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 drawing, sculpture, knitting, crochet, and macramé, and we will touch on some of these.
Braids, Cables, and Cells I: An Interesting Intersection of Mathematics, Computer Science, and Art
1. 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
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2. Braids and Cables Graphic Arts
“Knotwork” in graphic arts
Figure: Left: by A. Reed Mihaloew, Right: by Christian Mercat
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3. Braids and Cables Graphic Arts
“Knotwork” in historical manuscripts
Figure: Details from the “Book of Kells”, c. 800 CE
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4. 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
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5. Braids and Cables Fiber Arts
“Cables” in crochet
Figure: Both: Designed and crocheted by Jodi Euchner
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6. Braids and Cables Fiber Arts
“Traveling eyelets” in knitted lace
Figure: From Barbara Walker’s Charted Knitting Designs
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7. Braids and Cables Fiber Arts
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)
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8. 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)
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9. Braids and Cables Group Theory
Multiplying braids
You can multiply two braids by stacking them and then simplifying
× =
Figure: Multiplying braids (Wikipedia)
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10. 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
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11. Cellular Automata Rules and Examples
“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.
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12. Cellular Automata Rules and Examples
Example: A “Pulsar”
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13. Cellular Automata Rules and Examples
Example: A “Pulsar”
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14. Cellular Automata Rules and Examples
Example: A “Pulsar”
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15. Cellular Automata Rules and Examples
Example: A “Pulsar”
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16. Cellular Automata Rules and Examples
Example: A “Pulsar”
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17. Cellular Automata Rules and Examples
Example: A “Pulsar”
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18. Cellular Automata Rules and Examples
Example: A “Pulsar”
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19. Cellular Automata Rules and Examples
“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”
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20. Cellular Automata Rules and Examples
Example: “Rule 90”
Second dimension is used for “time”
´
Produces the Sierpinski triangle fractal
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21. 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.
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23. Cellular Automata Complex behavior
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”.)
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24. Cellular Automata Complex behavior
Rule 110 on a Single Cell Input
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25. Cellular Automata Complex behavior
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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26. Cellular Automata Complex behavior
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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27. Cellular Automata Complex behavior
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)
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29. 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
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30. 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?
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31. Braids and CAs Examples
Example of a braid CA
“Rule 47” (bottom-up, like knitting)
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32. Braids and CAs Examples
Cables
Figure: Left: Rule 0, Right: Rule 47
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33. Braids and CAs Examples
Knotwork
Figure: Left: Rule 0, Right: Rule 511
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34. Braids and CAs Examples
More knotwork
Figure: Left: Rule 47, Right: Rule 448
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35. 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
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36. Braids and CAs Questions and Results
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.
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37. Braids and CAs Questions and Results
Example of the proof
Figure: Rule 100 making a large repeat
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38. Braids and CAs Questions and Results
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
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39. Braids and CAs Questions and Results
Thanks for listening!
Figure: Design by Ada Fenick, knitted by Lana Holden
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