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N An Overview of this Session An Intuitive Visual Approach Stretching your Intuition An Easy Mathematical Approach Building a cube Mathematically Cubes within Cubes: Line sets Back to the Author
Most people do not know what an N-space cube is, much less how to draw one. These pages show you what they are and how to create them. The visual approach to drawing them can be understood by fourth graders, while the mathematical approach can be understood by freshmen in high school. I hope you have fun with N-space Cubes. They have provided me with much enjoyment and delight. Sincerely, Dennis Clark This material is copyrighted, but can be used as long as it is not altered and as long as the author retains credit.
N An Intuitive Visual Approach We say: “I See!” when we understand
A 0-space cube is a point. Here, we represent it with a black dot.
Two 0-space cubes with a line between the points makes a 1-space cube. (A line)
Two 1-space cubes with their corresponding points connected make a 2-space cube. (A square)
Two 2-space cubes with their corresponding corners connected form a 3-space cube. (a cube)
These two 3-space cubes appear to be overlapping. No problem.
Two 3-space cubes with their corresponding corners connected form a 4-space cube. (a tesseract)
So how do we create a 5-space cube?
Connect the corresponding corners of two 4-space cubes to form a 5-space cube.
A messy 5-space cube.
N Stretching your Intuition The brain needs to be Stretched and Exercised much like the muscles of your body
Even here we are violating the geometric definition of a cube. a
c
6-sides equal length lines 90 degree corners
b Angle b,a,c is not 90 degrees.
Whenever we represent a 3-dimensional object in 2 dimensions something gets distorted. Here it’s some of the angles.
For our purposes, the physical position of any point is irrelevant.
Only its connectivity is important. A point must remain connected to the same points regardless of its physical location.
e a g
c
f
b d
h
e h
a g
c
f b d
So this is also a cube.
This is still a cube. The physical location of any point is irrelevant. e h
c
g
a
f
b d
It is the connectivity that makes this a 3-space cube.
We pause for a neighborly definition: Adjacents Points b, c, and e are adjacent to point a.
e
a c
g b,c, and e are the adjacents of a.
b
f d
h
These are both 3-space cubes because their connectivity is correct. In fact, the right-hand cube is just a twisted version of the one on the left. a h
e
a c
b
g g c
b
f d
h
f
d e
Let’s untwist the Twisted Circle Cube Pretend the cube is a flat pancake.
a b
h
Cut the pancake in half along the dotted line. Flip the left half of the pancake g over while keeping each of the the points attached to their respective adjacents.
c
f
d e
Let’s untwist the Twisted Circle Cube
a
Now quarter the pancake.
b
e
Flip the G-H quarter over. Doing so will untwist EG and FH, but it will twist GC and HD.
f c
g
d h
Let’s untwist the Twisted Circle Cube
a
Now quarter the pancake.
b
e
Flip the G-H quarter over. Just when it looks like things have gotten worse...
f c
h
d g
Things finally get straightened out.
a
Our Twisted Circle Cube is now an
b
e
Untwisted Circle Cube. f d
h
c g
It is now an Untwisted Circle Cube a e
e
a c
b
g f d
b
f d
h
h
c g
N An Easy Mathematical Approach Math is a tool of Infinite Possibilities
a
d k r
i p
u
v
t
g
ad
w
q
h
ac ab
aa
o n
x
s
j b
y
l
c
f
m
e
Letters are a Cumbersome Naming Convention
z
ae
Every N-cube has 2N points (corners)
1
0 2
4
6
5 3
7
N space
Points in each cube
0 1 2 3 4 5 6 7
1 2 4 8 16 32 64 128
We label the corners using binary numbers 001
000
010
At first this may seem to be even more clumsy than using letters. 011
101 100 110
111
But it gives us one huge advantage we didn’t have before: Now, we can identify the adjacents mathematically.
The points of an N-space cube can be uniquely identified by N using 2 N-digit binary numbers. 001
000
010
00
01
10
11
011
101 100 110 N=3
111 N=2
3-digit numbers 2 N = (8) points
2-digit numbers N 2 = (4) points
Two points are adjacent if their binary digits differ by only one digit. Is adjacent to 00 00 01 01 10 10 11 11
01 10 00 11 00 11 01 10
00
01
10
11
A clarifying example: The adjacents of 1101001 are because each adjacent differs from 1101001 by one and only one digit.
