Enzymind Labs

Mark Dow

Geek art

Simple recursive systems and fractal patterns

 

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 pair of related sequences of symbols are generated by two of the most simple systems, and are prototypical of the fascinating properties of the 2-D patterns that can be 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 ("do an operation, 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.

    These pages are a visual exploration of variations of the simplest recurrent patterns, derived from the simplest sequences, formed by a particular class of symbol replacement systems (recursive block 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.

    Most of this material can be related directly to a single diagram, Simplest 2-D recurrent patterns. This is a table of recurrent patterns that shows the patterns in relation to their production rules and to each other. The early sections illustrate one-dimensional replacement rules and 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.

 

Motivation

"Like the artist who paints a canvas or the musician that plays a song, it is our purpose to create in an interesting and entertaining fashion." Anonymous comment on Michael Sharp's Kabbalistic cosmology.

"Invention is discernment, choice." Henri Poincaré in his essay "Mathematical creation"

[To Do: Reference a local copy of this image and others.]

    Recursive systems, and the resulting sequences, have curious and baffling qualities. The purpose of writing about and drawing them is primarily personal -- I want to organize some seemingly disparate ideas and observations. What keeps pulling me back to the core, the simplest possible systems, is the extraodinary zoo of forms derived from a few rules acting on a few symbols. The patterns are beautiful.

    In view of their simplicity, like the simplicity of a checkerboard, I am surprised that I haven't seen examples of many of these patterns. It is not hard to find examples of all periodic patterns, with symmetries of the wallpaper group, but most of the simplest recursive patterns are unexplored. I would think that nature would make use of these forms and humans would explore them, but I see only a small fraction of the possible patterns in natural systems or cultural artifacts (technology, art, etc.). Are these patterns hard to recognize, or are simple recurrent systems not common?

    There are several ways of thinking about how the patterns are generated, how they are related to each other, and how they are related to other topics. Some perspectives lead to number theoretical topics, some to physics, and some to analysis. This scope is appealing, and has given me an excuse to explore, and better understand, these topics.

    I know only a bit about the properties of the sequences and patterns, and I present only observations, not theorems, about them. Any novelty is in the presentation or juxtaposition of thoughts.

 

[To Do: How can a 1-D function be decomposed into Walsh functions, or the succesive generations of the periodic system? Do the succesive generations of the Thue-Morse and/or Period doubling sequence form a complete and unique basis set? ]

 

About the representation and range of systems considered

    The systems considered are, like numbers, abstract objects -- they exist independent of how they are represented. But, like numbers, it is useful to have representation as an aid to thinking about the systems. There are several elegant and sparse ways to write a description of the system using standard mathematical terminology and symbols.

[To Do: Give a couple examples of mathematical descriptions of the Thue-Morse system.]

    I tend to think in geometric and pictorial modes, so that's how most of the material will be presented. [To Do: About my graphical algorithmic and pattern representations.]

[To Do: About the number of symbols, orthogonal and rectilinear constraints.]

 

About scale symmetries

    Recursive systems generate patterns with scale symmetries -- whole patterns which are replicated within the pattern itself. When the output of a system is used as a component of the input, this is bound to happen. [To Do: Logarithmic spiral example.]

    Block replacement systems are ideal systems for exploring the range of ways that patterns can have scale symmetries. [See Symmetries of block replacement L-systems.]