How to Draw N-Space Cubes

How to Draw N-Space Cubes
An Overview of this Session An Intuitive Visual Approach Stretching your Intuition An Easy Mathematical Approach Building a cube Mathematically Since this session contains about 70 slides it has been divided into six smaller sections. At the end of each section you will be returned to this slide. Click on any of the blue boxes to go to that section. The sections are designed to be read sequentially. 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. Mailing address: Dennis Clark 469 Mahoney Road Oliver Springs, TN 37840

An Intuitive Visual Approach We say: “I See!” when we understand This first section gives you a visual idea of how N-space cubes can be created. There are many ways to draw N-Space cubes. You’ll see several of them and I will show you how you can create them yourself. First, I’ll take you through an intuitive visual method of creating them. Once you have done that, it is a simple matter to create your own versions of an N-Space cube. Then I’ll show you a mathematical approach so that you can appreciate some of the elegant symmetries involved.

Here, we represent it with a black dot.
A 0-space cube is a point. Here, we represent it with a black dot. We’ll start off with a zero-space cube. The only thing that can exist in 0-space (zero dimensions) is a point. So a 0-space cube must be a point which we represent here by a little black ball. If the ball were really a dimensionless point you couldn’t see it so I’m cheating and making it big enough to see.

line between the points makes a 1-space cube. (A line)
Two 0-space cubes with a line between the points makes a 1-space cube. (A line) If you clone a 0-space cube and then connect the two corners (end points) together, you get what we commonly call a line - which is a 1-space cube. Sometimes I refer to the end-points as corners (of the cubes) and sometimes as points and sometimes as end-points. I mean the same thing in each case.

points connected make a 2-space cube. (A square)
Two 1-space cubes with their corresponding points connected make a 2-space cube. (A square) If you clone a 1-space cube (a line) you get two lines. If you connect each end-point to its corresponding end-point on the other line, you get a 2-space cube - or what we know as a square. In this diagram I have indicated the connections between corresponding end-points with dotted lines.

corners connected form a 3-space cube. (a cube)
Two 2-space cubes with their corresponding corners connected form a 3-space cube. (a cube) If you clone a 2-space cube (square) and connect each point on each square to its corresponding point on the other square, you get a 3-space cube - what you would normally call a “cube”. Actually, what you see is a two-dimensional representation of a three dimensional object.

appear to be overlapping. No problem.
These two 3-space cubes appear to be overlapping. No problem. By now you are probably catching on to what will happen next. We just keep cloning whatever the current N-cube is, and connect corresponding corners with a line to get the next higher N-space cube. Here I’ve cloned the 3-space cube and colored it differently so you can recognize the original 3-cubes easily.

corners connected form a 4-space cube. (a tesseract)
Two 3-space cubes with their corresponding corners connected form a 4-space cube. (a tesseract) If you clone a 3-space cube and connect corresponding corners between each cube, you get a 4-space cube which is called a tesseract. Now the dotted lines are beginning to be useful. They clarify which corners correspond to each other.

So how do we create a 5-space cube?
Things are beginning to look more complicated although the process of creating the next higher N-cube remains simple. Let’s try it one more time. How do you create a 5-space cube?

Connect the corresponding corners of two 4-space cubes to form
a 5-space cube. To create a 5-space cube, clone a 4-space cube and connect their corresponding corners together with lines. It’s a simple idea which is now beginning to get complicated when we try to implement it.

A messy 5-space cube. Unfortunately, the picture is getting messy. Our intuitive visual approach is beginning to reach its limits. We already have two lines that are so close they appear to be one line. If we don’t do something different, 6, 7, and 8-space cubes are going to be incomprehensible blobs.

Stretching your Intuition
How to Draw N-Space Cubes N Stretching your Intuition The brain needs to be Stretched and Exercised much like the muscles of your body In this section we’ll free you from your eighth grade definition of a cube. So it’s time to throw away some of your mathematical training. Well, not really throw it away, but perhaps we’ll suspend our beliefs for a moment in order to stretch our intuition about cubes.

