Hypercomplex iterations : distance estimation and higher dimensional fractals / Yumei Dang, Louis H. Kauffman, Daniel Sandin.

Format:

Book

Author/Creator:

Dang, Yumei.

Contributor:

Kauffman, Louis H., 1945-

Sandin, Daniel J.

Series:

K & E series on knots and everything ; v. 17.

K & E series on knots and everything ; vol. 17

Language:

English

Subjects (All):

Iterative methods (Mathematics).

Quaternions.

Mandelbrot sets.

Fractals.

Physical Description:

1 online resource (163 p.)

Edition:

1st ed.

Other Title:

Distance estimation and higher dimensional fractals

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, c2002.

Language Note:

English

Summary:

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 it

Contents:

Contents ; Acknowledgements ; Preface ; Part 1 Introduction ; Chapter 1 Hypercomplex Iterations in a Nutshell ; Chapter 2 Deterministic Fractals and Distance Estimation ; 2.1. Fractals and Visualization ; 2.2. Deterministic Fractals Julia Sets and Mandelbrot Sets

2.3. Distance Estimation Part 2 Classical Analysis: Complex and Quaternionic ; Chapter 3 Distance Estimation in Complex Space ; 3.1. Complex Dynamical Systems ; 3.2. The Quadratic Family Julia Sets and the Mandelbrot Set ; 3.3. The Distance Estimation Formula

3.4. Schwarz's Lemma and an Upper Bound of the Distance Estimate 3.5. The Koebe 1/4 Theorem and a Lower Bound for the Distance Estimate ; 3.6. An Approximation of the Distance Estimation Formula ; Chapter 4 Quaternion Analysis ; 4.1. The Quaternions ; 4.2. Rotations of 3-Space

4.3. Quaternion Polynomials 4.4. Quaternion Julia Sets and Mandelbrot Sets ; 4.5. Differential Forms ; 4.6. Regular Functions ; 4.7. Cauchy's Theorem and the Integral Formula ; 4.8. Linear and Quadratic Regular Functions

4.9. Difficulties of the Quaternion Analytic Proof of Distance Estimation Chapter 5 Quaternions and the Dirac String Trick ; Part 3 Hypercomplex Iterations ; Chapter 6 Quaternion Mandelbrot Sets ; 6.1. Quaternion Mandelbrot Sets

6.2. The Distance Estimate for Quaternion Mandelbrot Sets

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9789812778604

9812778608

OCLC:

843333119