I am using the term **Wolfrule** to describe a collection of
geometrically-patterned art pieces that I am designing and making out of
wood. The geometrical patterns are derived from the operation of the
Wolfram Elementary Cellular Automaton. The term **Wolfrule** is a
contraction of the words in the phrase "**Wolf**ram Elementary
Cellular Automaton **Rule**." For a simple explanation of how things
work, check out Wolfrule Simplified.

I have a long-standing fascination with complex geometrical patterns. While studying European History at Plainview High School in the early 70's, I was introduced to the Alhambra in Spain, and was astonished by the intricate Moorish/Arabic/Islamic geometric patterns. Below are photos of the Alhambra. The first image is from Pattern in Islamic Art, an amazing web-site with thousands of photos of Islamic art, and the last three are from the Wikipedia article on the Alhambra. I have visited the Alhambra, and it was more amazing than I had imagined.

In the early 80's I read an article in **Scientific American**, by
Martin Gardner, entitled *"Extraordinary nonperiodic tiling that
enriches the theory of tiles"* (1977). This article introduced the
intriguing concept
of aperiodic
tilings – the most famous of which is
the Penrose
tiling. Below you see a photo of Roger Penrose followed by an
illustration of a Penrose tiling (both images borrowed from Wikipedia),
then a more complicated illustration of a Penrose tiling followed by a
photo of a Penrose tiling installation (attribution unknown for
both).

In 2002 or 2003, I saw the patterns generated by the Elementary
Cellular Automaton from Stephen Wolfram's amazing
book, *
A New Kind of Science*, and now I am compelled to do something
creative with these types of patterns. Some color images generated by
the Elementary Cellular Automaton, all borrowed
from
wolframscience.com, are shown below.

I have started designing and building, by hand, artistic models of interesting patterns I have found, using wood as my primary construction material. To help me discover interesting patterns, I wrote two computer programs that simulate the actions of the Wolfram Elementary Cellular Automaton. Both programs, presented here, are freely available for use by the public, so that others may experiment and create their own unique patterns.

This web site contains pages that showcase the patterns and photos of the Wolfrule Art. To observe the computations of an Elementary Cellular Automaton in action, watch the Wolfrule Animation, which demonstrates how one pattern can be computationally transformed into another, for a set of three different patterns (shown below).

For more specifics about Wolfrule, please continue to read the scientific explanations presented below. But before I get into the specifics, I'd like to say something about the names I've chosen for some of my art pieces. Iain M. Banks is one of my all-time favorite Science Fiction authors, I have all of his books, and he makes up the most delicious names for the artificially intelligent star-ships that inhabit his "Culture" civilization. I have borrowed some of these names for my art pieces, in tribute to Iain and his work. Iain, if you ever find out about Wolfrule and read this paragraph, contact me and I will gift you a Wolfrule art piece, for providing me with so much enjoyable reading time.

For a simple explanation of how things work, check out Wolfrule Simplified.

Wolfrule Art is based on patterns generated by an extremely primitive computing device, the cellular automaton [1], which, as postulated by Wolfram, may some day explain the basic principles of much of life and nature. These patterns are used in art constructions made from wood, a basic product of nature, producing art that uses computations to produce patterns that may form the basis of life itself. Both the background and the pattern are made from wood. The selection of contrasting woods, with variations in grain and color, provide a stark and striking exhibition of the geometric patterns computed by the cellular automaton, explained below.

The most primitive of computing devices is the cellular automaton (pl. cellular automata, abbrev. CA), first investigated by computing pioneer Jon van Neumann in the 1940s, for the purpose of providing insight into the logical requirements for machine self-replication [2]. Although primitive and almost trivial in concept, such CAs are universal computers, meaning they can perform any computation possible by modern computers. A simple description of a CA is that it sits in a cell on a grid, and changes state according to a set of rules by communicating with the CAs in its neighboring cells. Imagine checkers on a checker board having a conversation and changing color depending on their emotional state.

Starting in the 1980's Stephen Wolfram [3], inventor of Mathematica [4], did extensive research on the simplest of CAs, the 1-dimensional elementary cellular automaton [5]. Each cell has two states, 0 or 1 (white or black), and a cell's neighbors are only those to the right and left. Three cells form a neighborhood, and each cell can be either black or white, so there are 2^3=8 possible patterns for a neighborhood. The way these CAs compute is to decide, based on the pattern of each neighborhood, the value/color of the (middle) cell in the next row of the grid. There are 8 possible patterns, and each pattern can decide between two colors, so there are 2^8=256 possible distinct Rules, numbered from 0 to 255. Below is a graphical example of Rule 30 – three black cells produce a white one, as do two blacks and a white, and a white centered by blacks, but one black followed by two whites produces a black cell, and so forth. (For those of you who understand Base 2/Binary notation, notice that the cells on the bottom, with white interpreted as 0 and black as 1, represent the number 30 in Base 10. This is how Wolfram numbers his Rules.)

To understand how the Wolfram CA computes, imagine an empty grid. Color the cells in the top row any color you like, white or black. This selection of colors is called the "initial condition". Pretend the cells outside the edges of the top row of the grid are all white. Pick a Rule, and use it to compute the colors of all of the remaining cells in the grid, one row at a time. That's it. The simplest of computers, and probably the most boring, right? Below are some examples of grids (31 x 16) computed by some typical Rules, where the initial condition is one black cell centered in a row of white cells.

