Fractals and Dynamics in Mathematics, Science, and the Arts: Theory and Applications: Fractalize That! (ePub)

Fractalize That! A Visual Essay on Statistical Geometry brings a new class of geometric fractals to a wider audience of mathematicians and scientists. It describes a recently discovered random fractal space-filling algorithm. Connections with tessellations and known fractals such as Sierpinski are developed. And, the mathematical development is illustrated by a large number of colorful images that will charm the readers.The algorithm claims to be universal in scope, in that it can fill any spatial region with smaller and smaller fill regions of any shape. The filling is complete in the limit of an infinite number of fill regions. This book presents a descriptive development of the subject using the traditional shapes of geometry such as discs, squares, and triangles. It contains a detailed mathematical treatment of all that is currently known about the algorithm, as well as a chapter on software implementation of the algorithm.The mathematician will find a wealth of interesting conjectures supported by numerical computation. Physicists are offered a model looking for an application. The patterns generated are often quite interesting as abstract art. Readers can also create these computer-generated art with the advice and examples provided.Contents:Introduction: Space-Filling PatternsSquares and RectanglesTriangles and DiamondsDiscs and Circular ArcsMixed and Irregular ShapesExamplesThe Statistical Geometry AlgorithmThe Area–Perimeter AlgorithmStatistical Geometry in One DimensionDo It Yourself Readership: Students, mathematicians, physicists, professionals, and the general public who are interested in the fractal art of statistical geometry.Key Features:The book brings together information from the few published papers and unpublished material reported here for the first timeThis is the first book-length description of the subjectExamples of the use of the algorithm for decorative artA visual feast