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.

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. “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. “Cables” in crochet
Figure: Both: Designed and crocheted by Jody Euchner
Joshua Holden (RHIT) Braids, Cables, and Cells 3 / 17

4. “Traveling eyelets” in knitted lace
Figure: From Barbara Walker’s Charted Knitting Designs
Joshua Holden (RHIT) Braids, Cables, and Cells 4 / 17

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. 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. 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. 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. 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. Example of a braid CA
“Rule 47” (bottom-up, like knitting)
Joshua Holden (RHIT) Braids, Cables, and Cells 10 / 17

11. Cables
Figure: Left: Rule 0, Right: Rule 47
Joshua Holden (RHIT) Braids, Cables, and Cells 11 / 17

12. Knotwork
Figure: Left: Rule 0, Right: Rule 511
Joshua Holden (RHIT) Braids, Cables, and Cells 12 / 17

13. More knotwork
Figure: Left: Rule 47, Right: Rule 448
Joshua Holden (RHIT) Braids, Cables, and Cells 13 / 17

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. 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. Example of the proof
Figure: Rule 100 making a large repeat
Joshua Holden (RHIT) Braids, Cables, and Cells 16 / 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. Thanks for listening!
Figure: Design by Ada Fenick, knitted by Lana Holden
Joshua Holden (RHIT) Braids, Cables, and Cells 18 / 17