05126nam 2200709Ia 450 991077837290332120221206101903.0981-277-860-8(CKB)1000000000480120(EBL)1679565(OCoLC)843333119(SSID)ssj0000175592(PQKBManifestationID)11163824(PQKBTitleCode)TC0000175592(PQKBWorkID)10190170(PQKB)10648910(MiAaPQ)EBC1679565(WSP)00003625(Au-PeEL)EBL1679565(CaPaEBR)ebr10201328(CaONFJC)MIL491699(EXLCZ)99100000000048012020021023d2002 uy 0engur|n|---|||||txtccrHypercomplex iterations[electronic resource] distance estimation and higher dimensional fractals /Yumei Dang, Louis H. Kauffman, Daniel SandinSingapore ;River Edge, NJ World Scientificc20021 online resource (163 p.)K & E series on knots and everything ;vol. 17Description based upon print version of record.981-02-3296-9 Includes bibliographical references and index.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 Sets2.3. 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 Formula3.4. 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-Space4.3. 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 Functions4.9. 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 Sets6.2. The Distance Estimate for Quaternion Mandelbrot Sets 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 itK & E series on knots and everything ;v. 17.Iterative methods (Mathematics)QuaternionsMandelbrot setsFractalsIterative methods (Mathematics)Quaternions.Mandelbrot sets.Fractals.514.742Dang Yumei1577527Kauffman Louis H.1945-57757Sandin Daniel J1577528MiAaPQMiAaPQMiAaPQBOOK9910778372903321Hypercomplex iterations3856169UNINA