axes labelled "i,j" imitating me doing it on the back of my chessboard. Also a 300o circular arc with arrow. Once people have done their cubes I draw attention to this fact. That clockwise is just anticlockwise from the other side. Same as a mirror image.) Key idea to paste in somewhere else. You can stand on either side of a 1-d space (line) and reverse the orientation. You can stand on either side of a 2-d space (window) and reverse the orientation, but a 3-d space, there’s no obvious sense of there being ”another side” – yet one can meaningfully discuss reversal of orientation. Keep orientation and dimensionality as your key themes. 06 – under the footbridge.

Matthew gets everyone to return their pencils, point their "i"'s upward and see how the group splits. [note that now Queens Crescent is 'out', there is no need to split the group - unless we can find somewhere to do it on way back to 82 ] Talk about the duality again - this is something trans-dimensional: left-right in 1-d, clockwise/anti-clockwise in 2-d, left and right handed helicities in 3-d. Use your stickers (or a pair of gloves?) on Phil's fingers to illustrate left and right-handed orientations in 2 and 3-d Remind people about Hamilton as you head to spot under bridge, how he came up with his quaternions. chalk it up under the bridge: Explain that this is their i,j,k. That the i we saw in the car park i x i = -1. Same kind of thing, but Hamilton came up with a system involving three of these things. carved under the Brougham bridge in Dublin which he was walking along Royal Canal to meeting of the Irish Academy! . i x i = j x j = k x k = i x j x k = -1 i x j = k, j x k = i, k x i = j j x i = -k, k x j = -i, i x k = -j so i x j = -j x i, etc. (draw up the cycle i -> j -> k -> )

In 1 dimension we have one "unit" - the number 1 In 2 dimensions we have two units. These are often considered 1 and i. Hamilton's quaternions are effectively 3 units for 3-d space. You may ask why quaternions, and it's cos he was thinking 1,i,j,k. But the 1 plays a different kind of role as WC elucidated. So in 3 dimensions we have these 3 units i,j,k. They 'span' the space, and we can multiply them like a game. (we could actually pull out some i,j,k stones, multiply them according to the rule). Then explain that Clifford generalised this - chalk up some equations with e1,e2, etc....explain this wasn't his notation, but the one commonly used now. Explain how these would be the edges of Debenhams, the room and walls would correspond to bivectors, but these could be negative or positive depending on "which way round you were considering them".

Phil and Matthew get their bags out and play the C0,3 (quaternionic) Clifford algebra game with a +/flat stone we can flip. Either of us can multiply in stones. Get across the idea that you can't get higher than e1e2e3. Explain that even tho' it's a silly game, the essence is used in robotic and space vehicle design. Phil should point out i,j,k is the same as e1, e2, e3. Relate the vectors to the cube edges (3 possible, 2 directions), the bivectors to the cube faces (there are 3 possible, with two possible directions).The trivector - only one possible. Represents the cube. Need to get across that there are other vectors than e1, e2, e3 - this is just a representative set in terms of which all the others can be represented. They're called a basis? By stretching and adding these 3 vectors we can produce any vector. Just like 1 and i in the plane. You can add and multiply any vectors, not just the basis vectors. And we get all sorts of other bivectors and trivectors than the basic ones that will show up in the game. But every bivector can be expressed in terms of these basis bivectors, every trivector...

