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The Plaid Model
-
Richard Evan Schwartz
Language:
English
Published/Copyright:
2019
About this book
Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz’s Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites.
Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics.
The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.
Author / Editor information
Richard Evan Schwartz is the Chancellor’s Professor of Mathematics at Brown University. He is the author of Spherical CR Geometry and Dehn Surgery and Outer Billiards on Kites (both Princeton).
Reviews
"[An] enjoyable excursion into a new field."---Meghan De Witt, MAA Reviews
Topics
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Frontmatter
i -
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Contents
v -
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Preface
xi -
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Introduction
1 - Part 1. The Plaid Model
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Chapter 1. Definition of the Plaid Model
15 -
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Chapter 2. Properties of the Model
25 -
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Chapter 3. Using the Model
35 -
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Chapter 4. Particles and Spacetime Diagrams
45 -
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Chapter 5. Three-Dimensional Interpretation
57 -
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Chapter 6. Pixellation and Curve Turning
71 -
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Chapter 7. Connection to the Truchet Tile System
81 - Part 2. The Plaid PET
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Chapter 8. The Plaid Master Picture Theorem
93 -
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Chapter 9. The Segment Lemma
103 -
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Chapter 10. The Vertical Lemma
111 -
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Chapter 11. The Horizontal Lemma
119 -
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Chapter 12. Proof of the Main Result
125 - Part 3. The Graph PET
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Chapter 13. Graph Master Picture Theorem
135 -
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Chapter 14. Pinwheels and Quarter Turns
143 -
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Chapter 15. Quarter Turn Compositions and PETs
153 -
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Chapter 16. The Nature of the Compactification
161 - Part 4. The Plaid-Graph Correspondence
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Chapter 17. The Orbit Equivalence Theorem
173 -
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Chapter 18. The Quasi-Isomorphism Theorem
185 -
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Chapter 19. Geometry of the Graph Grid
195 -
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Chapter 20. The Intertwining Lemma
199 - Part 5. The Distribution of Orbits
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Chapter 21. Existence of Infinite Orbits
213 -
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Chapter 22. Existence of Many Large Orbits
219 -
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Chapter 23. Infinite Orbits Revisited
227 -
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Chapter 24. Some Elementary Number Theory
239 -
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Chapter 25. The Weak and Strong Case
245 -
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Chapter 26. The Core Case
253 -
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References
265 -
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Index
267
Publishing information
Pages and Images/Illustrations in book
eBook published on:
January 29, 2019
eBook ISBN:
9780691188997
Pages and Images/Illustrations in book
eBook ISBN:
9780691188997
Keywords for this book
Theorem; Rectangle; Parameter; Polytope; Diagram (category theory); Outer billiard; Cartesian coordinate system; Dot product; Translational symmetry; Bijection; Intersection (set theory); Special case; Convex polytope; Equation; Subset; Homeomorphism; Metric space; Line segment; Lattice path; Symbolic dynamics; Integer; Extreme point; Automorphism; Y-intercept; Parity (mathematics); Big O notation; Fundamental domain; Pairwise; Parallelogram; Fiber bundle; Renormalization; Calculation; Polygon mesh; Convex hull; Cantor set; Unit square; Disjoint sets; Summation; Mathematical induction; Hyperplane; Hyperbola; Geometry; Right angle; Parallelepiped; Tensor product; Function composition; Right half-plane; Piecewise; Correspondence theorem (group theory); Without loss of generality; Diameter; Pixelation; Rotational symmetry; Unit interval; Equivalence class; Graph partition; Topology; Natural number; Coprime integers; Covering space; Comparison theorem; Rational number; Affine transformation; Compact space; Lexicographical order; Rhombus; Degeneracy (mathematics); Absolute value; Sanity check; Tessellation
Audience(s) for this book
For an expert adult audience, including professional development and academic research
