本书是我们工作室引进的一部不错科普著作英文影印版。本书由道格拉斯·霍夫施塔特作序,是有史以来本讲述“霍夫施塔特蝴蝶”的书.“霍夫施塔特蝴蝶”是一个美丽而迷人的图形,位于物质量子理论的核心,本书讲述了蝴蝶的故事和它与量子霍尔效应的联系,它揭示了霍尔电阻惊人的准确量子化背后的秘密是如何被编码在数学的一个叫作拓扑的分支中的.拓扑学揭示了一个球体和一个立方体之间隐藏的数字量,同时将它们与甜甜圈和咖啡杯区分开来.量子霍尔效应背后的深层拓扑现象是物理学的一个抽象版本,它构成了傅科摆日常活动的基础;它可以被认为是这种进动量子的表亲,被称为贝里相。
Summary
About the author
Preface
Prologue
Prelude
Part I The butterfly fractal
0 Kiss precise
0.1 Apollonian gaskets and integer wonderlands
Appendix: An Apollonian sand painting--the world's largest artwork
References
1 The fractal family
1.1 The Mandelbrot set
1.2 The Feigenbaum set
1.2.1 Scaling and universality
1.2.2 Self-similarity
1.3 Classic fractals
1.3.1 The Cantor set
1.3.2 The Sierpinski gasket
1.3.3 Integral Apollonian gaskets
1.4 The Hofstadter set
1.4.1 Gaps in the butterfly
1.4.2 Hofstadter meets Mandelbrot
1.4.3 Concluding remarks: A mathematical, physical, and poetic magt
Appendix: Harper's equation as an iterative mapping
References
Geometry, number theory, and the butterfly: Friendly numbers and kissing circles
2.1 Ford circles, the Farey tree, and the butterfly
2.1.1 Ford circles
2.1.2 Farey tree
2.1.3 The saga of even-denominator and odd-denominator fractions
2.1.4 The sizes of butterflies
2.2 A butterfly at every scale--butterfly recursions
2.3 Scaling and universality
2.3.1 Flux scaling
2.3.2 Energy scaling
2.3.3 Universality
2.4 The butterfly and a hidden trefoil symmetry
2.5 Closing words: Physics and number theory
Appendix A: Hofstadter recursions and butterfly generations
Appendix B: Some theorems of number theory
Appendix C: Continued-fraction expansions
Appendix D: Nearest-integer continued fraction expansion
Appendix E: Farey paths and some comments on universality
References
3 The Apollonian-butterfly connection (ABC)
3.1 Integral Apollonian gaskets (IAG) and the butterfly
3.1.1 A duality transformation
3.1.2 Illustrating the Apollonian-butterfly connection
3.2 The kaleidosoopic effect and trefoil symmetry
3.2.1 Seeing an Apollonian gasket as a kaleidoscope