Even here we are violating the geometric definition of a cube. 6-sides
equal length lines 90 degree corners Angle b,a,c is not 90 degrees. a c b Once upon a time you memorized a definition for a cube in your geometry class. Something like “A Cube is a six sided three dimensional solid with all sides of equal length and all corners being 90 degrees. Well, I started violating the geometric descriptions at the very start of this exercise. I showed you a black circle where it should have been a dimensionless point. I showed you a black line where the line should have had no width. I violated the 90 degree rule when I showed you a picture of a cube. When I showed you a tesseract, I violated the equal length sides rule. No problem. That definition is for geometry. We’re not talking geometry here. We’re talking topology! 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 Since we have already violated most of the geometric definition of a cube, let’s go further. We’re not interested in angles and line lengths here, we’re only interested in the connectivity of the points. This means that the figure above is still a cube, even though we have stretched the lines attached to the point H which we have moved. As long as the point H is still connected to each of its original three neighbors (points D, F, and G) we haven’t changed the connectivity of the cube and that is sufficient for our purposes. So let’s stretch your brain further! h d

e h a g c f b d So this is also a cube.
This is still a cube because we haven’t changed the connectivity of the points. It doesn’t matter where any of the points are located. As long as they are connected to the proper neighbors, we still have a cube. d So this is also a cube.

The physical location of any point is irrelevant.
This is still a cube. The physical location of any point is irrelevant. e h c g a f b Still a cube. Move any or all of the corners around, and as long as you keep them all connected to the correct neighbors, it’s still a cube (for our purposes). However, I wouldn’t call this a pretty picture. It is useful only to the extent that it stretches your intuition. But what do I mean by “neighbors”? It is the connectivity that makes this a 3-space cube. d

We pause for a neighborly definition: Adjacents
Points b, c, and e are adjacent to point a. b,c, and e are the adjacents of a. a e c g b f Neighboring points are those which have a single line between them. Lines are connected only to points. So there is no connection between the two lines CD and BF. Two points are said to be adjacent to each other if they are connected by a single line. So point A is adjacent to B,C, and E, but not D,F,G, or H. Adjacents are close to each other. They’re neighbors. 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 b h a e c g g c b f Given the freedom to place points wherever we want, we can say that the figure on the right is a cube because each of its points is connected to the same neighbors (adjacents) as its corresponding point on the figure at the left which we recognize as a cube. For instance, point A is connected to points B, C, and E in each case. Note: the lines in the figure on the right do not connect at the center, they merely cross each other much like CD and BF on the left. d h f d e

Pretend the cube is a flat pancake. Cut the pancake in half along the
Let’s untwist the Twisted Circle Cube Pretend the cube is a flat pancake. Cut the pancake in half along the dotted line. Flip the left half of the pancake over while keeping each of the the points attached to their respective adjacents. a b h g c I call this cube a Twisted Circle Cube. It’s a circle cube because all of the points are arranged along the circumference of a circle shown here with a dotted line. It’s twisted because so many lines cross each other. And, it can be untwisted. More mental gymnastics. Cut the pancake in half and flip the left half over as if you were cooking it on both sides. If we keep the lines connected to their original points, then it looks as if we have untwisted the four lines HD, GC, FB, and EA. f d e

Now quarter the pancake. Flip the G-H quarter over.
Let’s untwist the Twisted Circle Cube Now quarter the pancake. Flip the G-H quarter over. Doing so will untwist EG and FH, but it will twist GC and HD. a b e f c It looks a little less twisted now, but there are still lines that are twisted. Let’s untwist them too. Cut the pancake into quarters and flip the GH quarter over oops... It untwists two lines but in the process twists two other lines. g d h

Now quarter the pancake. Flip the G-H quarter over.
Let’s untwist the Twisted Circle Cube Now quarter the pancake. Flip the G-H quarter over. Just when it looks like things have gotten worse... a b e f c But if we now flip the DC quarter pancake over, not only will the HD and GC lines be untwisted, but at the same time we will untwist the AC and BD lines. h d g

