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The plaid model / Richard Evan Schwartz.

By: Schwartz, Richard Evan [author.].
Material type: materialTypeLabelBookSeries: Annals of mathematics studies: no. 198.Publisher: Princeton, New Jersey : Princeton University Press, [2019]Copyright date: �2019Description: 1 online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9780691188997; 0691188998.Subject(s): Differentiable dynamical systems | Combinatorial dynamics | Geometry | Number theory | Dynamique diff�erentiable | Orbites p�eriodiques (Math�ematiques) | G�eom�etrie | Th�eorie des nombres | geometry | MATHEMATICS -- Essays | MATHEMATICS -- Pre-Calculus | MATHEMATICS -- Reference | MATHEMATICS -- Geometry -- General | Number theory | Geometry | Differentiable dynamical systems | Combinatorial dynamics | MathematicsGenre/Form: Electronic books.Additional physical formats: Print version:: Plaid model.DDC classification: 515/.39 Online resources: Click here to access online
Contents:
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 --
Summary: 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.
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Includes bibliographical references and index.

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Frontmatter -- Contents -- Preface -- Introduction -- Part 1. The plaid model -- Chapter 1. Definition of the plaid model -- Chapter 2. Properties of the model -- Chapter 3. Using the model -- Chapter 4. Particles and spacetime diagrams -- Chapter 5. Three-dimensional interpretation -- Chapter 6. Pixellation and curve turning -- Chapter 7. Connection to the Truchet tile system -- Part 2. The plaid PET -- Chapter 8. The plaid master picture theorem -- Chapter 9. The segment lemma -- Chapter 10. The vertical lemma -- Chapter 11. The horizontal lemma -- Chapter 12. Proof of the main result -- Part 3. The graph PET -- Chapter 13. Graph master picture theorem -- Chapter 14. Pinwheels and quarter turns -- Chapter 15. Quarter turn compositions and PETs -- Chapter 16. The nature of the compactification -- Part 4. The plaid-graph correspondence -- Chapter 17. The orbit equivalence theorem -- Chapter 18. The quasi-isomorphism theorem -- Chapter 19. Geometry of the graph grid -- Chapter 20. The intertwining lemma -- Part 5. The distribution of orbits -- Chapter 21. Existence of infinite orbits -- Chapter 22. Existence of many large orbits -- Chapter 23. Infinite orbits revisited -- Chapter 24. Some elementary number theory -- Chapter 25. The weak and strong case -- Chapter 26. The core case -- 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.

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