Braids, Cables, and Cells: An intersection of Mathematics, Computer Science, and Fiber Arts

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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 knitting and crochet, where braids are called “cables”. We will view some examples of braids and their mathematical representations in these media.

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Braids, Cables, and Cells: An intersection of Mathematics, Computer Science, and Fiber Arts

  1. 1. Braids, Cables, and Cells: An intersection of Mathematics, Computer Science, and Fiber Arts Joshua Holden Rose-Hulman Institute of Technology http://www.rose-hulman.edu/~holden Joshua Holden (RHIT) Braids, Cables, and Cells 1 / 17
  2. 2. “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 2 / 17
  3. 3. “Cables” in crochet Figure: Both: Designed and crocheted by Jody Euchner Joshua Holden (RHIT) Braids, Cables, and Cells 3 / 17
  4. 4. “Traveling eyelets” in knitted lace Figure: From Barbara Walker’s Charted Knitting Designs Joshua Holden (RHIT) Braids, Cables, and Cells 4 / 17
  5. 5. “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 5 / 17
  6. 6. 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 6 / 17
  7. 7. Example of a cellular automaton Grid is one-dimensional Two states, “white” and “black” Neighborhood includes self and one cell on each side “Rule 90” (Stephen Wolfram) Second dimension is used for “time” Joshua Holden (RHIT) Braids, Cables, and Cells 7 / 17
  8. 8. 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 8 / 17
  9. 9. 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 9 / 17
  10. 10. Example of a braid CA “Rule 47” (bottom-up, like knitting) Joshua Holden (RHIT) Braids, Cables, and Cells 10 / 17
  11. 11. Cables Figure: Left: Rule 0, Right: Rule 47 Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 17
  12. 12. Knotwork Figure: Left: Rule 0, Right: Rule 511 Joshua Holden (RHIT) Braids, Cables, and Cells 12 / 17
  13. 13. More knotwork Figure: Left: Rule 47, Right: Rule 448 Joshua Holden (RHIT) Braids, Cables, and Cells 13 / 17
  14. 14. 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 −1 n 2 − 1 crossings, the maximum repeat might reach n 2 rows. 2 Joshua Holden (RHIT) Braids, Cables, and Cells 14 / 17
  15. 15. 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 15 / 17
  16. 16. Example of the proof Figure: Rule 100 making a large repeat Joshua Holden (RHIT) Braids, Cables, and Cells 16 / 17
  17. 17. 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 17 / 17
  18. 18. Thanks for listening! Figure: Design by Ada Fenick, knitted by Lana Holden Joshua Holden (RHIT) Braids, Cables, and Cells 18 / 17

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