N-space Snakes are special maximal length loops through an N-space cube. They’re full of intriguing symmetries, puzzles and surprises. They’re simple structures that baffle us with their complexities.

Fascinating creatures.

Let’s go find some Snakes.

In this session:

We’ll define what a Snake is,Search for 3,4, and 5-space snakes by hand,Identify snakes with binary names,Identify snakes by their column changes,Find the unique snakes up through 6-space,Look at a snake’s physique-l makeup, and

ask some questions that maybe you will answer.

So,

what IS an N-space Snake?

Me.

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A “Snake” is a closed path (loop) through an N-space cube. But, the path must follow one special rule.You must understand that rule in order to create valid snakes.

The green lines form a valid 3-space snake of length 6.

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That special rule is:No point on a snake (other than the preceding and succeedingpoints on the snake) can be within one line length of any otherpoint on the snake.

This is an invalid snake because point 011 is one lengthaway from 010, and both points are already part of the snake.

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Every point (b) on the snake has one point that comes before it (a), and one that comes after it (c).Points a and c are one length away from b.

No other point on the snake can be just one length away frompoint b. If it is, the snake is invalid. That’s the case here. Point 101 is one length away from point 001.

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A point is “adjacent” to another if it is one line length away. The“adjacents” of a point are those points that are one line length away.

The points a, c, and d are adjacent to point b.a, c, and d are the adjacents of point b.

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We will be looking for maximal length snakes which I callGreat Snakes.

The snake shown here is valid, but is not a Great Snake because it is not the longest snake possible in 3-space.

This is a valid 3-space snake of length 4.It is not a maximal length (Great) snake.

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The longest snake possible in a 3-space cubeis a snake of length 6.

This is a valid 3-space Great Snake.

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The longest snake in a 4-space cube is of length 8. You maywish to print this page and try to find a 4-space Great Snakeon your own.

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This is an invalid 4-space snake. Do you see why?

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It is invalid because points 0010 and 0110 (which are already on the snake) are within one line length of each other.

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Do you see why this snake is invalid?

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Actually, there are two problems here. The point 1010 is adjacent to both 0010 and 1011 which are part of the snake.

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Is this a valid 4-space snake? Is it a Great Snake?

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This snake is a valid 4-space Great Snake.

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Here’s another 4-space Great Snake. From now on, when I say “snake”, I will usually be talking about Great Snakes.

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Once you know the rules for finding a snake, it is trivial to find a 3-space snake and easy to find a 4-space snake.

5-space snakes take a little more work,although most people can find several without too much trouble.

Give it a try...

Find me...

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A 5-space cube Maximal length snake = 14

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Here’s a 5-space Great Snake

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To become more familiar with our snakes, we have to uniquely identify them .

We have to name them.

My name is Joe Finklesnake III

One way to name a snake is to list the points that make up the snake. They must be listed in order; otherwise they won’t be a valid snake.

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But since there is no head or tail to the snake, you can start anywhere on the snake, and list the points as you follow the path back to your starting point.

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Although the two “lists” are different, they are really the same snake. They just start at different points and go in opposite directions.

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Start

Start

So a single snake can have many different binary names. Since these particular lists appear to rotate vertically, they are called “vertical rotations” of each other.

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Pretend that our 4-cube is a round transparent Christmas tree ornament suspended by a red ribbon from the 0000 point.

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There are other rotations too.

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If we slowly twirl the ornament, some of the points would appear to change places with other points on the same level and the snake would appear to move around the ornament.

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If you twirled just the snake, and not the ornament, you could make an intuitive leap and call the resulting snakes “horizontal rotations” of each other.

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The horizontally rotated list of points looks very different, so you might thinkthat you have a new, different snake. But, it’s really the same old snake rotated.

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ColumnsRotate the 4 Column to the right hand side.

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Old Snake

Rotated Snake

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In fact, if you exchange any column of a given snake with any other column of the samesnake, you have an intermixed rotation of the snake, and it is really the same snake as before even though the list of points is very different.

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ColumnsHorizontally inter-mixed Columns

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Old Snake

New Snake

There are other intriguing ways to name our snakes.

My name is Joe Finklesnake III

You can call me Joe

This picture shows colored linesets as well as points of a 4-space cube. 0000

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1 2 3 4

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2 31 4 1 3 421 42 3

3 42 421 3 1 31 24

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Instead of using the points to name the snake, we can use the column number between each of the snake’s 8 points. This snake’s name would then be:

1 2 3 4 1 2 3 4

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It turns out that the column-change naming convention is a more effective, efficient, easymethod of naming snakes. And it highlights something we might not have seen otherwise.

ColumnsSnake named by its points

Snake named by column changes

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Snake named by its points

Snake named by column changes

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This snake appears to be made from two “identical” halves.

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The column-change naming convention reveals structures within the snake that we did not expect to find.

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Now, we can name this 5-space snake two different ways.

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Binary snake name

Column-change name

2 3 4 5 3 1 4 2 3 4 5 3 1 4

Symmetry, symmetry, everywhereand what a lot to think.

2345314 2345314A 4-space cube

A 4-space cube00000

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A 5-space Cube

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A 4-space cube

A 4-space cube

5-space cubeThis gives us a clue as to how we might construct N-space snakes from (N-1)-space snakes.

Just how big do these snakes get?

0-space 0 1-space 1 2-space 4 3-space 6 4-space 8 5-space 14 6-space 26 7-space 48

We don’t know how big they are above 7-space.

This Big

Now, it might be informative to catalog all of the snakes in an N-space cube to see how each of them is constructed. That could give us a clue as to how to construct snakes in higher N-space cubes. However, a lot of the snakes are just transformations of each other. The N-cubes appear to be infested with snakes!

