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Braids, Cables, and Cells I: An Interesting Intersection of Mathematics, Computer Science, and Art

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The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension.

Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and macramé, and we will touch on some of these.

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• Reed: Thanks for posting the sample so I could use it! Yes, Rule 47 does seem to be one of the most artistic --- most of the rules only a mathematician could love. :)

Modeling knots that 'bounce' off the top and bottom of the pattern, or otherwise do anything besides basically going from the bottom to the top of the design, seems like it would require a very different mathematical idea than the one I've been using. Possibly a 2-D cellular automaton would be the way to handle it, but I haven't thought about that yet. Maybe in the future....

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Braids, Cables, and Cells I: An Interesting Intersection of Mathematics, Computer Science, and Art

1. 1. Braids, Cables, and Cells: An Interesting Intersection of Mathematics, Computer Science, and Art Joshua Holden Rose-Hulman Institute of Technology http://www.rose-hulman.edu/~holden Joshua Holden (RHIT) Braids, Cables, and Cells 1 / 29
2. 2. Braids and Cables Graphic Arts “Knotwork” in graphic arts Figure: Left: by A. Reed Mihaloew, Right: by Christian Mercat Joshua Holden (RHIT) Braids, Cables, and Cells 2 / 29
3. 3. Braids and Cables Graphic Arts “Knotwork” in historical manuscripts Figure: Details from the “Book of Kells”, c. 800 CE Joshua Holden (RHIT) Braids, Cables, and Cells 3 / 29
4. 4. Braids and Cables Fiber Arts “Cables” in knitting Figure: Left: Design by Barbara McIntire, knitted by Lana Holden Figure: Right: Design by Betty Salpekar, knitted by Lana Holden Joshua Holden (RHIT) Braids, Cables, and Cells 4 / 29
5. 5. Braids and Cables Fiber Arts “Cables” in crochet Figure: Both: Designed and crocheted by Jodi Euchner Joshua Holden (RHIT) Braids, Cables, and Cells 5 / 29
6. 6. Braids and Cables Fiber Arts “Traveling eyelets” in knitted lace Figure: From Barbara Walker’s Charted Knitting Designs Joshua Holden (RHIT) Braids, Cables, and Cells 6 / 29
7. 7. Braids and Cables Group Theory “Braids” in group theory Two braids which are the same except for “pulling the strands” are considered equal All strands are required to move from bottom to top Figure: Two equal braids (Wikipedia) Joshua Holden (RHIT) Braids, Cables, and Cells 7 / 29
8. 8. Braids and Cables Group Theory Multiplying braids You can multiply two braids by stacking them and then simplifying × = Figure: Multiplying braids (Wikipedia) Joshua Holden (RHIT) Braids, Cables, and Cells 8 / 29
9. 9. Cellular Automata Rules and Examples Cellular automata Finite number of cells in a regular grid Finite number of states that a cell can be in Each cell has a well-deﬁned ﬁnite neighborhood Time moves in discrete steps State of each cell at time t is determined by the states of its neighbors at time t − 1 Each cell uses the same rule Joshua Holden (RHIT) Braids, Cables, and Cells 9 / 29
10. 10. Cellular Automata Rules and Examples “The Game of Life” Invented by John Conway Grid is two-dimensional Two states, “live” and “dead” Neighborhood is the eight cells which are directly horizontally, vertically, or diagonally adjacent Any live cell with two or three live neighbors stays live. Any other live cell dies. Any dead cell with exactly three live neighbors becomes a live cell. Any other dead cell stays dead. Joshua Holden (RHIT) Braids, Cables, and Cells 10 / 29
11. 11. Cellular Automata Rules and Examples Example: A “Pulsar” Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 29
12. 12. Cellular Automata Rules and Examples Example: A “Pulsar” Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 29
13. 13. Cellular Automata Rules and Examples Example: A “Pulsar” Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 29
14. 14. Cellular Automata Rules and Examples Example: A “Pulsar” Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 29
15. 15. Cellular Automata Rules and Examples Example: A “Pulsar” Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 29
16. 16. Cellular Automata Rules and Examples Example: A “Pulsar” Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 29
17. 17. Cellular Automata Rules and Examples Example: A “Pulsar” Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 29
18. 18. Cellular Automata Rules and Examples “Elementary” Cellular Automata Popularized by Stephen Wolfram (A New Kind of Science) Grid is one-dimensional Two states, “white” and “black” Neighborhood includes self and one cell on each side Example: “Rule 30” Joshua Holden (RHIT) Braids, Cables, and Cells 12 / 29
