This book is based on the authors' research on rendering images of higher dimensional fractals by a distance estimation technique. It is self-contained, giving a careful treatment of both the known techniques and the authors' new methods. The distance estimation technique was originally applied to Julia sets and the Mandelbrot set in the complex plane. It was justified, through the work of Douady and Hubbard, by deep results in complex analysis. In this book the authors generalise the distance estimation to quaternionic and other higher dimensional fractals, including fractals derived from iteration in the Cayley numbers (octonionic fractals). The generalization is justified by new geometric arguments that circumvent the need for complex analysis. This puts on a firm footing the authors' present work and the second author's earlier work with John Hart and Dan Sandin. The results of this book will be of great interest to mathematicians and computer scientists interested in fractals and computer graphics.
Contents:
- Introduction:
- Hypercomplex Iteractions in a Nutshell
- Deterministic Fractals and Distance Estimation
- Classical Analysis: Complex and Quaternionic:
- Distance Estimation in Complex Space
- Quaternion Analysis
- Quaternions and the Dirac String Trick
- Hypercomplex Iteractions:
- Quaternion Mandelbrot Sets
- Distance Estimation in Higher Dimensional Spaces
- Inverse Iteraction, Ray Tracing and Virtual Reality:
- Inverse Iteraction: An Interactive Visualization
- Ray Tracing Methods by Distance Estimation
- Quaternion Deterministic Fractals in Virtual Reality
Readership: Mathematicians and computer scientists.
“This book is well written and self-contained. A CD-ROM comes with the book which provides striking animated illustrations of such fractals.”
Mathematical Reviews
“The subject is very interesting and very modern. The book is very well written and contains various coloured fractal images.”
Zentralblatt MATH
Louis H Kauffman, Professor at the University of Illinois, Chicago. Professor Kauffman is a renowned theorist in the field of Knot theory. He is the recipient of many awards for his contributions. These awards include: Alternative Natural Philosophy Association Award (1996), Warren McCulloch Memorial Award for Outstanding Contributions to the field of Cybernetics (1993), awarded by the American Society for Cybernetics Lester R Ford Award of the Mathematical Association of American (1978). He was Woodrow Wilson Honorary Fellow(1968), National Science Foundation Fellow (1966-69) and Princeton National Fellow (1966-69). He is the Managing Editor of the Journal of Knot Theory and its Ramifications and the Series of Knots and Everything, both published by World Scientific Publishing. The journal is in its 12 volumes now and there are 32 volumes in the series published so far. He has contributed more than 100 papers in scientific journals and proceedings. Professor Kauffman is also the author/co-author of 14 books.
CD-ROM
9789812778604SM01.iso (645118 KB)
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