Clifford algebras somehow manage to deal with 1, 2, 3, and higher dimensional objects all at the same time. In the sort of adding-apples-to-pears kind of way. There's also a special kind of inbuilt duality which distinguishes Clifford algebras from other algebras. You need to get that across with your mirrors, window tricks, etc. Once people get the idea of the rules, they will get the idea that we can produce a new e4 and just accord it the same rules. Your e1,e2,e3 were like quaternions, with negative signature, so they 'flip annihilate'. The e4 will just annihilate. It's the time dimension. But it could have been the other way. We have to specify our algebra, there are options. Perhaps chalk up 1,2,3 with -,-,- under it, and then when you add the 4 you put a + under it. Phil: I’m a bit shocked about this, you know. I’d imagined that if I ever found myself working in 4 dimensions… then the screen would go all wavy, or I’d fragment into little pixels and shoot off into space or be transported back into medieval Exeter… but this passing into 4 dimensions turns out to be connected to how the real world works… rather than just the one we see and experience… because we can’t see get around the back of 3 dimensions… with a one dimensional line, we step off the line… like we did in the car park… and with the two dimensional plane of the chess board or the window we could be on both sides of it… but how do we get outside the three dimensions of Debenhams when we’re in the same three dimensions as it is? And this is the answer – it’s not a

geographical answer, it’s a mathematical one… and it’s not a trick, or a marginal bit of absurdity… it’s a key to standing outside of massive three dimensional events… as J. J. Gibson helps us to see the way that two dimensional lines and planes in three dimensions make up our perception of the world, so Clifford makes the next leaps – (hold two of the stones up to our eyes) – and gives us mathematical eyes to see how the threedimensional world operates from the viewpoint of four dimensions. Matthew: explain that we've just extended the algebra from 2 to 3 to 4 dimensions. But the fact that the 2-d algebra seems to perfectly match our sense of 2-d space, and a 3-d algebra our sense of 3-d space, this 4-d algebra makes you think "Maybe there's some kind of 4-dim space in some sense which this algebra similarly describes." It exists in the minds, the collective imagination of mathematicians, at the very least. JRSR says (p.160) "It is always hard to explain why some discoveries are not immediately appreciated by the scientific word. One reason for the neglect of CA was that the idea of 'higher-dimensional spaces' was not

fashionable in the nineteenth century. Indeed, some contemporaries of Clifford argued strongly that we live in a 3-dim world, and that it was meaningless to talk of spaces of four, five, and higher dimensions." But once we go above 3, it's hard to visualise (there are some limited tricks in 3-d whereby you can model 4-d goings on) - refer back to Bishops Move image Refer back to the BM chessboard - just as you can have a 2-dim plane at a certain 'slant in 3-space', this would be a particular bivector, once we could agree on an orientation. Similarly once you're in 4-d, you can have a 3-dim space embedded therein at a particular 'slant' - it would be represented with a trivector - almost impossible to imagine this. IMPORTANT TO STRESS: All scientists now familiar with vectors, but only the small number of Clifford-minded ones know about bivectors and trivectors. Weirdness of 4-dimensions... Can get into Einstein's special relativity - 4 dimensional spacetime continuum. Clifford C1,3 is ideal. Important to mention that the same Clifford Algebra turns out to describe both space-time and the fundamental particles of nature. Explain how this idea was largely ignored right up through the 60's and the publication of Hestenes’ book in '66, but now becoming fashionable.

line segments, squares, cubes, hypercubes - the sorts of things you could deal with very easily in Clifford algebras describing 4-dimensional geometric objects and calculating hypervolumes, etc. Consider circle (1-d in 2-d), sphere (2-d in 3-d) Back to the corkscrew. Phil describes leap from 2 dim to 3 dim motion. – (Matthew, I’m not sure where this comes from?) (Phil: motion is important here – Gibson studied the world through senses in motion… would it be right to say that Clifford’s cosmos is not one that has a state of rest? ) Matthew: I mention Clifford's leap from 3 to 4 dimensions was something of a 'mental corkscrew', and 'helicity' is a recurrent theme in his work...perhaps show them Penrose's diagram as an example of 'Clifford parallels' on a 'hypersphere' in 4-dimensions... 07 – The helix. Go inside. Point out we have a helix. Point out that the helix immediately suggests an orientation. Pull out mirror to evoke the opposite orientation. Clifford was particularly keen on helices. If there's one iconic image it would be this. (Maybe Phil could be up on the road part of the helix bringing something down? The following is explained as

