This book shows how to turn computer generated number sequences
into intricate visual patterns. The sequences are strings of the
binary numbers 0 and 1 which are translated into drawing
instructions to produce beautiful patterns. These patterns provide
a glimpse of the hidden platonic world of mathematics.
The book starts with Christoffel words and Sturmian sequences
which are derived from the continued fraction expansion of rational
and irrational numbers. How this is done is explained very clearly
in the book and very little mathematical background is required
from the reader.
The book moves on to Automatic sequences such as the Thue-Morse
and Rudin-Shapiro sequences which are various ways of calculating
digital roots of the integers. The first part of the book ends with
sequences generated by folding paper.
Translating a sequence into drawing instructions is done using a
finite automaton. This is a very general method for translating
sequences that allows the same sequence to produce many different
patterns. No prior experience with finite automata is necessary.
All the background needed is explained in the book.
The second part of the book is devoted to L-systems which is
another way of producing a string of drawing instructions. Here the
strings are produced by an iterative symbol substitution process.
The images produced often have a self similar fractal structure. It
is possible to create many images that resemble plants. The book
shows how to use an automaton and context free grammars to
systematically look at all L-systems of a particular type.
All software used to create the sequences and images in the book
are free for readers to download from the book's website at: http:
//www.abrazol.com/books/patterngen/
The software consists of small programs written in the C
programming language that can be run on all major operating
systems. Inside the book are 327 images serving as inspiration for
the kinds of images you can create. There are an infinite variety
of images you can generate using the software that comes with this
book, providing a computational image generation lab.
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