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

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 / 33

Braids and Cables Graphic Arts

“Knotwork” in graphic arts

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


Braids and Cables Graphic Arts

“Knotwork” in historical manuscripts

Figure: Details from the “Book of Kells”, c. 800 CE


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



“Cables” in crochet

Figure: Both: Designed and crocheted by Jodi Euchner



“Traveling eyelets” in knitted lace

Figure: From Barbara Walker’s Charted Knitting Designs



Weaving patterns

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

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


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)


Braids and Cables Group Theory

Multiplying braids

You can multiply two braids by stacking them and then simplifying

× =

Figure: Multiplying braids (Wikipedia)


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



“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.


Example: A “Pulsar”



“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”



Example: “Rule 90”

Second dimension is used for “time”
´
Produces the Sierpinski triangle fractal


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.


Rule 30



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”.)



Rule 110 on a Single Cell Input



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.


Cyclic Tag System Example

Production rules Data string

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



Simulating a Cyclic Tag System with Rule 110
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)



Rule 110 Performing (Part of) a Computation


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


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 state for edge cells?
Cylindrical?
Reflection around edge of cells?
Reflection around center of cells?


Braids and CAs Examples

Example of a braid CA
“Rule 47” (bottom-up, like knitting)



Cables

Figure: Left: Rule 0, Right: Rule 47



Knotwork




More knotwork



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



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.


Example of the proof

Figure: Rule 100 making a large repeat



Future work

More work on repeats
“Properly” implement reﬂection
Topology which changes over time
Add cell itself to neighborhood?
Add curved strands
7 (or 8?) types of cells
“A few” more different rules
Add vertical “strands”
16 types of cells
Many more different rules
Which braids can be represented? (In the sense of braid groups)
Which rules are “reversible”?
Two-dimensional grids with time as the third dimension



Thanks for listening!

Figure: Design by Ada Fenick, knitted by Lana Holden


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

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Mehr von Joshua Holden

Mehr von Joshua Holden (14)

Kürzlich hochgeladen

Kürzlich hochgeladen (20)

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