Phil is instructed by Matthew to walk up and down the helical ramp to illustrate the ideas.) His motion around the lamppost would be most accurately described in terms of Clifford algebras as particularly relevant for spins, twists, rotations Clifford generalised vectors, came up with 'rotors' and writes in CSOE about 'twist', using corkscrew to illustrate a point about an axis rotating, combined with a linear motion - used in describing rotating bodies generally. Dirac's equation for relativistic electron - 1928 - now would be described involving 'spinors', which have come out of Clifford's thinking. It's all about the spin of the electron. From the 30's to the 80's this was described in terms of 'Pauli spin matrices'...possibly do soup plate trick to illustrate how a spin1/2 particle can work. Powers of -1. Parity. Fermions and bosons. Supersymmetry. JSRC: In 1928, Paul Dirac published his revolutionary theory of the electron, a theory which rapidly became a cornerstone of all modern theoretical physics. To describe the electron mathematically, Dirac invented a mathematical structure known as 'Dirac algebra'. He did not realise that this algebra was very closely related to Clifford's algebra describing four dimensional space, discovered half a century earlier. It was another 25 years before physicists began to appreciate that hte algebra Dirac had invented was just a particular one of a whole range of Clifford algebras.

The coincidence of the close relationship between Clifford's and Dirac's algebras is important. Remember that each CA corresponds to its own 'space'. Dirac's algebra is precisely the CA associated with the space-time of Einstein's Special...Relativity...So out of a myriad of possible mathematical structures, the same Clifford algebra turns out to describe both space-time and the fundamental particles of nature." Penrose's more recent work on 'twistors' takes the whole thing further. WKC would certainly have recognised this fundamental role of spin, twist, spatial orientation in these descriptions of the fundamental structure of the Universe. Show them a blow-up of Penrose's twistor picture and tell them a bit about where it came from. Clifford's approach as unifying. He sketched out some ideas in his later lectures about how linear motions and rotations (seemingly very different things) could be unified into a single framework involving the combination of helical motions. Coils of wire. Electrical (vector) and magnetic (bivector) fields. Spatial orientation directly relevant to the +/- polarity in electrical theory. Dynamos and generators. Reverse the +/- wires to an electric motor and it spins the opposite way. Hestenes (should have just mentioned him) showing how electromagnetism could be unified in a CA framework. Gyroscope and mirror. Gyroscope, or any spinning wheel has angular momentum. This is a combination of how fast it's spinning (really how much 'momentum' it has) and it also involves a sense of the plane in which it is spinning.

Traditionally this would have been described with a vector. The length relating to the momentum, and the direction being at a right angle to the plane of spin. Hestenes has shown how a Clifford bivector is much more ideal (already describes a plane, has a 'magnitude' AND and orientation show gyroscope in mirror). Pull out Hestenes printout and show the Omega. Clifford all about unifying (quote Hestenes) - unifying language of the sciences, organic unity of the sciences (embracing the whole 'human world') Perhaps you could reassure people about the emergence of all this terminology vector, rotor, motor, spinor, twistor, tensor (there are probably others) rather like, electron, proton, neutron, boson, fermion, muon, pion, gluon, tachyon, etc... Something that does something rather than is something. Perhaps mention that Left Hand, Right Hand book (perhaps try to GET it, see if Clifford is mentioned. 08 - Curved lawn Phil: Before us is the re-built church of St Sidwells, William’s local parish church – a traditionally High Church, AngloCatholic place of worship and the scene shortly after his birth of the Surplice Riots in which thousands came out onto the streets to protest against the priest here wearing a ritualistic white vestment – a general passionate engagement with religion that began to fade away during William’s life – but even so, the stories of those events along with his early High Church faith

