braids, cables, and cells: an intersection of mathematics, computer science, and fiber arts

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

Joshua Holden

Rose-Hulman Institute of Technologyhttp://www.rose-hulman.edu/~holden

Joshua Holden (RHIT) Braids, Cables, and Cells 1 / 17

http://www.rose-hulman.edu/~holden

“Cables” in knitting

Figure: Left: Design by Barbara McIntire, knitted by Lana Holden

Figure: Right: Design by Betty Salpekar, knitted by Lana Holden


“Cables” in crochet

Figure: Both: Designed and crocheted by Jody Euchner


“Traveling eyelets” in knitted lace

Figure: From Barbara Walker’s Charted Knitting Designs


“Braids” in group theory

Two braids which are the same except for “pulling the strands” areconsidered equalAll strands are required to move from bottom to top

Figure: Two equal braids (Wikipedia)


Cellular automata

Finite number of cells in a regular gridFinite number of states that a cell can be inEach cell has a well-defined finite neighborhoodTime moves in discrete stepsState of each cell at time t is determined by the states of itsneighbors at time t − 1Each cell uses the same rule


Example of a cellular automaton

Grid is one-dimensionalTwo states, “white” and “black”Neighborhood includes self and one cell on each side“Rule 90” (Stephen Wolfram)Second dimension is used for “time”


CAs and Fiber Arts

Figure: Left: Designed and crocheted by Jake Wildstrom

Figure: Right: Knitted by Pamela Upright, after Debbie New


Representing braids using CAs

Five types of cells:Neighborhood only cells on either sideRestricted rule set:

Must “follow lines”Only choice is direction of crossings29 different rules possible

Edge conditions?

Infinite?Special kind of edge cell?Cylindrical?Reflection around edge of cells?Reflection around center of cells?


Example of a braid CA

“Rule 47” (bottom-up, like knitting)


Cables

Figure: Left: Rule 0, Right: Rule 47


Knotwork



More knotwork



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

For a given (even) width n, no repeat can be longer than n 2n2−1 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 thereare n

2 crossings the maximum repeat is ≤ 2n2 rows, but if there are

n2 − 1 crossings, the maximum repeat might reach n 2

n2−1 rows.


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 arepeat (modulo cyclic shift) which is a multiple of 2k if n > 2k .

RemarkFor n ≤ 10, this is sharp.

For large n, neither this upper bound nor this lower bound seemsespecially likely to be sharp.


Example of the proof

Figure: Rule 100 making a large repeat


Future work

More work on repeatsProperly implement reflectionAdd cell itself to neighborhood?Add vertical “strands”

16 types of cells29681 different rules(?)

Which braids can be represented? (In the sense of braid groups)Which rules are “reversible”?


Thanks for listening!

Figure: Design by Ada Fenick, knitted by Lana Holden


braids, cables, and cells: an intersection of mathematics, computer science, and fiber arts

Education

timejoshua holden rhit

rulejoshua holden rhit

edge of cells

center of cells

n rows

width n

large n

proposition n