1101000 1101011 1101101 1100001 1111001 1001001 0101001
1101001 1101001 1101001 1101001 1101001 1101001 1101001 1101000 1101011 1101101 1100001 1111001 1001001 0101001
The XOR binary operation 0
1
0
0
1
1
1
0
0 0 1 1
xor xor xor xor
0=0 1=1 0=1 1=0
1101001 1101001 1101001 1101001 1101001 1101001 1101001 0000001 0000010 0000100 0001000 0010000 0100000 1000000 1101000 1101011 1101101 1100001 1111001 1001001 0101001
N Build a Cube Mathematically XORbitantly Large N-Space Cubes
000 001 001
000
001
000 001 001
000 010 010
000
001
010
000 001 001
000 010 010
001
000 100 100
000
010
100
001 001 000
001 010 011
000
001
010
011
100
001 001 000
001 010 011
001 100 101
000
001
010
011
101
100
010 001 011
010 010 000
000
001
010
011
101
100
010 001 001
010 010 000
010 100 110
000
001
010
100
011
101
110
100 001 001
100 010 110
000
001
010
100
011
101
110
100 001 001
100 010 110
100 100 000
000
001
010
100
011
101
110
011 001 010
011 010 001
011 100 111
000
001
010
100
011
101
110
111
101 001 101
101 010 111
101 100 001
000
001
010
100
011
101
110
111
110 001 111
110 010 100
110 100 110
000
001
010
100
011
101
110
111
000
001
010
100
011
101
110
111
0000 0001
0011
0101
0111
0010
0100
1001
0110
1011
1101 1111
1000
1010
1110
1100
1 4
6 0011
4 1
0000 0001
0101
0111
0010
0100
1001
0110
1011
1101 1111
1000
1010
1110
1100
1 1 1 1 1 1 1 1
1 2 3 4 5 6 7
Pascal’s Triangle 1 3 1 6 4 1 10 10 5 1 15 20 15 6 1 21 35 35 21 7 1
00000
00001
00010
00100
01000
10000
00011
00101
01001
10001
00110
01010
10010
01100
10100
11000
00111
01011
10011
01101
10101
11001
01110
10110
11010
11100
01111
10111
11011
11111
11101
11110
N Lineset Symmetries Pretty N-cube Pictures
0000 1
0001
0010
0100
1
1
1
0011
0101
1001
0110
1
0111
1000
1010
1
1011
1
1101
1110 1
1111
1100
0000 2
0001
0010
0100
2
0011
2
0101 2
0111
1001
0110
1000 2
1010
2
1100 2
1011
1101 2
1111
1110
0000 3
0001
0010
0100
3
0011
0101
3
1001
3
3
0110 3
0111
1000
1011
3
1101 3
1111
1010
1110
1100
0000 4
0001
0010
0100
4
0011
1000
4
0101
1001
4
0110
4
0111
4
1010 4
1011
1101
4
1111
1110
1100
0000 1
0001 2
0011
31
3
0010 4
1
0101 3
2
421
0111
1000
421
42
0110 3
4
1101 3
2
1111
3
1010
1
1011 4
0100 3
1001
42
4
31
1110 1
1100 2
0 (t+2)
(t+2)
=
1
=
T + 2
n
1 (t+2) 2 (t+2)
2 =
T
3 (t+2)
3 =
T
6 =
T
7 (t+2)
4
3
=
T
+ 12 T
+ 14 T
=
T
+ 16 T
+ 80 T
+ 84 T
+ 80 T + 32 2
+ 160 T
5
+ 240 T
4 + 280 T
6 + 112 T
2 3
+ 60 T
7
+ 32 T + 16
4
6
8
+ 40 T
5
7
8 (t+2)
+ 24 T
+ 10 T
6
2
+ 8 T
5 =
+ 12 T + 8
3
T
5
(t+2)
+ 6 T
4 =
(t+2)
2
T
4 (t+2)
+ 4 T + 4
3 + 560 T
5 + 448 T
+ 192 T + 64 2
+ 672 T
4 + 1120 T
+ 448 T + 128
3 + 1792 T
2 + 1792 T
+ 1024 T + 256
6
6
5
4
3
2
(T+2) = T + 12T + 60T + 160T + 240T + 192T + 64
A 6-space cube contains: (1)
6-space cube
12
5-space cubes
60
4-space cubes
160
3-space cubes
240
2-space cubes
192
1-space cubes
64
0-space cubes
The coefficients of the algebraically expanded expression N
(T+2) contain some curious information.