Things finally get straightened out. Our Twisted Circle Cube is now an
Untwisted Circle Cube. a b e f d It doesn’t look exactly like a “traditional” cube, but like I said, a warped imagination helps.. h c g

It is now an Untwisted Circle Cube a b e a e c g f d b f d h h c g
So are these two cubes really the same? Do both cubes have the same connectivity? In each cube A is connected to B,C, and E; B is connected to A,D, and F; etc. They are essentially the same cube. d h h c g

N How to Draw N-Space Cubes An Easy Mathematical Approach
Math is a tool of Infinite Possibilities Since our intuitive visual approach started getting messy, we need to supplement it with some other, very powerful tools. Math has some particularly elegant applications here in N-space.

Letters are a Cumbersome Naming Convention e
y l a d c k x s r ad w j v u ac b i q h We have been labeling our points with letters. While this is useful, it doesn’t convey or contain a whole lot of information. Unless you see the picture, you wouldn’t know the adjacents for any given point. And as we get to the larger N-space cubes, we run out of letters and have to improvise. p t ab g aa o ae f n z

Every N-cube has 2 points (corners)
N Points in space each cube 0 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 1 2 6 4 5 On the other hand, numbers provide us with an astonishingly beautiful modeling tool. To start out with, you may have noticed that each successive N-cube has twice as many points as the one before. 3 7

We label the corners using binary numbers
001 000 At first this may seem to be even more clumsy than using letters. But it gives us one huge advantage we didn’t have before: Now, we can identify the adjacents mathematically. 010 011 101 100 If, instead of decimal numbers, we label each corner with binary numbers, we are given a simple but powerful mechanism for identifying which corners are close to each other. We can identify the adjacents mathematically. 110 111

The points of an N-space cube can be uniquely identified by
using N-digit binary numbers. N 000 100 010 101 011 111 110 001 00 01 Each N-space cube has 2**N points, so each N-space cube point can be uniquely identified by an N-digit binary number. The points in a 2-space cube can be identified by the four two-digit binary numbers (00, 01, 10, 11). The 2**3=8 points in a 3-space cube can be identified by the 8 three digit binary numbers (000, 001, 010, 011, 100, 101, 110, 111). The 2**4=16 points in a 4-space cube can be identified by 4 digit binary numbers. Etc... 10 11 N = 3 N = 2 3-digit numbers 2 = (8) points 2-digit numbers 2 = (4) points N N

Two points are adjacent
if their binary digits differ by only one digit. 00 01 Is adjacent to So how do we tell which points are adjacent to each other? Points are adjacent to each other if their binary digits differ from each other by one and only one binary digit. Note: the point 00 is not adjacent to 11 because you would have to change two digits in order to get the binary number 11. Likewise, 10 is not adjacent to 01. 10 11

A clarifying example: The adjacents of 1101001 are
because each adjacent differs from by one and only one digit. If you have studied binary numbers and the binary operations that can be performed on binary numbers, you may recognize the binary operation that we use to figure out the adjacents mathematically. We are performing an Exclusive OR operation. Sometimes it is called an EOR operation and sometimes an XOR operation. They both mean the same thing. We’ll use XOR here.

The XOR binary operation
1 0 xor 0 = 0 0 xor 1 = 1 1 xor 0 = 1 1 xor 1 = 0 1 1 1 The XOR operation truth table is shown here. If you perform an exclusive OR operation on two zeroes, the result is zero. A zero XORed with a one yields one. A one XORed with zero yields one. And a one XORed with a one yields a zero. If you perform an XOR on the top number ( ) with one of the middle numbers (like ) you get one of the seven adjacents that differ from by one digit only. ( )

Build a Cube Mathematically XORbitantly Large N-Space Cubes Now, you’re ready to build a 3-space cube using the XOR method of creating an N-space cube. We can use the binary numbered corners to create much larger N-space cubes mathematically. We don’t have to depend on a visual method which was nearing its limitations. In fact, the binary point numbering system can provide us with pictures that look neater in appearance with fewer line crossings. Let’s see how it works...