19. 19. Cellular Automata Rules and Examples Example: “Rule 90” Second dimension is used for “time” ´ Produces the Sierpinski triangle fractal Joshua Holden (RHIT) Braids, Cables, and Cells 13 / 29
20. 20. Cellular Automata Complex behavior Aperiodic behavior Conjecture (Wolfram, 1984) The sequence of colors produced by the cell at the center of Rule 30 is aperiodic. This sequence is used by the pseudorandom number generator in the program Mathematica. The center and right portions of Rule 30 appear to have some of the characteristics of “chaotic” systems. Theorem (Jen, 1986 and 1990) (a) At most one cell of Rule 30 produces a periodic sequence of colors. (b) The sequence of color pairs produced by any two adjacent cells of Rule 30 is aperiodic. Joshua Holden (RHIT) Braids, Cables, and Cells 14 / 29
21. 21. Cellular Automata Complex behavior Rule 30 Joshua Holden (RHIT) Braids, Cables, and Cells 15 / 29
22. 22. Cellular Automata Complex behavior Universality Theorem (Cook, 1994+) Rule 110 can be used to simulate any Turing machine. This is important because of the widely accepted: Church-Turing Thesis Anything that can be computed by an algorithm can be computed by some Turing machine. And for complexity geeks: Theorem (Neary and Woods, 2006) Rule 110 can be used to simulate any polynomial time Turing machine in polynomial time. (I.e., it is “P-complete”.) Joshua Holden (RHIT) Braids, Cables, and Cells 16 / 29
23. 23. Cellular Automata Complex behavior Rule 110 on a Single Cell Input Joshua Holden (RHIT) Braids, Cables, and Cells 17 / 29
24. 24. Cellular Automata Complex behavior Rule 110 Performing a Computation Joshua Holden (RHIT) Braids, Cables, and Cells 18 / 29
25. 25. Braids and CAs Motivation CAs and Fiber Arts Figure: Left: Designed and crocheted by Jake Wildstrom Figure: Right: Knitted by Pamela Upright, after Debbie New Joshua Holden (RHIT) Braids, Cables, and Cells 19 / 29
26. 26. Braids and CAs Model Representing braids using CAs Five types of cells: Neighborhood only cells on either side Restricted rule set: Must “follow lines” Only choice is direction of crossings 29 different rules possible Edge conditions? Inﬁnite? Special kind of edge cell? Cylindrical? Reﬂection around edge of cells? Reﬂection around center of cells? Joshua Holden (RHIT) Braids, Cables, and Cells 20 / 29
27. 27. Braids and CAs Examples Example of a braid CA “Rule 47” (bottom-up, like knitting) Joshua Holden (RHIT) Braids, Cables, and Cells 21 / 29
28. 28. Braids and CAs Examples Cables Figure: Left: Rule 0, Right: Rule 47 Joshua Holden (RHIT) Braids, Cables, and Cells 22 / 29
29. 29. Braids and CAs Examples Knotwork Figure: Left: Rule 0, Right: Rule 511 Joshua Holden (RHIT) Braids, Cables, and Cells 23 / 29
30. 30. Braids and CAs Examples More knotwork Figure: Left: Rule 47, Right: Rule 448 Joshua Holden (RHIT) Braids, Cables, and Cells 24 / 29
31. 31. Braids and CAs Questions and Results Repeats: Upper bound Since the width is ﬁnite, the pattern must eventually repeat. Question For a given width, how long can a repeat be? Proposition n For a given (even) width n, no repeat can be longer than n 2 2 −1 rows. Proof. After n rows, all of the strands have returned to their original positions. The only question is which strand of each crossing is on top. If there n are n crossings the maximum repeat is ≤ 2 2 rows, but if there are 2 n n −1 2 − 1 crossings, the maximum repeat might reach n 2 rows. 2 Joshua Holden (RHIT) Braids, Cables, and Cells 25 / 29
32. 32. Braids and CAs Questions and Results Repeats: Lower bound Proposition For a given (even) n ≥ 2k , the maximum repeat is at least lcm(2k , n) rows long. Proof. Consider the starting row with one single strand and n − 1 crosses, e.g.: . Rule 100 acts on this with a repeat (modulo cyclic shift) which is a multiple of 2k if n > 2k . Remark For n ≤ 10, this is sharp. For large n, neither this upper bound nor this lower bound seems especially likely to be sharp. Joshua Holden (RHIT) Braids, Cables, and Cells 26 / 29
33. 33. Braids and CAs Questions and Results Example of the proof Figure: Rule 100 making a large repeat Joshua Holden (RHIT) Braids, Cables, and Cells 27 / 29
34. 34. Braids and CAs Questions and Results Future work More work on repeats Properly implement reﬂection Add cell itself to neighborhood? Add vertical “strands” 16 types of cells 29 681 different rules(?) Which braids can be represented? (In the sense of braid groups) Which rules are “reversible”? Joshua Holden (RHIT) Braids, Cables, and Cells 28 / 29
35. 35. Braids and CAs Questions and Results Thanks for listening! Figure: Design by Ada Fenick, knitted by Lana Holden Joshua Holden (RHIT) Braids, Cables, and Cells 29 / 29