1. 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.

2. 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.

3. So, what IS an N-space Snake? Me.

4. 000 100010001 111 110101011 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.

5. 000 100010001 111 110101011 That special rule is: No point on a snake (other than the preceding and succeeding points on the snake) can be within one line length of any other point on the snake. This is an invalid snake because point 011 is one length away from 010, and both points are already part of the snake.

6. 000 100010001 111 110101011 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 from point b. If it is, the snake is invalid. That ’ s the case here. Point 101 is one length away from point 001. b a c

7. 000 100010001 111 110101011 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. b a c d

8. 000 100010001 111 110101011 We will be looking for maximal length snakes which I call Great 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. b a c d

9. 000 100010001 111 110101011 The longest snake possible in a 3-space cube is a snake of length 6. This is a valid 3-space Great Snake. b a c d

10. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 The longest snake in a 4-space cube is of length 8. You may wish to print this page and try to find a 4-space Great Snake on your own.

11. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 This is an invalid 4-space snake. Do you see why?

12. 0000 10000001 11001010100100110101 1111 1110110110110111 It is invalid because points 0010 and 0110 (which are already on the snake) are within one line length of each other. 00100100 0110

13. 0000 10000001 11001010100100110101 1111 1110110110110111 Do you see why this snake is invalid? 00100100 0110

14. 0000 10000001 11001010100100110101 1111 1110110110110111 Actually, there are two problems here. The point 1010 is adjacent to both 0010 and 1011 which are part of the snake. 00100100 0110

15. 0000 10000001 11001010100100110101 1111 1110110110110111 Is this a valid 4-space snake? Is it a Great Snake? 00100100 0110

16. 0000 10000001 11001010100100110101 1111 1110110110110111 This snake is a valid 4-space Great Snake. 00100100 0110

17. 0000 10000001 11001010100100110101 1111 1110110110110111 Here ’ s another 4-space Great Snake. From now on, when I say “ snake ”, I will usually be talking about Great Snakes. 00100100 0110

18. 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...

19. A 5-space cubeMaximal length snake = 14

20. 100000010000001 1011101111 11111 0011111100101101100101011 10100100100001111000011000101000101 Here ’ s a 5-space Great Snake 00000 0100000010 111101110111011 11010011101010110011 00110 01101 10001 01001

21. To become more familiar with our snakes, we have to uniquely identify them. We have to name them. My name is Joe Finklesnake III

22. 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. 0000 10000001 11001010100100110101 1111 1110110110110111 00100100 0110 0000 0001 0011 0111 1111 1110 1100 1000

23. 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. 0000 0001 0011 0111 1111 1110 1100 1000 1111 0111 0011 0001 0000 1000 1100 1110 0000 10000001 11001010100100110101 1111 1110110110110111 00100100 0110

24. 0000 0001 0011 0111 1111 1110 1100 1000 Although the two “ lists ” are different, they are really the same snake. They just start at different points and go in opposite directions. 0000 10000001 11001010100100110101 1111 1110110110110111 00100100 0110 1111 0111 0011 0001 0000 1000 1100 1110 Start

25. 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. 0000 0001 0011 0111 1111 1110 1100 1000 0011 0111 1111 1110 1100 1000 0000 0001 0111 1111 1110 1100 1000 0000 0001 0011 1111 1110 1100 1000 0000 0001 0011 0111 1110 1100 1000 0000 0001 0011 0111 1111 1100 1000 0000 0001 0011 0111 1111 1110 1000 0000 0001 0011 0111 1111 1110 1100 0001 0011 0111 1111 1110 1100 1000 0000

26. Pretend that our 4-cube is a round transparent Christmas tree ornament suspended by a red ribbon from the 0000 point. 0000 10000001 11001010100100110101 1111 1110110110110111 00100100 0110 0000 0001 0011 0111 1111 1110 1100 1000 There are other rotations too.

27. 0000 0001 0011 0111 1111 1110 1100 1000 0000 10000001 11001010100100110101 1111 1110110110110111 00100100 0110 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.

28. 0000 0001 1001 1101 1111 1110 0110 0010 0000 10000001 11001010100100110101 1111 1110110110110111 00100100 0110 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.

29. 0000 0001 0011 0111 1111 1110 1100 1000 0000111100001111 0001111000011110 0011110000111100 0111100001111000 43214321 The horizontally rotated list of points looks very different, so you might think that you have a new, different snake. But, it ’ s really the same old snake rotated. 0000111100001111 0001111000011110 0011110000111100 0111100001111000 4321 Columns Rotate the 4 Column to the right hand side. 0000 0010 0110 1110 1111 1101 1001 0001 3214 Old Snake Rotated Snake

30. 0000 0001 0011 0111 1111 1110 1100 1000 0000111100001111 0001111000011110 0011110000111100 0111100001111000 43214321 In fact, if you exchange any column of a given snake with any other column of the same snake, 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. 0000111100001111 0001111000011110 0011110000111100 0111100001111000 4321 Columns Horizontally inter-mixed Columns 0000 0001 1001 1011 1111 1110 0110 0100 2431 Old Snake New Snake

31. There are other intriguing ways to name our snakes. My name is Joe Finklesnake III You can call me Joe

32. This picture shows colored linesets as well as points of a 4-space cube. 0000 1000010000100001110010100110100100110101 1111 1110110110110111 1234 4321 231413421423 3 421313124