000 001 000 001 Let’s build a 3-space cube using binary numbers and their ability to generate adjacents by applying the XOR operation. We could start with any corner (numbers 000 through 111) but let’s start with There are three numbers that differ from 000 by one digit. They are 001, 010, and Place them on an imaginary line below 000.

000 001 010 The numbers at the top left of the picture represent the XOR operations we are performing to get the adjacents of XORed with 001 yields the first adjacent XORed with 010 yields the second adjacent (010).

000 001 010 100 000 XORed with 100 yields the third adjacent (100). Bear with me, it gets more interesting shortly. The first three cases were trivial. Now, we want to create the adjacents for the numbers 001, 010, and We’ll put these new adjacents on a line below 001, 010 and 100.

000 001 010 100 We’ll start with If we XOR it with 001, we get the first adjacent of 001 which is That adjacent is already connected. Next we XOR 001 with 010 to get the second adjacent of 001 which is 011. 011

000 001 010 100 Then we XOR 001 with 100 to get the third adjacent 101 which we again place on the line below. Now we’re ready to get the adjacents for 010. 011 101

000 001 010 100 If we XOR 010 with 001, we get 011 which is already on the cube. We connect the two with a line. We then XOR 010 with 010 which gives us already on the cube and connected. 011 101

000 001 010 100 XORing 010 with 100 yields 110 which we place on the cube. Then we are ready to create the adjacents for the number 100. 011 101 110

000 001 010 100 100 XORed with 001 yields 101. 011 101 110

000 001 010 100 100 XORed with 010 yields Already on the cube - just connect a line between them XORed with 100 yields already there, already connected. 011 101 110

000 001 010 100 Next we drop down to the number 011 and create its adjacents - two of which are already connected XORed with 001 yields XORed with 010 yhields XORed with 100 yields 111 which we put on a line below 001. 011 101 110 111

000 001 010 100 We do the same thing for XORed with 001 yields XORed with 010 yields 111, and 101 XORed with 100 yields 001. 011 101 110 111

000 001 010 100 We now create the adjacents for the next to last point: XORed with 001 yields XORed with 010 yields XORed with 100 yields All of the points have been created, so we don’t have to worry about the last point (111). 011 101 110 111

000 001 010 100 This process can be used to create larger N-space cubes. The only difference is the number of binary digits each N-space cube will use. 011 101 110 111

0000 0001 0010 0100 1000 0011 0101 1001 0110 1010 1100 A 4-space cube will use 16, 4-digit binary numbers. Their adjacents can be created in the same manner as the 3-space cube: It just takes a little more time. I figure you know the procedure by now, so we won’t go through it again. 0111 1011 1101 1110 1111

1 4 6 0000 1000 0100 0010 0001 1100 1010 0110 1001 0011 0101 1111 1110 1101 1011 0111 Here’s another interesting mathematical pearl. You’ll notice that in a 3-space cube there is one point at the top, the next level down contains 3 points, the next level down has 3 points and the bottom level has one point. The 4-space cube here has one point at the top, 4 on the next level, 6 on the next, 4 on the next, and 1 on the bottom. Do you recognize the sequence?

Pascal’s Triangle It’s Pascal’s Triangle. Each number is the sum of the two numbers above it (above and left - in this representation). Because of this interesting phenomenon I call the kind of cube that we just generated a Pascal’s Triangle Cube.

00000 10000 01000 00100 00010 00001 11110 11101 11011 10111 01111 11111 11010 01110 10101 10011 00111 11100 10110 11001 01101 01011 10100 10010 00110 01001 00011 11000 01100 01010 10001 00101 When we generate an N-space cube using the XOR method, each level of the cube has the same number of points as the Nth level of Pascal’s Triangle. So a 5-space cube has 1, 5, 10, 10, 5, and 1 points in each of its six levels.

Lineset Symmetries Pretty N-cube Pictures Now, you’ll see some pretty pictures.

