Braids, Cables, and Cells II: Representing Art and Craft with Mathematics and Computer Science

Braids, Cables, and Cells
Representing Art and Craft with Mathematics and Computer
Science

Joshua Holden

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

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“Knotwork” has been used in visual arts for many
centuries.

Left: Detail from Roman mosaic at Woodchester, c. 325 CE

Right: Detail from the “Book of Kells”, c. 800 CE

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“Knotwork” has been used in visual arts for many
centuries.

Left: by A. Reed Mihaloew, Right: by Christian Mercat

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In knitting, raised knot-like designs are known as
“cables”.

Left: Design by Meredith Morioka, knitted by Lana Holden

Right: Design by Julie Levy, knitted by Lana Holden

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“Cables” can also be done in crochet.

Both: Designed and crocheted by Lisa Naskrent
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A somewhat similar effect is given by “traveling
eyelets” in knitted lace.

From Barbara Walker’s Charted Knitting Designs

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And of course, there are many actual weaving
patterns.

Left: 2/2 twill weave, woven by Sarah, a.k.a. Aranel

Right: “Noonday Sun” pattern, woven by Peggy Brennan
(Cherokee Nation)

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Vandermonde was interested in the mathematical
study of knots and braids.

From “Remarques sur les problèmes de situation”, 1771

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So was Gauss.

From Page 283 of Gauss’s Handbuch 7, c. 1825?

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Today, braids are studied from the perspectives of
topology and group theory.

Two equal braids (Wikipedia)

Two braids which are the same except for “pulling the
strands” are considered equal.
All strands are required to move from bottom to top.

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You can make braids into a group by “multiplying”
them.

× =

Multiplying braids (Wikipedia)

You multiply two braids by stacking them and then
simplifying.

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A Cellular Automaton is a mathematical construct
modeling a system evolving in time.
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

The “von Neumann neighborhood”
(Wikibooks)

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“The Game of Life” is an example you might know.
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.
(Figures by Paul Callahan, from www.math.com)
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Example: A “Pulsar”

(Wikipedia)

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Another well-known class of automata are the
“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

“Rule 30” (Mathworld)

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Example: “Rule 90”
Second dimension is used for “time”
´
Produces the Sierpinski triangle fractal
An “additive” rule

(Mathworld)
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Cellular automata can exhibit 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.

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Rule 30

(Mathworld)
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Cellular automata are also “computationally universal”.

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”.)

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Rule 110 on a Single Cell Input

(Mathworld)
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How is this possible?
1. Use Rule 110 to simulate a “cyclic tag system”.
A cyclic tag system has:
A data string
A cyclic list of “production rules”
To perform a computation:
If the first data symbol is 1, add the production rule to the
end of the data string. If the first data symbol is 0 do
nothing.
Delete the first data symbol.
Move to the next production rule.
Repeat until the data string is empty.

2. Show that any Turing machine can be simulated by a cyclic
tag system.

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Cyclic Tag System Example

Production rules Data string

010 11001
000 1001010
1111 001010000
010 01010000
000 1010000
1111 010000000
010 10000000
.
. ..
. .

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To simulate a cyclic tag system with Rule 110, you
need:
a representation of the data string (stationary)
a representation of the production rules (left-moving)
“clock pulses” (right-moving)

(Wikipedia)
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Rule 110 Performing (Part of) a Computation

(Wikipedia)
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CAs have been used in ﬁber arts before.

Left: Designed and crocheted by Jake Wildstrom
Right: Knitted by Pamela Upright, after Debbie New
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Each of our cells will store 4 bits of information in 8
states.
upright slanted

no strands

left only

right only

both

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The neighborhood will be a “brick wall” neighborhood.

(Time moves from bottom to top, like a knitting pattern.)

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The CA rule system can actually be thought of as four
simpler CAs. The ﬁrst two just control whether strands
are present or not.
no left left

no right

right
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The CA rule system can actually be thought of as four
simpler CAs. The ﬁrst two just control whether strands
are present or not.
no left left

no right

right
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The third CA controls whether the strands are upright
or diagonal, speciﬁed by a numbered rule.

“Turning Rule 39”
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And the fourth CA controls which strand is on top if the
strands cross, also speciﬁed by a numbered rule.

“Crossing Rule 39”
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There are several possible choices for what to do at
the edges of the grid.

Make the grid inﬁnite?
Have a special kind of state for edge cells?
Make the grid cylindrical? (“Periodic boundary conditions”)
Reﬂect cells at the edges? (Where to put the axis?)

I have so far only implemented the cylindrical case.

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The rules can produce fractal patterns, . . .

Rules 68 and 0 33 / 43

. . . weaving patterns, . . .

Left: Rules 0 and 47, Right: Rules 0 and 448

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. . . traditional braids, . . .

’
Left: Wikipedia, Right: Rules 333 and 39
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. . . slightly less traditional braids, . . .

Left: backstrapweaving.wordpress.com
Right: Rules 333 and 99
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. . . and other sorts of “cable” patterns.

Left: Rules 47 and 0, Right: Rules 201 and 39
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If all strands are present and only one rule is active,
previously known results on “elementary” CA’s apply.

Rules 68 and 0 give the same result as Wolfram’s Rule 90
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Since the width is ﬁnite, the pattern must eventually
repeat. For a given width, how long can a repeat be?
Proposition
Assume only the crossing rule is active. For a given width m, no
repeat can be longer than m2m − 2m rows.

Proof.

After 2m rows, all of the strands
have returned to their original
positions. The only question is
which strand of each crossing
is on top.

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Proposition

Proof.
If there are m crossings then
there are 2m possible
arrangements of the crossings
but only 2 different ways the
row can be shifted. So the
maximum repeat is the lcm of a
number ≤ 2m and a
number ≤ 2.

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Proposition

Proof.

If there are m − 1 crossings,
then there are 2m−1 possible
arrangements but 2m different
shifts, so the maximum repeat
is the lcm of a number ≤ 2m−1
and a number ≤ 2m.

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Proposition
Assume only the crossing rule is active. For a given width
m ≥ 2k , the maximum repeat is at least lcm(2k +1 , 2m) rows
long.

Proof.

Consider the starting row with
one single strand and m − 1
crossings. Crossing Rule 100
(which is additive) acts on this
with a repeat (modulo cyclic
shift) of 2k +1 if m > 2k . The
cyclic shift gives the 2m.

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Proposition
Assume only the crossing rule is active. For a given width
m ≥ 2k , the maximum repeat is at least lcm(2k +1 , 2m) rows
long.

Remark
For m ≤ 5, this is sharp. For
m = 23, m crossings and
Crossing Rule 257 (which is
also additive) does better.

For large m, neither the upper
bound above nor this lower
bound seems especially likely
to be sharp.

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If only one or two strands are present then the
maximum length of a repeat can be determined.

Proposition
Assume only one or two strands are
present. For a given width m ≤ 5, the
maximum repeat is (2m)(2m + 1) rows long.

Proof.
This is achieved by Turning Rule 97 and two
strands.

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There is much future work to be done.

If all strands are present and both rules are active, then we
have two “elementary” CA’s where one can “overwrite” the
other.
The length of a maximum repeat in other cases is open.
What is the computational complexity of predicting things
that the CA might do?
More work can be done with different edge conditions.
Which braids can be represented? (In the sense of braid
groups)
Which rules are “reversible”?
Two-dimensional grids with time as the third dimension

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Thanks for listening!

“Barolo”, designed and knitted by Lana Holden

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Braids, Cables, and Cells II: Representing Art and Craft with Mathematics and Computer Science

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