00000
00001
00010
00100
01000
10000
00011
00101
01001
10001
00110
01010
10010
01100
10100
11000
00111
01011
10011
01101
10101
11001
01110
10110
11010
11100
01111
10111
11011
11111
11101
11110
Most people do not know what an N-space cube is, much less how to draw one. These pages show you what they are and how to create them. The visual approach to drawing them can be understood by fourth graders, while the mathematical approach can be understood by freshmen in high school. I hope you have fun with N-space Cubes. They have provided me with much enjoyment and delight. Sincerely, Dennis Clark This material is copyrighted, but can be used as long as it is not altered and as long as the author retains credit.
N An Intuitive Visual Approach We say: “I See!” when we understand
A 0-space cube is a point. Here, we represent it with a black dot.
Two 0-space cubes with a line between the points makes a 1-space cube. (A line)
Two 1-space cubes with their corresponding points connected make a 2-space cube. (A square)
Two 2-space cubes with their corresponding corners connected form a 3-space cube. (a cube)
These two 3-space cubes appear to be overlapping. No problem.
Two 3-space cubes with their corresponding corners connected form a 4-space cube. (a tesseract)
So how do we create a 5-space cube?
Connect the corresponding corners of two 4-space cubes to form a 5-space cube.
A messy 5-space cube.
N Stretching your Intuition The brain needs to be Stretched and Exercised much like the muscles of your body
Even here we are violating the geometric definition of a cube. a
c
6-sides equal length lines 90 degree corners
b Angle b,a,c is not 90 degrees.
Whenever we represent a 3-dimensional object in 2 dimensions something gets distorted. Here it’s some of the angles.
For our purposes, the physical position of any point is irrelevant.
Only its connectivity is important. A point must remain connected to the same points regardless of its physical location.
e a g
c
f
b d
h
e h
a g
c
f b d
So this is also a cube.
This is still a cube. The physical location of any point is irrelevant. e h
c
g
a
f
b d
It is the connectivity that makes this a 3-space cube.
We pause for a neighborly definition: Adjacents Points b, c, and e are adjacent to point a.
e
a c
g b,c, and e are the adjacents of a.
b
f d
h
These are both 3-space cubes because their connectivity is correct. In fact, the right-hand cube is just a twisted version of the one on the left. a h
e
a c
b
g g c
b
f d
h
f
d e
Let’s untwist the Twisted Circle Cube Pretend the cube is a flat pancake.
a b
h
Cut the pancake in half along the dotted line. Flip the left half of the pancake g over while keeping each of the the points attached to their respective adjacents.
c
f
d e
Let’s untwist the Twisted Circle Cube
a
Now quarter the pancake.
b
e
Flip the G-H quarter over. Doing so will untwist EG and FH, but it will twist GC and HD.
f c
g
d h
Let’s untwist the Twisted Circle Cube
a
Now quarter the pancake.
b
e
Flip the G-H quarter over. Just when it looks like things have gotten worse...
f c
h
d g
Things finally get straightened out.
a
Our Twisted Circle Cube is now an
b
e
Untwisted Circle Cube. f d
h
c g
It is now an Untwisted Circle Cube a e
e
a c
b
g f d
b
f d
h
h
c g
N An Easy Mathematical Approach Math is a tool of Infinite Possibilities
a
d k r
i p
u
v
t
g
ad
w
q
h
ac ab
aa
o n
x
s
j b
y
l
c
f
m
e
Letters are a Cumbersome Naming Convention
z
ae
Every N-cube has 2N points (corners)
1
0 2
4
6
5 3
7
N space
Points in each cube
0 1 2 3 4 5 6 7
1 2 4 8 16 32 64 128
We label the corners using binary numbers 001
000
010
At first this may seem to be even more clumsy than using letters. 011
101 100 110
111
But it gives us one huge advantage we didn’t have before: Now, we can identify the adjacents mathematically.
The points of an N-space cube can be uniquely identified by N using 2 N-digit binary numbers. 001
000
010
00
01
10
11
011
101 100 110 N=3
111 N=2
3-digit numbers 2 N = (8) points
2-digit numbers N 2 = (4) points
Two points are adjacent if their binary digits differ by only one digit. Is adjacent to 00 00 01 01 10 10 11 11
01 10 00 11 00 11 01 10
00
01
10
11
A clarifying example: The adjacents of 1101001 are because each adjacent differs from 1101001 by one and only one digit.
