LEADER 05396nam 22009135 450 001 9910438141003321 005 20200630090556.0 010 $a0-8176-8382-8 024 7 $a10.1007/978-0-8176-8382-5 035 $a(CKB)2670000000359237 035 $a(EBL)1204427 035 $a(SSID)ssj0001068003 035 $a(PQKBManifestationID)11600991 035 $a(PQKBTitleCode)TC0001068003 035 $a(PQKBWorkID)11092014 035 $a(PQKB)11760409 035 $a(DE-He213)978-0-8176-8382-5 035 $a(MiAaPQ)EBC6310508 035 $a(MiAaPQ)EBC1204427 035 $a(Au-PeEL)EBL1204427 035 $a(CaPaEBR)ebr10969129 035 $a(OCoLC)841482616 035 $a(PPN)169134687 035 $a(EXLCZ)992670000000359237 100 $a20130423d2013 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 12$aA Tale of Two Fractals /$fby A.A. Kirillov 205 $a1st ed. 2013. 210 1$aNew York, NY :$cSpringer New York :$cImprint: Birkhäuser,$d2013. 215 $a1 online resource (138 p.) 300 $aDescription based upon print version of record. 311 $a0-8176-8381-X 320 $aIncludes bibliographical references & index. 327 $aIntroduction -- Part 1. The Sierpi?ski Gasket -- Definition and General Properties -- The Laplace Operator on the Sierpi?ski Gasket.- Harmonic Functions on the Sierpi?ski 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. 330 $aSince 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 Sierpi?ski 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. 606 $aMathematics 606 $aVisualization 606 $aSpecial functions 606 $aGeometry 606 $aMathematical analysis 606 $aAnalysis (Mathematics) 606 $aAlgebra 606 $aApplied mathematics 606 $aEngineering mathematics 606 $aVisualization$3https://scigraph.springernature.com/ontologies/product-market-codes/M14034 606 $aSpecial Functions$3https://scigraph.springernature.com/ontologies/product-market-codes/M1221X 606 $aGeometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21006 606 $aAnalysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12007 606 $aAlgebra$3https://scigraph.springernature.com/ontologies/product-market-codes/M11000 606 $aApplications of Mathematics$3https://scigraph.springernature.com/ontologies/product-market-codes/M13003 615 0$aMathematics. 615 0$aVisualization. 615 0$aSpecial functions. 615 0$aGeometry. 615 0$aMathematical analysis. 615 0$aAnalysis (Mathematics). 615 0$aAlgebra. 615 0$aApplied mathematics. 615 0$aEngineering mathematics. 615 14$aVisualization. 615 24$aSpecial Functions. 615 24$aGeometry. 615 24$aAnalysis. 615 24$aAlgebra. 615 24$aApplications of Mathematics. 676 $a514.742 686 $a00A09$a26A30$a28A80$a57S$2msc 700 $aKirillov$b A.A$4aut$4http://id.loc.gov/vocabulary/relators/aut$0603877 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910438141003321 996 $aA Tale of Two Fractals$92494437 997 $aUNINA