0000 1000 0100 0010 0001 1100 1010 0110 1001 0011 0101 1111 1110 1101 1011 0111 1 1 1 1 1 1 1 Another interesting way to view N-cubes is to look at the sets of lines that are used to create the N-cube. In this picture, you see the set of lines that connect points which differ by one digit in the right-most column. 1

0000 1000 0100 0010 0001 1100 1010 0110 1001 0011 0101 1111 1110 1101 1011 0111 2 2 2 2 2 2 2 This is the set of lines that differ in the second (from the right) column. So I have labeled these lines with a 2. 2

0000 1000 0100 0010 0001 1100 1010 0110 1001 0011 0101 1111 1110 1101 1011 0111 3 3 3 3 3 3 3 These lines connect points that differ in the third (from the right) column. They are labeled with a 3. Notice also that I have colored the different sets of lines different colors. 3

0000 1100 1010 0110 1001 0011 0101 1111 1110 1101 1011 0111 4 1000 0100 0010 0001 4 4 4 4 4 4 This is the last set of lines. They differ in the fourth column by one and only one digit. They are labeled with a 4 and are again, a different color. 4

0000 1000 0100 0010 0001 1100 1010 0110 1001 0011 0101 1111 1110 1101 1011 0111 1 2 3 4 2 31 4 1 3 421 42 3 42 421 1 31 2 4 Together, all four numbered, colored sets of lines make up a 4-space cube. Now, for the fun part! 4 3 2 1

First off, since we will be viewing for fun, I’m going to change the binary numbered “points” into black dots. We won’t be worrying about the “names” of the corners.

It turns out that any two sets of lines (blue and yellow, here) form sets of squares. True, they’re not “square” squares, but remember, we’re interested in connectivity - not a strict geometric interpretation.

Again, ANY two sets of lines create squares.

Sometimes the squares overlap
Sometimes the squares overlap. I know it’s kind of weird calling that thing in the upper left corner a “square”. It helps to have a warped imagination.

Here’s another set of overlapping squares
Here’s another set of overlapping squares. In this case they are formed by the “4” and “2” linesets.

These squares were created with the 1 and 3 linesets.

In all, there are six ways to combine the four sets of lines in groups of two. We just covered them all.

Now, whereas any two sets of lines create squares, any three sets of lines create cubes. These cubes are made from the 1, 2, and 3 linesets.

In the previous picture, the cubes did not overlap
In the previous picture, the cubes did not overlap. In this one, they do but are still easy to separate in your mind. These cubes were made with the (2,3,4) linesets.

These two cubes are much harder to visualize.

They are easier to see when separated by color, but if I do that, I don’t see the uniquely colored linesets.

Perhaps you might ask “How many cubes are there in a tesseract
Perhaps you might ask “How many cubes are there in a tesseract?” Math not only gives us that particular answer, but it tells us much more!

(t+2) n (t+2) = 1 1 (t+2) = T + 2 2 2 (t+2) = T + 4 T + 4 3 3 2
(t+2) = 1 (t+2) = T + 2 (t+2) = T + 4 T + 4 (t+2) = T + 6 T T + 8 (t+2) = T + 8 T T T + 16 (t+2) = T T T T T + 32 (t+2) = T T T T T T + 64 (t+2) = T T T T T T T + 128 (t+2) = T T T T T T T T + 256 It turns out that the coefficients of the equation (t+2)**n tell us how many points, lines, squares, cubes, tesseracts, and in general how many sub-cubes there are in any N-space cube.

The coefficients of the algebraically expanded expression
(T+2) = T + 12T + 60T + 160T + 240T + 192T + 64 The coefficients of the algebraically expanded expression (T+2) contain some curious information. 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 N It is fascinating that math can encode or encapsulate large amounts of information in such small descriptions as the formula (T+2)**n.

00000 10000 01000 00100 00010 00001 11110 11101 11011 10111 01111 11111 11010 01110 10101 10011 00111 11100 10110 11001 01101 01011 10100 10010 00110 01001 00011 11000 01100 01010 10001 00101 Congratulations! You’ve graduated from N-cube school. Now you know what N-space cubes are, and you can draw them. So you’re properly prepared and licensed to hunt Great Snakes in N-space Cubes. That’s another session. Cheers, Dennis Clark

How to Draw N-Space Cubes

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