Summary -- 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 artwork1. The fractal family -- 1.1. The Mandelbrot set -- 1.2. The Feigenbaum set -- 1.3. Classic fractals -- 1.4. The Hofstadter set -- Appendix. Harper's equation as an iterative mapping2. Geometry, number theory, and the butterfly : friendly numbers and kissing circles -- 2.1. Ford circles, the Farey tree, and the butterfly -- 2.2. A butterfly at every scale--butterfly recursions -- 2.3. Scaling and 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 universality3 The Apollonian-butterfly connection (ABC) -- 3.1 Integral Apollonian gaskets (IAG) and the butterfly -- 3.2 The kaleidoscopic effect and trefoil symmetry -- 3.3 Beyond Ford Apollonian gaskets and fountain butterflies -- Appendix. Quadratic Diophantine equations and IAGs4. Quasiperiodic 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. Quasicrystalspart II. Butterfly in the quantum world -- 5. The quantum world -- 5.1. Wave or particle--what is it? -- 5.2. Quantization -- 5.3. What is waving?--The Schr?odinger picture -- 5.4. Quintessentially quantum -- 5.5. Quantum effects in the macroscopic world6. 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 [phi] -- 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 [phi] 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 equationpart 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 magnetic field : cyclotron orbits and their quantization -- 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 conductivity8. 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 -- 8.3. 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 monopoles9. 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 effect10. The kiss precise and precise quantization -- 10.1. Diophantus gives us two numbers for each swath in the butterfly -- 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 outlookpart IV. Catching the butterfly -- 11. The art of tinkering -- 11.1. The most beautiful physics experiments12. 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 release13. The butterfly gallery : variations on a theme of Philip G Harper -- 14. Divertimento -- 15. Gratitude 15-1 -- 16. Poetic math & science -- 17. Coda.