## The Beauty of Fractals: Images of Complex Dynamical SystemsIn 1953 I realized that the straight line leads to the downfall of mankind. But the straight line has become an absolute tyranny. The straight line is something cowardly drawn with a rule, without thought or feeling; it is the line which does not exist in nature. And that line is the rotten foundation of our doomed civilization. Even if there are places where it is recognized that this line is rapidly leading to perdition, its course continues to be plot ted . . . Any design undertaken with the straight line will be stillborn. Today we are witnessing the triumph of rationalist knowhow and yet, at the same time, we find ourselves confronted with emptiness. An esthetic void, des ert of uniformity, criminal sterility, loss of creative power. Even creativity is prefabricated. We have become impotent. We are no longer able to create. That is our real illiteracy. Friedensreich Hundertwasser Fractals are all around us, in the shape of a mountain range or in the windings of a coast line. Like cloud formations and flickering fires some fractals under go never-ending changes while others, like trees or our own vascular systems, retain the structure they acquired in their development. To non-scientists it may seem odd that such familiar things have recently become the focus of in tense research. But familiarity is not enough to ensure that scientists have the tools for an adequate understanding. |

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### Contents

FRONTIERS OF CHAOS | 1 |

SPECIAL SECTIONS | 23 |

Julia Sets and Their Computergraphical Generation | 27 |

Sullivans Classification and Critical Points | 53 |

The Mandelbrot Set | 56 |

External Angles and Hubbard Trees | 63 |

Newtons Method for Complex Polynomials Cayleys Problem | 93 |

Newtons Method for Real Equations | 103 |

Renormalization | 142 |

147 | |

INVITED CONTRIBUTIONS | 151 |

Julia Sets and the Mandelbrot Set | 161 |

Freedom Science and Aesthetics | 175 |

Refractions of Science into Art | 181 |

DO IT YOURSELF | 189 |

DOCUMENTATION | 193 |

### Other editions - View all

The Beauty of Fractals: Images of Complex Dynamical Systems, Page 377 Heinz-Otto Peitgen,Peter H. Richter No preview available - 1986 |

### Common terms and phrases

aesthetic algorithm attractive cycle attractive fixed points attractor basin of attraction beautiful bifurcation binary decomposition called Cantor set cardioid chaos chaotic color complex numbers complex plane components computer graphics conjugation converges corresponding critical point domain of attraction Douady dynamical systems equation equipotential lines example experiments external angles external arguments field lines finite fractal geometry Hubbard trees infinite number infinity initial point inverse orbit iteration Julia and Fatou Julia set Julia-like sets lattice level sets locally connected look magnetic Mandelbrot set mathematical mathematicians nature Newton's method nonlinear obtained parabolic parameter partition function period doubling periodic orbit periodic point phase transitions physicists physics polynomial preimages preperiodic problem rational mapping real axis region renormalization renormalization transformation result root scientific scientists self-similar sequence shows Siegel disk solutions Special Section structure temperature theory tion Verhulst process Yang-Lee zeros