may partly account for his sense of the importance of the symbolical. Behind us here we’ve got the Hair Emporium with its ironically used nineteenth century name: if Clifford were here now the fashionable braiding of hair might remind him of his ideas about the nature of matter. For the common idea of matter when he was alive was that there was some kind of background medium – called aether - in which all matter ‘swam’ – this didn’t satisfy Clifford at all: "… whatever may turn out to be the ultimate nature of the ether and of molecules…” he wrote “…we know that to some extent at least they obey the same dynamic laws, and that they act upon one another in accordance with these laws. Until, therefore, it is absolutely disproved, it must remain the simplest and most probable assumption that they are finally made of the same stuff - that the material molecule is some kind of knot or coagulation of ether." Of course, the idea of an aether was abandoned after the Mitchelson experiments to measure it found that there wasn’t anything there to measure, but the idea of knots in matter continues today with the use by quantum physicists of braided algebras – whose ‘commutation relations’ are related to the geometric structure of certain kinds of braiding– incidentally resurrecting the work of various knot enthusiasts of the 19th century! (Draw quick diagram on the ground in chalk – Matthew to do this)

Phil: Maybe when he was thinking about molecules he remembered those sheep and shepherd tangled in his kite string. Or the hay that grew into a thatch of hair in The Giant’s Shoes. Wherever Clifford got the idea from, knot theory takes us back to that celtic artwork – an expression, perhaps of something more fundamental than decoration…now, such speculation is not entirely inappropriate here, on such an august event supported by the British Association… Clifford was never afraid to speculate beyond the boundaries of what he could know, but when he did, he speculated according to principles he could observe elsewhere… for him describing what we know is essentially a technical procedure, scientific knowledge is being able to predict from that technical knowledge how what we don’t know might turn out to be. And it is the prescience of his speculations that maybe suggests to us that there is something in the deep structures of his thought that is continuing to make his work relevant to researchers into the fundamentals of matter… “solving the universe”… Matthew: Do his remarks on the unification of space and matter, prefigure general relativity? Matter as curvature of empty space. We're getting more into his speculative/philosophical stuff now, but he questioned the usefulness of 'matter' and 'force', arguing that it should all be unified in a uniform geometric context. Very much precognising general relativity. (We walk the noncommutative parallelogram thing. Phil and Matthew should just 'act this out' without regard to accuracy.)

Matthew: Point to an example of curved 2-d space in 3dimensions. Here is the gap that reveals the curvature. Clifford was able to realise that our space might be curved in some sense...He was aware of the emerging 'nonEuclidean geometry' which described perfectly consistent spaces which were utterly counterintuitive (give examples). He was heavily involved in promoting these ideas, and perhaps the first to suggest that our space might be curved...that we have no evidence that it isn't. Just as the adult Clifford would challenge and reject the supernatural doctrines of Christianity, and all supernatural religions indeed… Phil: He wasn’t beyond mugging mediums during séances to prevent them using their trick props and strings… Matthew: … he also challenged the secular ‘faith’ in Euclidean geometry…( From Newman CSOE intro:) Now this enthronement was unfortunate in two respects. First, as a description of actual space Euclidean geometry could not pretend to universality: its postulates and theorems having been tested only in a most limited range might not be valid outside that range - in the domain of the very small or of the very large –. Second, by turning Euclidean postulates into commandments, the freedom of mathematical inquiry was more effectively throttled than by ecclesiastical ban. For there appeared to be no possibility of constructing new geometries based upon non-Euclidean postulates, such postulates being, clearly, "contrary to nature." Any discussion of space not known to the sense was thus forbidden.

To the second part - Clifford’s passion for upsetting the Church meant he had no inhibitions about upsetting Euclideans. To the first part – he speculated that not only might space be curved on an immense scale, but at a small scale it might be non-continuous… The gap… "In WC's private papers is an unpublished note discussing 'revolutions in science' and listing a wide variety of physical and chemical phenomena. The last sentence of this note reads: “All these things must come out of a knowledge of the form of atoms and their relation to the aether. What is pointed to is therefore a connection between kinetic and undulatory theory.”." This might be too much weirdness at once (It will only work if you have you mentioned the Planck scale?), also not that much to do with WKC himself. Noncommutativity in itself wasn't really his thing....but you could chalk Schrodinger's equation up on pavement and explain about the fact that in QM normal variables of mechanics get replaced by operations, and that PQ - QP would normally be 0, but in fact there's a little gap. It's magnitude is h/2π, where h is the Planck length and π is familiar, but it's also multiplied by i.