Rule 0 | Rule 4 | Rule 16 | Rule 37 | Rule 97 | Rule 178 | Rule 230 | Rule 255 |

As expected, boring and unexciting. But look what happens when Rule 30 is used on larger grid with the same initial condition:

Wow! Amazingly complex and unpredictable behavior, from such a simple conceptual device. Other Rules produce interesting patterns, some of which are shown below. The last row shows patterns generated when the initial condition is random. (Each Rule shown below is used in Wolfrule Art.)

Rule 30 | Rule 57 | Rule 73 | Rule 102 | Rule 105 | Rule 109 | Rule 151 | Rule 193 |

In his book *A New Kind of Science* [6], Stephen Wolfram
claims "it suddenly becomes possible to make progress on a remarkable
range of fundamental issues that have never successfully been
addressed by any of the existing sciences before." [7] He claims these
fundamental issues include the randomness found in the universe,
biological complexity, the nature of space and time, a "theory of
everything," and the scope and limitations of mathematics. Wolfram
even claims his insights can be used to tackle the ancient paradoxes
of free will and determinism, and the nature of intelligence. [8]
Wolfram has his critics and skeptics, and most of his claims remain
unproven, but still, there are amazing similarities between the
actions of the Wolfram CA and the patterns and behaviors found in
nature.

The term **Wolfrule** [9] comes from **Wolf**ram's Elementary
Cellular Automaton **Rule**. Wolfrule takes patterns generated by
Wolfram's CA, and incorporates these patterns into art made from wood.
Wolfrule Art must obey one fundamental stricture: each art piece must
contain a pattern that is computable. This stricture is self-enforced
by computing the patterns for the art using open-source software [10]
written by the artist and freely available for general use. The
stricture does not limit patterns to those resulting from individual
CA computations. There are many transformational meta-operations that
may be applied to patterns, including erasing cells, extracting
sub-images, flipping, mirroring (regular & 45°), quadrellation
(this term, and seemingly new word, courtesy of friend Leo Marcus),
rotating, stitching patterns together, etc. Below are examples of
each such transformation, making use of the original Rule 73 pattern
from which the "W" in the **Wolfrule Logo** was
generated.

Erase | Extract | Flip | Mirror | Mirror 45° | Quadrellate | Rotate | Stitch | ||||

| |||||||||||

**Note:** Around 2000, Matthew Cook published a proof of a 1985
conjecture by Stephen Wolfram by proving that Rule 110 is Turing
complete, i.e., capable of universal computation. [11] So even the
Wolfrule CAs, perhaps the most primitive of all computational devices,
are capable of computing the same results as modern computers. Rule
193, a "Class 4" rule like Rule 110, is used in Wolfrule #28.

[1] http://en.wikipedia.org/wiki/Cellular_automata

[2] http://en.wikipedia.org/wiki/Von_Neumann_cellular_automatonp

[3] http://en.wikipedia.org/wiki/Stephen_Wolfram

[4] http://www.wolfram.com/mathematica/

[5] http://en.wikipedia.org/wiki/Cellular_automata#History

[6] Wolfram, Stephen. 2002. A New Kind of Science. Wolfram Media, Inc. (http://www.wolframscience.com/)

[7] Wolfram, Stephen. 2002. A New Kind of Science. Wolfram Media, Inc., 1.

[8] http://en.wikipedia.org/wiki/Rule_110

[9] http://wolfrule.com/

[10] http://wolfrule.com/free.html

[11] Naiditch, David, 2003, Divine Secrets Of the Ya-Ya Universe (http://www.theeway.com/skepticc/wolfram.htm)

**Wolfrule Note**: On March 15, 2010 (before the existence
of this web site), a Google search of the term "wolfrule" returned no
hits involving this term; all hits contained the individual words "wolf"
and "rule" in close proximity.

**Background Note**: The background image on this page was
extracted from a GIF generated by a computation performed by
the **Wolfrule Perl Calculator**, using Rule
30, where the initial condition was one black cell centered in a row of
white cells, with a grid size of 600 by 600, and a cell size of 3
pixels. The command used to perform the computation was "wolfrule.pl -g
-c3faintgray 600 600 30" (the color "faintgray" is RGB[230,230,230] or
#E6E6E6), after which I used a photo image editing program to crop a
1600x1000 pixel selection from the lower portion of the generated image.
On various computers (PC/Windows, PC/Linux, Mac/OS X) under various
environments (Windows Cmd, Cygwin, Darwin Terminal, Fedora 13) and
various versions of Perl (v5.6.1, v5.10.0, v5.10.1), the compute time
for this command has ranged from 5 to 110 seconds, and the image
creation time from 4 to 38 seconds (the Linux machine, a dual-boot
Windows PC, was hands-down the fastest). For grids of size 100 cells on
side, the compute time is typically less than a second, size 200 less
than 5 seconds, size 300 about 10 seconds, etc. Of course,
YMMV.

Jeffrey V. Cook

12 June 2011

12 June 2011