1101000 1101011 1101101 1100001 1111001 1001001 0101001
1101001 1101001 1101001 1101001 1101001 1101001 1101001 1101000 1101011 1101101 1100001 1111001 1001001 0101001
The XOR binary operation 0
1
0
0
1
1
1
0
0 0 1 1
xor xor xor xor
0=0 1=1 0=1 1=0
1101001 1101001 1101001 1101001 1101001 1101001 1101001 0000001 0000010 0000100 0001000 0010000 0100000 1000000 1101000 1101011 1101101 1100001 1111001 1001001 0101001
N Build a Cube Mathematically XORbitantly Large N-Space Cubes
000 001 001
000
001
000 001 001
000 010 010
000
001
010
000 001 001
000 010 010
001
000 100 100
000
010
100
001 001 000
001 010 011
000
001
010
011
100
001 001 000
001 010 011
001 100 101
000
001
010
011
101
100
010 001 011
010 010 000
000
001
010
011
101
100
010 001 001
010 010 000
010 100 110
000
001
010
100
011
101
110
100 001 001
100 010 110
000
001
010
100
011
101
110
100 001 001
100 010 110
100 100 000
000
001
010
100
011
101
110
011 001 010
011 010 001
011 100 111
000
001
010
100
011
101
110
111
101 001 101
101 010 111
101 100 001
000
001
010
100
011
101
110
111
110 001 111
110 010 100
110 100 110
000
001
010
100
011
101
110
111
000
001
010
100
011
101
110
111
0000 0001
0011
0101
0111
0010
0100
1001
0110
1011
1101 1111
1000
1010
1110
1100
1 4
6 0011
4 1
0000 0001
0101
0111
0010
0100
1001
0110
1011
1101 1111
1000
1010
1110
1100
1 1 1 1 1 1 1 1
1 2 3 4 5 6 7
Pascal’s Triangle 1 3 1 6 4 1 10 10 5 1 15 20 15 6 1 21 35 35 21 7 1
00000
00001
00010
00100
01000
10000
00011
00101
01001
10001
00110
01010
10010
01100
10100
11000
00111
01011
10011
01101
10101
11001
01110
10110
11010
11100
01111
10111
11011
11111
11101
11110
N Lineset Symmetries Pretty N-cube Pictures
0000 1
0001
0010
0100
1
1
1
0011
0101
1001
0110
1
0111
1000
1010
1
1011
1
1101
1110 1
1111
1100
0000 2
0001
0010
0100
2
0011
2
0101 2
0111
1001
0110
1000 2
1010
2
1100 2
1011
1101 2
1111
1110
0000 3
0001
0010
0100
3
0011
0101
3
1001
3
3
0110 3
0111
1000
1011
3
1101 3
1111
1010
1110
1100
0000 4
0001
0010
0100
4
0011
1000
4
0101
1001
4
0110
4
0111
4
1010 4
1011
1101
4
1111
1110
1100
0000 1
0001 2
0011
31
3
0010 4
1
0101 3
2
421
0111
1000
421
42
0110 3
4
1101 3
2
1111
3
1010
1
1011 4
0100 3
1001
42
4
31
1110 1
1100 2
0 (t+2)
(t+2)
=
1
=
T + 2
n
1 (t+2) 2 (t+2)
2 =
T
3 (t+2)
3 =
T
6 =
T
7 (t+2)
4
3
=
T
+ 12 T
+ 14 T
=
T
+ 16 T
+ 80 T
+ 84 T
+ 80 T + 32 2
+ 160 T
5
+ 240 T
4 + 280 T
6 + 112 T
2 3
+ 60 T
7
+ 32 T + 16
4
6
8
+ 40 T
5
7
8 (t+2)
+ 24 T
+ 10 T
6
2
+ 8 T
5 =
+ 12 T + 8
3
T
5
(t+2)
+ 6 T
4 =
(t+2)
2
T
4 (t+2)
+ 4 T + 4
3 + 560 T
5 + 448 T
+ 192 T + 64 2
+ 672 T
4 + 1120 T
+ 448 T + 128
3 + 1792 T
2 + 1792 T
+ 1024 T + 256
6
6
5
4
3
2
(T+2) = T + 12T + 60T + 160T + 240T + 192T + 64
A 6-space cube contains: (1)
6-space cube
12
5-space cubes
60
4-space cubes
160
3-space cubes
240
2-space cubes
192
1-space cubes
64
0-space cubes
The coefficients of the algebraically expanded expression N
(T+2) contain some curious information.
00000
00001
00010
00100
01000
10000
00011
00101
01001
10001
00110
01010
10010
01100
10100
11000
00111
01011
10011
01101
10101
11001
01110
10110
11010
11100
01111
10111
11011
11111
11101
11110
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