If we throw out all of the duplicate snakes, how many are left? How many UNIQUE snakes are there in each N-cube?

First, you have to find them all. How do you do that?

One way is to write a computer program that exhaustively searches for them.

I wrote one and named it

TailWagger

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You could find all of the snakes in an N-space cube if you tried all of the possible paths. This is called the BFI orBrute Force and Ignorance method.

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TailWagger starts at point 0000. It chooses one of fourpossible points. It then has three more choices, chooses one and checks to see if the snake has violated any rules.

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If TailWagger chooses a point that violates a rule, it backtracksand tries one of the other points.

0010 would have to link with 0000but the snake is still too small.

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If no rules have been violated, it continues choosing new points.If all three choices violate a rule, it backtracks to the previouspoint and chooses another point there.

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When it finds a valid snake it prints it out. Then it backtracks (as if it had found an error) and chooses other points that haven’t been tried.

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Eventually, it backtracks all the way to the third node where theprogram stops. Do you see why it isn’t necessary to backtrack to the first point to try all of the possibilities there?

Once TailWagger found all of the snakes (up through 6-space) all of the duplicate snakes had to be thrown out in order to determine the number of unique snakes and their composition.

The matter required a bit of careful thought.

X X X

Are these two snakes the same?

1 2 3 4 1 2 4 31 2 4 3 1 2 3 4

They are if the second snake is a vertical, horizontal, or intermixed rotation of the first snake.

Yes, the second snake is a rotation of the first.

1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 31 2 4 3 1 2 3 4

Here, we duplicated the first snake (red numbers) and shifted the second snake to the right. The numbers match.The snakes are the same.

Are these two snakes the same?

1 2 3 4 1 2 4 33 2 1 3 4 2 1 4

They are if the second snake is a vertical, horizontal, or intermixed rotation of the first snake.

Yes, the second snake is a rotation of the first.

1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 34 1 2 4 3 1 2 3

Here, we duplicated the first snake (red numbers),turned the second snake around (32134214 to 41243123)and shifted the second snake to the right. The numbers match.The snakes are the same.

Are these two snakes the same?

1 2 3 4 1 2 4 34 1 3 4 2 1 3 2

They are if the second snake is a vertical, horizontal, or intermixed rotation of the first snake.

Yes, the second snake is a shifted, inter-mixed rotation of the first.

1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 3 first snake

4 1 3 4 2 1 3 2 second snake

In the second snake we changed every 2 to a 3 and every 3 to a 2.Then we shifted it to the right.The numbers match.The snakes are the same.

4 1 2 4 3 1 2 3 second snake with 3s and 2s swapped

4 1 2 4 3 1 2 3 second snake shifted right

1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 3 first snake

I promised you a third way to name snakes.

My name is Joe Finklesnake III

I’m from the class of 65

Snakes can be partially described by using the following trick.

2345314234531423453142345314.....1......1......1......1.2......2......2......2.......3..3...3..3...3..3...3..3....4...4..4...4..4...4..4...4...5......5......5......5...

1 7 7 1 occurs every 7th number

2 7 7 2 occurs every 7th number

3 3 4 3 4 3 occurs every 3rd, 4th, 3rd, 4th number 4 4 3 4 3 4 occurs every 4th, 3rd, 4th, 3rd number 5 7 7 5 occurs every 7th number

Because transformations or rotations of snakes are equivalent,the following two snakes are in the same class.The are the same snake.

Snake 11 7 72 7 73 3 4 3 44 4 3 4 35 7 7

Snake 21 7 72 7 73 7 74 4 3 4 35 3 4 3 4

Snake 1 2 3 4 5 3 1 4 2 3 4 5 3 1 4Snake 2 2 5 4 3 5 1 4 2 5 4 3 5 1 4

In order to unmask the unique snakes, every snake in an N-space cube must be compared to every other snake in the N-space cubeto see whether they are

forward, backward (vertical) and / or intermixed rotations of each other.

Will the Real Unique Snakes Please Step forward ?

These are unique snakes for N < 7.

3-space 1 2 3 1 2 3

4-space 1 2 3 4 1 2 3 41 2 3 4 1 2 4 31 2 3 4 2 1 4 3

5-space 1 2 3 4 5 2 1 4 2 3 4 5 2 4 1 2 3 4 5 2 3 1 2 4 3 2 5 3 1 2 3 4 5 2 4 1 2 3 4 5 2 4

6-space 1 2 3 4 5 6 1 2 5 4 1 5 6 1 2 3 6 5 4 1 2 5 6 1 5 4 1 2 3 4 5 6 1 2 5 4 2 3 4 1 2 5 4 3 6 1 2 3 4 2 5 4 1 2 3 4 5 6 3 4 2 3 5 4 1 5 3 6 2 5 6 4 3 5 6 2 5 3 1 2 3 4 5 6 3 4 2 3 5 4 3 1 2 3 4 5 6 3 4 2 3 5 4 3 1 2 3 4 5 6 3 4 2 3 5 4 3 1 2 4 3 5 6 3 4 2 3 5 4 3

How long did it take to find every snake in 7-space?

About 30 years.

Why so long?

3-space 3**6 = 7.2x10**2 = 729

4-space 4**8 = 6.5x10**4 = 65536

5-space 5**14 = 6.1*10**9 = 6103515625

6-space 6**26 = 1.7*10**20 = 170581728179578208256

7-space 7**48 = 3.6x10**40 = 36703368217294125441230211032033660188801

So, we’ve come to the end with lots of questions.

How long are the snakes in any N-space cube? What are the unique snakes in an N-space cube?What governs the construction of snakes?Are there equations that describe all of these things?

We don’t know…

yet.