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Birkhäuser - Birkhäuser Mathematics | A Tale of Two Fractals

A Tale of Two Fractals

Kirillov, A.A.

2013, XIII, 138 p. 44 illus., 4 illus. in color.

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  • First book to provide a rigorous study of fractals usable by undergraduates
  • Written by a renowned authority in the field and expert at communicating mathematical ideas to students
  • An excellent means for young mathematicians to acquire basic tools from many different areas
  • Abundant figures and exercises provide valuable clarity and practice

Since Benoit Mandelbrot's pioneering work in the late 1970s, scores of research articles and books have been published on the topic of fractals. Despite the volume of literature in the field, the general level of theoretical understanding has remained low; most work is aimed either at too mainstream an audience to achieve any depth or at too specialized a community to achieve widespread use. Written by celebrated mathematician and educator A.A. Kirillov, A Tale of Two Fractals helps bridge this gap, providing an original treatment of fractals that is at once accessible to beginners and sufficiently rigorous for serious mathematicians. The work is designed to give young, non-specialist mathematicians a solid foundation in the theory of fractals.

As its title suggests, this book focuses primarily on two fractals: the Sierpiński gasket and the Apollonian gasket. Over the course of the book, they are developed and discussed in various contexts. Along with fundamental definitions and properties, some of the key concepts and approaches covered include

* the Laplace operator
* harmonic functions
* generalized numerical systems
* Descartes' theorem
* rational paramaterizations
* group action on fractals
* generalization to multiple dimensions

In addition to its explicit goal of providing undergraduate and graduate students with a sound foundation in fractal theory, A Tale of Two Fractals serves to enhance their overall understanding of mathematics by drawing on a wide variety of techniques from other subfields. Furthermore, by virtue of the subject matter, it provides a unique opportunity for students to develop their capacity for recognizing patterns and formulating interesting questions. It is therefore a valuable text not only for any course on fractals or hyperbolic geometry, but also for any survey course with an aim of honing creative-problem-solving skills.

Content Level » Upper undergraduate

Keywords » Apollonian gasket - Hausdorff dimension - Sierpinski gasket - fractals - harmonic functions - hyperbolic geometry - sphere packing

Related subjects » Birkhäuser Mathematics

Table of contents 

Introduction.- Part 1. The Sierpiński Gasket.- Definition and General Properties.- The Laplace Operator on the Sierpiński Gasket.- Harmonic Functions on the Sierpiński Gasket.- Part 2. The Apollonian Gasket.- Introduction.- Circles and Disks on Spheres.- Definition of the Apollonian Gasket.- Arithmetic Properties of Apollonian Gaskets.- Geometric and Group-Theoretic Approach.- Many-Dimensional Apollonian Gaskets.- Bibliography.

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