
The Plaid Model
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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.
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Content
- Cover
- Title
- Copyright
- Contents
- Preface
- Introduction
- 0.1 Part 1: The Plaid Model and its Properties
- 0.2 Part 2: The Plaid PET
- 0.3 Part 3: The Graph PET
- 0.4 Part 4: Plaid-Graph Correspondence
- 0.5 Part 5: The Distribution of Orbits
- 0.6 Companion Program
- PART 1. THE PLAID MODEL
- Chapter 1. Definition of the Plaid Model
- 1.1 Chapter Overview
- 1.2 Basic Quantities and Notation
- 1.3 Six Families of Lines
- 1.4 Capacity, Mass, and Sign
- 1.5 Light Points
- 1.6 Transverse Directions for the Light Points
- 1.7 Main Definition
- Chapter 2. Properties of the Model
- 2.1 Chapter Overview
- 2.2 Symmetries
- 2.3 The Number of Intersection Points
- 2.4 The Meaning of Capacity
- 2.5 A Subtle Symmetry
- Chapter 3. Using the Model
- 3.1 Chapter Overview
- 3.2 The Big Polygon
- 3.3 Hierarchical Information
- 3.4 A Subdivision Algorithm
- 3.5 Comparing Different Parameters
- Chapter 4. Particles and Spacetime Diagrams
- 4.1 Chapter Overview
- 4.2 Remote Adjacency
- 4.3 Horizontal Particles
- 4.4 Vertical Particles
- 4.5 Spacetime Diagrams and Their Symmetries
- 4.6 The Bad Tile Lemma
- Chapter 5. Three-Dimensional Interpretation
- 5.1 Chapter Overview
- 5.2 Stacking the Blocks
- 5.3 Pixelated Spacetime Diagrams
- 5.4 Tile Compatibility
- 5.5 Spacetime Plaid Surfaces
- 5.6 Discussion and Speculation
- Chapter 6. Pixellation and Curve Turning
- 6.1 Chapter Overview
- 6.2 Orienting the Worldlines
- 6.3 The Sparseness of Worldlines
- 6.4 Curve Turning Theorem: Vertical Case
- 6.5 Curve Turning Theorem: Horizontal Case
- 6.6 Two Applications
- Chapter 7. Connection to the Truchet Tile System
- 7.1 Chapter Overview
- 7.2 Truchet Tilings
- 7.3 The Truchet Comparison Theorem
- 7.4 The Fundamental Surface
- 7.5 A Result from Elementary Number Theory
- 7.6 Proof of the Truchet Comparison Theorem
- PART 2. THE PLAID PET
- Chapter 8. The Plaid Master Picture Theorem
- 8.1 Chapter Overview
- 8.2 The Spaces
- 8.3 The Checkerboard Partition
- 8.4 The Classifying Map
- 8.5 The Main Result
- Chapter 9. The Segment Lemma
- 9.1 Chapter Overview
- 9.2 The Anchor Point
- 9.3 A Computational Tool
- 9.4 The Vertical Case
- 9.5 The Horizontal Case
- Chapter 10. The Vertical Lemma
- 10.1 Chapter Overview
- 10.2 Using Symmetry
- 10.3 Translating the Picture
- 10.4 Some Useful Formulas
- 10.5 The Undirected Result
- 10.6 Determining the Directions
- Chapter 11. The Horizontal Lemma
- 11.1 Chapter Overview
- 11.2 Using Symmetry
- 11.3 Translating the Picture
- 11.4 Two Easy Technical Lemmas
- 11.5 The Undirected Result
- 11.6 Determining the Directions
- Chapter 12. Proof of the Main Result
- 12.1 Chapter Overview
- 12.2 Prism Structure
- 12.3 Some Extra Symmetry
- 12.4 The Vertical Case
- 12.5 The Horizontal Case
- PART 3. THE GRAPH PET
- Chapter 13. Graph Master Picture Theorem
- 13.1 Chapter Overview
- 13.2 Special Orbits
- 13.3 The Arithmetic Graph
- 13.4 A Preliminary Result
- 13.5 The PET Structure
- 13.6 The Fundamental Polytopes
- Chapter 14. Pinwheels and Quarter Turns
