Enzymind Labs

Introduction

The subject of these pages of images and explanatory text are simple discrete recursive systems, where the output of the system is used as feedback to the input of the system. The following two related sequences of symbols are generated by two simple systems, and are prototypical of the fascinating properties of the patterns that are generated:

the Period doubling, or First Feigenbaum symbolic, sequence:
a b a a a b a b a b a a a b a a a b a a a b a b a b a a a b a b ...
or

and the Thue-Morse sequence:
a b b a b a a b b a a b a b b a b a a b a b b a a b b a b a a b ...
or

These sequences are simple in the sense that their description is very short, in the same way that constant or repeating sequences can be briefly described ("append a set of symbols, repeat ad infinitum"). The symbols used to represent the sequences is irrelevant -- any two objects can replace the a's and b's, or the black and white squares.

Simple sequences make pleasing and useful patterns. Human hearing is finely tuned to recognize simple repeating patterns in sound, both pitch and harmony, as well as to appreciate variations from strict regularity. Human vision is similarly sensitive to two-dimensional patterns in images.

This is a visual exploration of variations of the simplest recurrent patterns, derived from the simplest sequences, formed by a particular class of symbol replacement systems. The expectation that simple and redundant patterns are not interesting is strikingly incorrect. To demonstrate this, a range of patterns will be presented along with an explanation of how they are created. Some short formal notation (mathematical description) will be provided, primarily to help make sense of referenced links and supplementary material.

Much of this material can be related directly to a single diagram, Simplest 2-D block replacement L-systems. This is a table of recurrent patterns with respect to their production rules and to each other. The first sections illustrate one-dimensional replacement rules, the recurrent sequences they generate, and how these are related to 2-D recurrent patterns. The later sections illustrate the properties and variations of the 2-D recurrent patterns.

The material is a work in progress, slow progress. Some of the meatiest sections are sub-pages of:

Simplest 2-D reccurent systems

2x2 symmetric L-systems

Thue-Morse sequence tilings