Exploring Scale Symmetry

Scale symmetry, as the author describes it, is a branch of geometry that studies shapes that look the same under at least one shape-preserving transformation. Shape-preserving is defined as a transformation under which small spheres remain spheres and includes conformal transformations and reflections, as well as scaling. 

 

This book is a visual exploration of the landscape of shapes in two and three dimensions that appear when one considers the multitude of structures that exist in the natural world or can be constructed as scale-symmetric. The author is a computer scientist now working in robotics who has a history of working with 3D computer games and animation. He intends his book for mathematicians, programmers, artists and hobbyists, and suggests that a background in mathematics and geometry would be helpful. His book is an enthusiastic investigation of the many kinds of geometrical objects that can be assembled or understood as shape symmetric and a website is provided with C++ source code to generate the examples he uses.

 

Scale symmetry never gets a rigorous mathematics-style definition. Instead, the term is used flexibly to incorporate diverse features that include multiple-scale symmetry, self-similarity, approximate scale symmetry, and statistical scale symmetry. The structures of interest to the author include fractals (including several variations of Mandelbrot and Julia set structures) as well as structures created by spherical inversion and cellular automata. Recursive shapes built from progressively smaller copies of a base structure, especially tree-like shapes, are also considered. What these have in common is that they are assembled from combinations of reflections, rotations, translations, and scaling. A few structures from physics and dynamical systems are also explored.

 

The author describes his exploration of the field as seeking out the new, beautiful and interesting. He says that many structures described in his book are new and have not been previously characterized. Among the new ones is a three-dimensional “Mandelbox”. This is a structure assembled using folds and dilation; its corresponding Julia set is shape-preserving. While much of the work here is exploration for the sake of exploration, the author does suggest some broad categories of potential applications.

 

The mathematical level of the book ranges widely. Much of it is accessible with a very basic background, but there are occasional more detailed references to conformal mappings, dimension theory, the complex geometry of the Ford Sphere set, and the like. The book is probably best suited for those interested in exploring similar structures on their own using suggestions from the book and software that the author has provided.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.