目錄
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
3.2.2 How nested butterflies are related to kaleidoscopes
3.2.3 ABC and trefoil symmetry
3.3 Beyond Ford Apollonian gaskets and fountain butterflies
Appendix: Quadratic Diophantine equations and IAGs
References
4 Quasiperioflic patterns and the butterfly
4.1 A tale of three irrationals
4.2 Self-similar butterfly hierarchies
4.3 The diamond, golden, and silver hierarchies, and Hofstadter recursions
4.4 Symmetries and quasiperiodicities
Appendix: Quasicrystals
A.1 One-dimensional quasicrystals
A.2 Two-dimensional quasicrystals: Quasiperiodic tiles
A.3 A brief history of the discovery of quasicrystals
A.4 Excerpts from the ceremony of the Nobel Prize in chemistry in 2011
References
Part II Butterfly in the quantum world
5 The quantum world
5.1 Wave or particle--what is it?
5.1.1 Matter waves
5.2 Quantization
5.3 What is waving？--The Schr6dinger picture
5.4 Quintessentially quantum
5.4.1 The double-slit experiment, first hypothesized and finally realized
5.4.2 The Ehrenberg-Siday-Aharonov-Bohm effect (ESAB)
5.5 Quantum effects in the macroscopic world
5.5.1 Central concepts of condensed-matter physics
5.5.2 Summary
References
6 A quantum-mechanical marriage and its unruly child
6.1 Two physical situations joined in a quantum-mechanical marriage
6.2 The marvelous pure number φ
6.3 Harper's equation, describing Bloch electrons in a magnetic field
6.4 Harper's equation as a recursion relation
6.5 On the key role of inexplicable artistic intuitions in physics
6.6 Discovering the strange eigenvalue spectrum of Harper's equation
6.7 Continued fractions and the looming nightmare of discontinuity
6.8 Polynomials that dance on several levels at once
6.9 A short digression on INT and on perception of visual patterns
6.10 The spectrum belonging to irrational values of φ and the "ten-martini problem"
6.11 In which continuity (of a sort) is finally established
6.12 Infinitely recursively scalloped wave functions: Cherries on the doctoral sundae
6.13 Closing words
Appendix: Supplementary material on Harper's equation
References
Part III Topology and the butterfly
7 A different kind of quantization: The quantum Hall effect
7.1 What is the Hall effect? Classical and quantum answers
7.2 A charged particle in a magne
7.2.2 Quantum picture
7.2.3 Semiclassical picture
7.3 Landau levels in the Hofstadter butterfly
7.4 Topological insulators
Appendix A: Excerpts from the 1985 Nobel Prize press release
Appendix B: Quantum mechanics of electrons in a magnetic field
Appendix C: Quantization of the Hall conductivity
References
8 Topology and topological invariants: Preamble to the
topological aspects of the quantum Hall effect
8.1 A puzzle: The precision and the quantization of Hall conductivity
8.2 Topological invariants
8.2.1 Platonic solids
8.2.2 Two-dimensional surfaces
8.2.3 The Gauss-Bonnet theorem
8.3 Anholonomy: Parallel transport and the Foucault pendulum
8.4 Geometrization of the Foucault pendulum
8.5 Berry magnetism--effective vector potential and monopoles
8.6 The ESAB effect as an example of anholonomy
Appendix: Classical parallel transport and magnetic monopoles
References
9 The Berry phase and the quantum Hall effect
9.1 The Berry phase
9.2 Examples of Berry phase
9.3 Chern numbers in two-dimensional electron gases
9.4 Conclusion: the quantization of Hall conductivity
9.5 Closing words: Topology and physical phenomena
Appendix A: Berry magnetism and the Berry phase
Appendix B: The Berry phase and 2 x 2 matrices
Appendix C: What causes Berry curvature? Dirac strings, vortices, and magnetic monopoles
Appendix D: The two-band lattice model for the quantum Hall effect
References
10 The kiss precise and precise quantization
10.1 Diophantus gives us two numbers for each swath in the butterfly
10.1.1 Quantum labels for swaths when φ is irrational
10.2 Chern labels not just for swaths but also for bands
10.3 A topological map of the butterfly
10.4 Apollonian-butterfly connection: Where are the Chern numbers'?
10.5 A topological landscape that has trefoil symmetry
10.6 Chern-dressed wave functions
10.7 Summary and outlook
References
Part IV Catching the butterfly
11 The art of tinkering
11.1 The most beautiful physics experiments
References
12 The butterfly in the laboratory
12.1 Two-dimensional electron gases, superlattices, and the butterfly revealed
12.2 Magical carbon: A new net for the Hofstadter butterfly
12.3 A potentially sizzling hot topic in ultracold atom laboratories
Appendix: Excerpts from the 2010 Physics Nobel Prize press release
References
13 The butterfly gallery: Variations on a theme of Philip G Harper
14 Divertimento
15 Gratitude
16 Poetic Math&Science
17 Coda
18 Selected bibliography
編輯手記
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