33. 0000 0001 0011 01111 11111 1011 0010 00 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 1234 4321 231413421423 3 421313124 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

34. 0000 0001 0011 0111 1111 1110 1100 1000 0000 000011110000011110 000111100000111100 001111000001111000 011110000011110000 43214321 It turns out that the column-change naming convention is a more effective, efficient, easy method of naming snakes. And it highlights something we might not have seen otherwise. Columns Snake named by its points Snake named by column changes 1234123412341234

35. 0000 0001 0011 0111 1111 1110 1100 1000 0000 4321 Snake named by its points Snake named by column changes 1234123412341234 This snake appears to be made from two “ identical ” halves. 1 2 3 4 and 1 2 3 4 The column-change naming convention reveals structures within the snake that we did not expect to find.

36. 00000 1000001000001000001000001 1111011101110111011101111 11111 11010011101010110011001111110010110110010110101011 10100100100011001001000111100001100010101000100101 Now, we can name this 5-space snake two different ways. 00000 00010 00110 01110 11110 11010 11011 10011 10001 10101 11101 01101 01001 01000 Binary snake name Column-change name 2345314234531423453142345314

37. Symmetry, symmetry, everywhere and what a lot to think. 2345314 2345314 A 4-space cube 00000 1000001000001000001000001 1111011101110111011101111 11111 11010011101010110011001111110010110110010110101011 10100100100011001001000111100001100010101000100101 A 5-space Cube

38. 23453142345314 00000 1000001000001000001000001 1111011101110111011101111 11111 11010011101010110011001111110010110110010110101011 10100100100011001001000111100001100010101000100101 A 4-space cube 5-space cube This gives us a clue as to how we might construct N-space snakes from (N-1)-space snakes.

39. Just how big do these snakes get?

40. 0-space0 1-space1 2-space4 3-space6 4-space8 5-space14 6-space26 7-space48 We don ’ t know how big they are above 7-space. This Big

41. 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!

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

43. 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

44. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 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 or Brute Force and Ignorance method.

45. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 TailWagger starts at point 0000. It chooses one of four possible points. It then has three more choices, chooses one and checks to see if the snake has violated any rules.

46. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 If TailWagger chooses a point that violates a rule, it backtracks and tries one of the other points. 0010 would have to link with 0000 but the snake is still too small.

47. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 If no rules have been violated, it continues choosing new points. If all three choices violate a rule, it backtracks to the previous point and chooses another point there.

48. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 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.

49. 0000 1000010000100001 110010100110100100110101 1111 1110110110110111 Eventually, it backtracks all the way to the third node where the program stops. Do you see why it isn ’ t necessary to backtrack to the first point to try all of the possibilities there?

50. 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. XXX

51. Are these two snakes the same? 1 2 3 4 1 2 4 3 1 2 4 3 1 2 3 4 They are if the second snake is a vertical, horizontal, or intermixed rotation of the first snake.

52. Yes, the second snake is a rotation of the first. 1 2 3 4 1 2 4 3 1 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.

53. Are these two snakes the same? 1 2 3 4 1 2 4 3 3 2 1 3 4 2 1 4 They are if the second snake is a vertical, horizontal, or intermixed rotation of the first snake.

54. Yes, the second snake is a rotation of the first. 1 2 3 4 1 2 4 3 4 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.

55. Are these two snakes the same? 1 2 3 4 1 2 4 3 4 1 3 4 2 1 3 2 They are if the second snake is a vertical, horizontal, or intermixed rotation of the first snake.

56. 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

57. I promised you a third way to name snakes. My name is Joe Finklesnake III I ’ m from the class of 65

58. Snakes can be partially described by using the following trick. 23453142345314.....1......1......1......1. 2......2......2......2.......3..3...3..3....4...4..4...4...5......5... 17 7 1 occurs every 7th number 27 7 2 occurs every 7th number 33 4 3 4 3 occurs every 3rd, 4th, 3rd, 4th number 44 3 4 3 4 occurs every 4th, 3rd, 4th, 3rd number 57 7 5 occurs every 7th number

59. Because transformations or rotations of snakes are equivalent, the following two snakes are in the same class. The are the same snake. Snake 1 17 7 27 7 33 4 3 4 44 3 4 3 57 7 Snake 2 17 7 27 7 37 7 44 3 4 3 53 4 3 4 Snake 12 3 4 5 3 1 4 2 3 4 5 3 1 4 Snake 22 5 4 3 5 1 4 2 5 4 3 5 1 4

60. 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 cube to see whether they are forward, backward (vertical) and / or intermixed rotations of each other. Will the Real Unique Snakes Please Step forward ?

61. These are unique snakes for N < 7. 3-space1 2 3 1 2 3 4-space1 2 3 4 1 2 3 4 1 2 3 4 1 2 4 3 1 2 3 4 2 1 4 3 5-space1 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-space1 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

62. How long did it take to find every snake in 7-space? About 30 years. Why so long? 3-space3**6= 7.2x10**2= 729 4-space4**8= 6.5x10**4= 65536 5-space5**14= 6.1*10**9= 6103515625 6-space6**26= 1.7*10**20= 170581728179578208256 7-space7**48= 3.6x10**40= 36703368217294125441230211032033660188801

63. 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.