03937nam a22005055i 4500 978-0-8176-8382-5 DE-He213 20260521092035.0 cr nn 008mamaa 130423s2013 xxu| s |||| 0|eng d 9780817683825 99780817683825 10.1007/978-0-8176-8382-5 doi 004 23 Kirillov, A.A. author. A Tale of Two Fractals [electronic resource] / by A.A. Kirillov. New York, NY : Springer New York : Imprint: Birkhäuser, 2013. XIII, 138 p. 44 illus., 4 illus. in color. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda 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. 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. MATHEMATICS. ALGEBRA. GLOBAL ANALYSIS (MATHEMATICS). FUNCTIONS, SPECIAL. VISUALIZATION. GEOMETRY. MATHEMATICS. VISUALIZATION. SPECIAL FUNCTIONS. GEOMETRY. ANALYSIS. ALGEBRA. APPLICATIONS OF MATHEMATICS. SpringerLink (Online service) Springer eBooks Printed edition: 9780817683818 http://dx.doi.org/10.1007/978-0-8176-8382-5 Ver el texto completo en las instalaciones del CICY ZDB-2-SMA ddc ER 35837 35837 0 0 ddc 0 0 LE CICY CICY EL 2025-10-06 0 004 2025-10-06 08:44:41 2025-10-06 ER