The plaid model /
Richard Evan Schwartz.
- 1 online resource
- Annals of mathematics studies ; Number 198 .
- Annals of mathematics studies ; no. 198. .
Includes bibliographical references and index.
The plaid model -- Definition of the plaid model -- Properties of the model -- Using the model -- Particles and spacetime diagrams -- Three-dimensional interpretation -- Pixellation and curve turning -- Connection to the Truchet tile system -- The plaid PET -- The plaid master picture theorem -- The segment lemma -- The vertical lemma -- The horizontal lemma -- Proof of the main result -- The graph PET -- Graph master picture theorem -- Pinwheels and quarter turns -- Quarter turn compositions and PETs -- The nature of the compactification -- The plaid-graph correspondence -- The orbit equivalence theorem -- The quasi-isomorphism theorem -- Geometry of the graph grid -- The intertwining lemma -- The distribution of orbits -- Existence of infinite orbits -- Existence of many large orbits -- Infinite orbits revisited -- Some elementary number theory -- The weak and strong case -- The core case -- Frontmatter -- Contents -- Preface -- Introduction -- Part 1. Chapter 1. Chapter 2. Chapter 3. Chapter 4. Chapter 5. Chapter 6. Chapter 7. Part 2. Chapter 8. Chapter 9. Chapter 10. Chapter 11. Chapter 12. Part 3. Chapter 13. Chapter 14. Chapter 15. Chapter 16. Part 4. Chapter 17. Chapter 18. Chapter 19. Chapter 20. Part 5. Chapter 21. Chapter 22. Chapter 23. Chapter 24. Chapter 25. Chapter 26. References -- Index.
Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behaviour even for 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 billiards. 'The Plaid Model', which is a self-contained sequel to Schwartz's 'Outer Billiards on Kites', provides a combinatorial model for orbits of outer billiards on kites. 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 very difficult to reach through traditional maths. The work includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.