A Concise Introduction to Hypercomplex Fractals

Andrzej Katunin

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9781138633421
• eBook ISBN: 9781351801201
• Topic: Computer Science
• Subtopic: Computer Graphics
• Index: Computer Science

This book presents concisely the full story on complex and hypercomplex fractals, starting from the very first steps in complex dynamics and resulting complex fractal sets, through the generalizations of Julia and Mandelbrot sets on a complex plane and the Holy Grail of the fractal geometry – a 3D Mandelbrot set, and ending with hypercomplex, multicomplex and multihypercomplex fractal sets which are still under consideration of scientists. I tried to write this book in a possibly simple way in order to make it understandable to most people whose math knowledge covers the fundamentals of complex numbers only. Moreover, the book is full of illustrations of generated fractals and stories concerned with great mathematicians, number spaces and related fractals. In the most cases only information required for proper understanding of a nature of a given vector space or a construction of a given fractal set is provided, nevertheless a more advanced reader may treat this book as a fundamental compendium on hypercomplex fractals with references to purely scientific issues like dynamics and stability of hypercomplex systems.

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Contents

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Preface

CHAPTER 1 ▪ Introduction to Fractals on a Complex Plane

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1.1 BIRTH OF COMPLEX FRACTALS – JULIA AND FATOU SETS

1.1.1 Preliminaries

1.1.2 Coloring Fractal Sets on a Complex Plane

1.1.3 Julia and Fatou Sets within the Polynomials of Various Orders

1.1.4 Other Variations of Julia and Fatou Sets on a Complex Plane

1.2 FRACTAL REVOLUTION – THE MANDELBROT SET

1.2.1 History and Concept of the Mandelbrot Set

1.2.2 Geometrical Structure and Atoms of the Mandelbrot Set

1.2.3 Mandelbrot Set within the Polynomials of Various Orders and Other Variations

1.3 SEARCHING FOR THE HOLY GRAIL-3-D MANDELBROT SET

1.4 WHAT’S NEXT? FURTHER EXTENSIONS OF THE MANDELBROT SET

CHAPTER 2 ▪ Fractal Beasts in Terms of Normed Division and Clifford Algebras

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2.1 QUATERNIONIC FRACTAL SETS

2.1.1 History and Preliminaries of Quaternion Calculus

2.1.2 Quaternionic Julia and Mandelbrot Sets

2.2 OCTONIONIC FRACTAL SETS

2.2.1 History and Preliminaries of Octonions

2.2.2 Octonionic Julia and Mandelbrot Sets

2.3 IS THERE ANYTHING ALIVE IN HIGHER-DIMENSIONAL HYPERCOMPLEX SPACES?

2.3.1 Preliminaries to Clifford Algebras

2.3.2 Cliffordean Hypercomplex Fractal Sets

2.4 THE PLACES (AND HYPERSPACES) WHERE FRACTALS CANNOT LIVE

CHAPTER 3 ▪ Tensor Product Fractal Mutants

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3.1 BICOMPLEX, TRICOMPLEX, … MULTICOMPLEX FRACTAL SETS

3.1.1 Julia and Mandelbrot Sets in Bicomplex Vector Space

3.1.2 Julia and Mandelbrot Sets in Tricomplex Vector Space

3.1.3 Julia and Mandelbrot Sets in Multicomplex Vector Space

3.2 TENSOR PRODUCT HYPERCOMPLEX AND MULTIHYPERCOMPLEX FRACTAL SETS

3.2.1 Julia and Mandelbrot Sets in Biquaternionic Vector Space

3.2.2 Beyond Biquaternionic Julia and Mandelbrot Sets

Bibliography

Index

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Preface

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Since I started working with fractals, my family, friends, colleagues, and students were puzzled hearing about hypercomplex fractals. This is mainly due to the specific nomenclature used both in the fractal theory as well as in the hypercomplex number theory, with all of their attractors, repellers, octonions, zero divisors, and nilpotents. These words naturally scare them‡ And at the same time, they love colorful images generated by my computer simulations. These images, however, are just nice pictures to them. I am convinced that many non scientists, who had contact with complex and hypercomplex fractals have the same feelings. Therefore, I decided to write a book which would present complex and hypercomplex fractals in a concise and comprehensible manner omitting mathematical formalism as much as possible. This idea has germinated in my mind for a few years, and finally I can place the fruit of a few years’ work into your hands.

This book concisely presents the complete story on complex and hypercomplex fractals, starting from the very first steps in complex dynamics and resulting complex fractal sets, through the generalizations of Julia and Mandelbrot sets on a complex plane and the Holy Grail of fractal geometry – a 3-D Mandelbrot set, and ending with hypercomplex, multicomplex, and multihypercomplex fractal sets that are still under consideration of scientists. I tried to write this book in a simpl...

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