The key here is to deal with the idea that, OK, Clifford was able to describe space, in a new and imaginative way which allows us to formulate large and small scale models of the Universe, but what about his philosophical questioning of his assumptions about space? 09 - Opposite St. Sidwells: Matthew: The bell visible by the church. (possibly mention - from CSOES or L&E - the bit about "bells on elastic stalks"...Grand piano... JSRC d175 "He was fascinated by the evidence of a structure of molecule, which he describes as 'at least as complicated as a grand piano' - a very apt simile ( vibrations – string theory… 10 – Edge of multi-storey Phil: Now we can see the back of 82 Longbrook Street. And from here we can get an idea of William Clifford’s view of the world. His vistas. The church is behind him – at 15 he leaves to study at King’s College, London and then on to Cambridge, entering an academic world where the Church is rapidly losing authority - influenced by the debates around the work of Darwin and Spencer, he rejects his Christian faith, and from then on, for him, consciousness is as finished in death as the bodies in the morgue of the funeral parlour over there. and this attitude seems to lasted beyond the grave…: Matthew: (Matthew’s seance story)

He began to seek answers to the fundamental questions of existence – what he called “solving the universe” – not in ingenious combinations of science and theology inspired by his reading of Aquinas, but in the shapes and lines that he seems to have been able to see everywhere… (referring to the geometric shapes on the car park below). And yet this is a world in which science was unsure of itself when it came to culture or consciousness and shied away from the vacuum left by the church. It’s a mark of the nature of the individual we’re dealing with here that he felt no such inhibition. Let’s find a suitable spot to take this a little further. (Leading the group towards the inner, undercover part of the car park.) Phil: (As the group is on the move) By the way I hope you’re enjoying the way that – if we take the cars parked along Longbrook Street as a linear one dimensional car park, and the plane of Howell Road car park as a two dimensional car park, well, now we are entering a three dimensional one – it’s a structure to our walk that partly we found and partly we placed upon it – But how can we stand inside or outside the dimensions of that structure – what algebra would be suitable for getting behind not just individual imagination, but the social sharing of it? Clifford had a theory for consciousness… so, for a moment, let’s imagine this three dimensional car park is a head, and we can… like

Clifford imagined doing with his audience – walk inside their brains and see the backs from the inside.

11 - Inside multistorey

Matthew (?) : As you can see we are in the tradition, here, of desperate editors of New Scientist and producers of BBC science programmes at their wits’ end when faced with presenting a piece on something physically unrepresentable – put a piece of geometric abstract art behind it. (Points to abstract mural on wall of the car park.)

Phil: Rather like that conventional attitude to matter – as something living in a background medium – the general view of consciousness in Clifford’s lifetime was that it was something separate from physical matter, but living in it – either an immortal soul or a personality capable of genius – for the scientists the best they were coming up with was a ghost in the brain machine. Just as Clifford challenged the duality of matter and aether, so he challenged the duality of consciousness and matter – and, as usual, he began his challenge by thinking hard about what he could actually observe. He looked at something outside himself – say, this geometric art… and he knew that he was able to create in his brain a mental configuration of it – something that was different from the art