- 14.1 Chapter Overview
- 14.2 The Pinwheel Map
- 14.3 Outer Billiards and the Pinwheel Map
- 14.4 Quarter Turn Compositions
- 14.5 The Pinwheel Map as a QTC
- 14.6 The Case of Kites
- Chapter 15. Quarter Turn Compositions and PETs
- 15.1 Chapter Overview
- 15.2 A Result from Linear Algebra
- 15.3 The Map
- 15.4 Compactifying Shears
- 15.5 Compactifying Quarter Turn Maps
- 15.6 The End of the Proof
- Chapter 16. The Nature of the Compactification
- 16.1 Chaper Overview
- 16.2 The Singular Directions
- 16.3 The First Parallelotope
- 16.4 The Second Parallelotope
- 16.5 The General Master Picture Theorem
- 16.6 Structure of the PET
- 16.7 The Case of Kites
- PART 4. THE PLAID-GRAPH CORRESPONDENCE
- Chapter 17. The Orbit Equivalence Theorem
- 17.1 Chapter Overview
- 17.2 The Prisms
- 17.3 The Map
- 17.4 Characterizing the Image
- 17.5 The Clean Partition
- 17.6 The Main Proof
- 17.7 Computational Techniques
- 17.8 The Calculations
- Chapter 18. The Quasi-Isomorphism Theorem
- 18.1 Chapter Overview
- 18.2 The Canonical Affine Transformation
- 18.3 The Graph Grid
- 18.4 The Intertwining Lemma
- 18.5 The Correspondence of Orbits
- 18.6 The End of the Proof
- 18.7 The Projection Theorem
- 18.8 Renormalization Interpretation
- Chapter 19. Geometry of the Graph Grid
- 19.1 Chapter Overview
- 19.2 The Grid Geometry Lemma
- 19.3 The Graph Reconstruction Lemma
- Chapter 20. The Intertwining Lemma
- 20.1 Chapter Overview
- 20.2 A Resume of Transformations
- 20.3 Injectivity of the Map
- 20.4 Calculating a Single Point
- 20.5 Dissecting the Set
- 20.6 The Induction Step
- 20.7 Discussion
- 20.8 The Diagonal Case
- PART 5. THE DISTRIBUTION OF ORBITS
- Chapter 21. Existence of Infinite Orbits
- 21.1 Chapter Overview
- 21.2 Definedness Criterion
- 21.3 Spacetime Diagrams Revisited
- 21.4 Taking a Limit
- 21.5 Associated Paths
- 21.6 Sketch of an Alternate Proof
- Chapter 22. Existence of Many Large Orbits
- 22.1 Chapter Overview
- 22.2 Equidistribution Properties
- 22.3 The Ubiquity Lemma
- 22.4 The Rectangle Lemma
- 22.5 Proof of the Main Result
- 22.6 The Continued Fraction Length
- 22.7 The End of the Proof
- Chapter 23. Infinite Orbits Revisited
- 23.1 Chapter Overview
- 23.2 The Approximating Sequence
- 23.3 The Copy Theorem
- 23.4 The End of the Proof
- 23.5 The Copy Lemma
- 23.6 Proof of the Box Theorem
- 23.7 Proof of the Copy Theorem
- 23.8 Hidden Symmetries
- Chapter 24. Some Elementary Number Theory
- 24.1 Chapter Overview
- 24.2 A Structural Result
- 24.3 Unfinished Business
- Chapter 25. The Weak and Strong Case
- 25.1 Chapter Overview
- 25.2 The First Two Statements
- 25.3 A Technical Lemma
- 25.4 The Mass and Capacity Sequences
- 25.5 Vertical Intersection Points
- 25.6 A Matching Criterion
- 25.7 Verifying the Matching Criterion
- Chapter 26. The Core Case
- 26.1 Chapter Overview
- 26.2 The First Two Statements
- 26.3 Geometric and Arithmetic Alignment
- 26.4 Geometric Alignment
- 26.5 Alignment of the Capacity Sequences
- 26.6 A Technical Lemma
- 26.7 The Mass Sequences: Central Case
- 26.8 The Mass Sequences: Peripheral Case
- 26.9 The End of the Proof
- Appendix References
- Index
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