An Overview of this Session

N

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.

An Intuitive Visual Approach

We say:

“I See!”

when we understand

N

A 0-space cube is a point.

Here, we represent it witha black dot.

Two 0-space cubes with a line between the points makesa 1-space cube. (A line)

Two 1-space cubes withtheir correspondingpoints connected make a2-space cube. (A square)

Two 2-space cubes withtheir correspondingcorners connected forma 3-space cube.(a cube)

These two 3-space cubesappear to be overlapping.

No problem.

Two 3-space cubes withtheir correspondingcorners connected forma 4-space cube.(a tesseract)

So how do we createa 5-space cube?

Connect the corresponding corners of two 4-space cubes to forma 5-space cube.

A messy 5-space cube.

Stretching your Intuition

The brain needs to be

Stretched and Exercised

much like the muscles of your body

N

Even here we are violating thegeometric definition of a cube.

6-sidesequal length lines90 degree corners

Angle b,a,c is not 90 degrees.

a

b

c

Whenever we represent a 3-dimensional object in 2 dimensionssomething gets distorted. Here it’s some of the angles.

c

ea

b

d

f

g

h

For our purposes, the physical position of any point is irrelevant.

Only its connectivity isimportant. A point must remainconnected to the samepoints regardless of itsphysical location.

c

ea

b

d

f

g

h

So this is also a cube.

c

e

a

b

d

f

g

h

It is the connectivity that makesthis a 3-space cube.

This is still a cube.The physical location of any point is irrelevant.

a

b

c

e

d

f

g

h

Points b, c, and e are adjacent to point a.

b,c, and e are the adjacents of a.

We pause for a neighborly definition:Adjacents

a

b

c

e

d

f

g

h

a

b

c

d

e

f

g

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 oneon the left.

a

b

c

d

e

f

g

h

Let’s untwist the Twisted Circle Cube

Pretend the cube is a flat pancake.

Cut the pancake in half along thedotted line. Flip the left half of the pancake over while keeping each of thethe points attached to theirrespective adjacents.

a

b

c

d

h

g

f

e

Let’s untwist the Twisted Circle Cube

Now quarter the pancake.

Flip the G-H quarter over.

Doing so will untwist EGand FH, but it will twistGC and HD.

a

b

c

d

g

h

f

e

Let’s untwist the Twisted Circle Cube

Now quarter the pancake.

Flip the G-H quarter over.

Just when it looks like things have gottenworse...

Things finally get straightenedout.

Our Twisted Circle Cubeis now an

Untwisted Circle Cube.

a

b

d

c

g

h

f

e

a

b

d

c

g

h

f

e

It is now anUntwisted Circle Cube

a

b

c

e

d

f

g

h

An Easy Mathematical Approach

Math is a tool

of Infinite Possibilities

N

Letters are a CumbersomeNaming Convention

a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

p

r

s

q

t

u v

w

x

y

z

aa

ab

ac

ad

ae

0

4

2

1

5

6

7

Every N-cube has 2 points (corners)

N Points inspace each cube

0 11 22 43 84 165 326 647 1283

N

000

100

010

101

011

111

We label the corners using binary numbers

At first this may seemto be even more clumsythan using letters.

But it gives us one hugeadvantage we didn’t havebefore:

Now, we can identify theadjacents mathematically.

110

001

The points of an N-space cube can be uniquely identified byusing 2 N-digit binary numbers.

000

100

010

101

011

111110

00100

10 11

01

N = 2

2-digit numbers2 = (4) points

N = 3

3-digit numbers2 = (8) points

N

N N

Two points are adjacentif their binary digits differ by only one digit.

00

10 11

01Is adjacent to00 0100 1001 0001 1110 0010 1111 0111 10

A clarifying example:

The adjacents of 1101001 are

because each adjacent differs from 1101001 byone and only one digit.

1101001 1101001 1101001 1101001 1101001 1101001 11010011101000 1101011 1101101 1100001 1111001 1001001 0101001

1101000110101111011011100001111100110010010101001

1101001 1101001 1101001 1101001 1101001 1101001 11010010000001 0000010 0000100 0001000 0010000 0100000 10000001101000 1101011 1101101 1100001 1111001 1001001 0101001

The XOR binary operation

0

0 1

1

0 1

1 0

0 xor 0 = 00 xor 1 = 11 xor 0 = 11 xor 1 = 0

Build a Cube Mathematically

XORbitantly Large

N-Space Cubes

N

000

001

000001001

000

010001

000 000001 010001 010

000

100010001

000 000 000001 010 100001 010 100

000

100010001

011

001 001001 010000 011

000

100010001

101011

001 001 001001 010 100000 011 101

000

100010001

101011

010 010001 010011 000

000

100010001

110101011

010 010 010001 010 100001 000 110

000

100010001

110101011

100 100001 010001 110

000

100010001

110101011

100 100 100001 010 100001 110 000

000

100010001

111

110101011

011 011 011001 010 100010 001 111

000

100010001

111

110101011

101 101 101001 010 100101 111 001

000

100010001

111

110101011

110 110 110001 010 100111 100 110

000

100010001

111

110101011

0000

1000010000100001

11001010011010010011 0101

1111

1110110110110111

0000

1000010000100001

11001010011010010011 0101

1111

1110110110110111

1

4

6

4

1

1 Pascal’s1 1 Triangle1 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 1

00000

1000001000001000001000001

1111011101110111011101111

11111

1101001110101011001100111 1110010110110010110101011

1010010010001100100100011 1100001100010101000100101

Lineset Symmetries

Pretty N-cube

Pictures

N

0000

1000010000100001

11001010011010010011 0101

1111

1110110110110111

1

1

1 1 1

1 1 1

0000

1000010000100001

11001010011010010011 0101

1111

1110110110110111

2

2

2 2 2

2 2 2

0000

1000010000100001

11001010011010010011 0101

1111

1110110110110111

3

3

3 3 3

3 3 3

1000010000100001

0000

11001010011010010011 0101

1111

1110110110110111

4

4

4 4 4

4 4 4

0000

1000010000100001

11001010011010010011 0101

1111

1110110110110111

1 2 3 4

4 3 2 1

2 31 4 1 3 421 42 3

3 42 421 3 1 31 24

0(t+2) = 1

1 (t+2) = T + 2

2 2(t+2) = T + 4 T + 4

3 3 2(t+2) = T + 6 T + 12 T + 8

4 4 3 2(t+2) = T + 8 T + 24 T + 32 T + 16

5 5 4 3 2(t+2) = T + 10 T + 40 T + 80 T + 80 T + 32

6 6 5 4 3 2(t+2) = T + 12 T + 60 T + 160 T + 240 T + 192 T + 64

7 7 6 5 4 3 2(t+2) = T + 14 T + 84 T + 280 T + 560 T + 672 T + 448 T + 128

8 8 7 6 5 4 3 2(t+2) = T + 16 T + 112 T + 448 T + 1120 T + 1792 T + 1792 T + 1024 T + 256

(t+2)n

(T+2) = T + 12T + 60T + 160T + 240T + 192T + 646 6 5 4 3 2

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

(T+2) contain some curious information.

N

00000

1000001000001000001000001

1111011101110111011101111

11111

1101001110101011001100111 1110010110110010110101011

1010010010001100100100011 1100001100010101000100101