itself… and he called this configuration an ‘object’… this “object” was something that he could explore and examine. Now he also observed that other people also could look at this art and form their own mental configuration of it – their own object. But when he came to trying to form a mental configuration of another person’s mental configuration – then he felt that things were out of the loop – this wasn’t a n ‘object’ – this was an ‘eject’ – he invented his own name for it – this was something “thrown out of consciousness”. But this is Clifford – the universe is what it says it is – what it says on the packet – universal – “it seems that to follow that the belief in the existence of other men’s minds like our own, but not part of us, must be inseparably associated with every process whereby discrete impressions are built together into an object. I do not, of course, mean that it presents itself in consciousness as distinct, but I mean that as an object is formed in my mind (i.e. that complex of feeling about an external object), a fixed habit causes it to be formed as a social object, and insensibly embodies in it a reference to the minds of other men. And this sub-conscious reference to supposed ejects is what constitutes the impression of externality in the object, whereby it is described as not-me.” (p.75 vol 2, Essays) “A fixed habit”? Where does that suddenly come from? Clifford roots this “social habit” in physical matter. He calls it ‘mind-stuff’ and suggests it might be present in everything – indivisible units of matter that are not conscious in themselves,

but which possess the potential to become conscious, when forming in sufficient complexity:“…(W)hen molecules are so combined together as to form the film on the underside of a jellyfish, the elements of mind-stuff which go along with them are so combined as to form the faint beginning of Sentience…. When matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition.” Once again, universality was Clifford’s guiding principle – what held true in one part of the universe would hold true in every other part – not only in a jellyfish, but in a human brain …. “There is no … prima facie case against the dynamical uniformity of Nature; and I make no exception in favour of that slykick force which fills existing lunatic asylums and makes private houses into new ones” (p.78, vol 2, Essays) There is no magic to this consciousness, it is made up the same material as everything else and everything else has this material. I could say now: “It would be easy to join with his contemporaries and present day biographers in suggesting that this is one hypothesis that Clifford would soon have dropped.” But, actually, this is an edge… an edge of un-respectability where I feel comfortable… it’s a place where things happen, a fractal border, a third space… So I’m pretty suspicious of my own enjoyment of it. On the other hand, though… let me enjoy myself for two minutes here! – over 100 years before the development of

Consciousness Studies Clifford seems to have combined – or, indeed, refined – two of the main theories of consciousness emerging from contemporary Consciousness Studies; one: that consciousness is in everything, two: that consciousness is a product of complexity (in the literal sense) … … as well as the idea of consciousness being made of made up from irreducible units of consciousness – something like the memes of memetic theory… that, as yet, highly disreputable attempt to create a theory of the natural selection of consciousness… that I enjoy… In his introduction to an edition of ‘The Common Sense’, Bertrand Russell writes – “….over-emphasised .. quaternions” and, yet, now they are habitually used in Quantum Mechanics …. of course, that proves nothing about the validity of “mind-stuff” Matthew: As the theology dissolved, we were left, towards the end of the 20th century with a rather bleak, mechanistic view of the human being/brain. Rather like this car park. Functional, made of matter, with certain structures, pathways, etc. But with the rise of Quantum Mechanics (which he sort of foresaw) questions started to be asked about consciousness. This idea that there's another bit we can't capture with our mechanistic science. (1) As well as embracing Dirac's algebra (which is used in QM) it embraces Pauli's algebra of 'spin matrices'. Later in his life, Wolfgang Pauli collaborated with Carl Jung in an attempt to come up with a unifying theory for psyche and

matter. This is generally seen as a bit of non-starter, but more recent scholarship (looking at letters between Jung, Pauli, von Franz), suggest there's more to it. WC was even ahead of his time here - could be a link? (2) Penrose, as well as working on twistor theory, has been a major player in the emerging field of 'consciousness studies'. Emperor's New Mind was a watershed in this. This is a book WC would have eagerly read, as he was very concerned with the unification of mind and matter. Phil: I wonder if Clifford would enjoy the irony that the editor of the main Consciousness Studies Journal is a de-frocked Anglican priest living just outside Exeter.

12 – No. 82 Longbrook Street (On the way one of us goes into the New Horizon to pick up the Clifford Sign (the sculpture made by Tony Weaver). Phil: William Clifford would have enjoyed seeing the New Horizon café here – with its Egyptian proprietors serving Egyptian cuisine – he learned Arabic, and, as a republican and a resolute anti-Tory, he was a great admirer of what he saw as the non-absolutism in arab politics. (I’m minded to cut this next bit: At Algiers Clifford attended a lecture on Arabic at the Museum there and was particularly taken with a quotation from the Koran in support of a constitution that was “consultative and not absolute” and enjoyed a story recounting how a particular Caliph had boasted that he had never swerved from the path of

justice and was surprised when a soldier looked up and said: “Inshallah, and if you had our swords would have speedily brought you back!” (see p.57 volume 1 of Essays) ) It was a classically influenced arab culture he admired: as much scientific method as religion, a culture seen through the prism of his own desires … arising from the tradition of Pythagorus and the Stoics – in which a believer is exhorted to “Know, so far as is permitted thee, that Nature is in all things uniform” – this sense of order Clifford believed was the first step in the arts of life, and that “as far as is permitted thee” was a warning never to assume any ‘law’ of uniformity would be sure to hold forever and in every circumstance, but was, rather, a guide to human conduct… this devout science was the “light and right driven out of Europe by the Church… (that) found a home in the far East with the Omaiyad and Abbasside caliphs … (so) across the north of Africa came again the progressive culture of Greece and Rome, enriched with precious jewels of old world lore; it took firm ground in Spain, and the light and right were flashed back into Europe from the blades of Saracen swords.” (p.265-266 Cosmic Emotion, vol 2, Essays) (See the story of the Saracen’s Head in the Crab Man Document in Mythogeography.) I don’t remember that from school history lessons! (Matthew emerges with Tony’s sign and we all return to our starting point at 82 Longbrook Street)

Matthew: Clifford’s disappearance-and-subsequent renaissance of Clifford Algebras since the 80's - there are a lot of gaps… Phil: Our walk is full of gaps... missed opportunities that we have had to curtail for want of time… We wanted to walk along Sidwell Street – where the model shop would have linked us to William Froude in Paignton and his law of steamship comparison – with that theory’s links to the work of D’Arcy Thompson, applying ideas about the limits of scale and shape to evolution – suggesting a validity for shapes and patterns, which, for me anyway, should much more underpin the teaching of science than it does The video shop would have connected us to Oliver Heaviside in Paignton and his work on defining electro-magnetism as a single unified force – there’s a geography here spreading its connections across the county… Or we could have visited the Spiritualist Church in York Road and mugged one of their clairvoyants… our walk is full of gaps… And so is our biography … there was is another famous Clifford we haven’t even mentioned … his wife Lucy was a famous author in her own right – and shortly after William’s early death at the age of only 35 she wrote a sinister and disturbing children’s story called “The New Mother” in which two naughty children – who persist in naughtiness even when warned – find their mother replaced by a new mother with artificial electric eyes and a mechanical tail… having lost her husband so young

it’s as if she cannot bear her children to follow his naughty Faustian ways with its deadly risks… Matthew: Mention Clifford's early death, Lucy(?), the inconclusiveness of his life's work, only now rally starting to come into focus...Maybe pull out the robotics book, and evoke the image of horses and carts coming down Longbrook Street in the 1850's, young WC mentally laying the foundations for something then unimaginable. Show them Tony's sculpture, explain a bit about it... the corkscrew… the hyper-sphere…

Phil: the hyper-sphere is a ghost… of the corkscrew… just as that shadow of Clifford’s corkscrew on Tony’s sign is a ghost…

as if one could reach out…. (making a corkscrew motion with hand) … and take something from this place… Hand out cards of planning officers... explain that we're done but that there's an 'encore' if people want to join us: 13 - New Horizon ('encore') Coffee, snacks, informal discussion of Clifford's Arabist influences, Arabic origins of algebra and contributions to geometry, 'discovery' of New Scientist amongst the various newspapers and magazines...answering questions Need to arrange soundtrack with proprietor, forewarn him to time of influx, stash magazine, research Arab contributions to mathematics and Clifford’s interest in Arabism, Islam.