Richard Evan Schwartz
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691181387
- eISBN:
- 9780691188997
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691181387.003.0008
- Subject:
- Mathematics, Educational Mathematics
This chapter fixes some even rational parameter p/q as usual. It shows that the pixelated spacetime slices of capacity 2p are combinatorially equivalent to certain of the tilings from P. Hooper's ...
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This chapter fixes some even rational parameter p/q as usual. It shows that the pixelated spacetime slices of capacity 2p are combinatorially equivalent to certain of the tilings from P. Hooper's Truchet tile system [H]. Section 7.2 describes the Truchet tile system. Section 7.3 states the main result, the Truchet Comparison Theorem. One can view the Truchet Comparison Theorem as a computational tool for understanding some of the pixelated spacetime diagrams. Section 7.4 uses the Truchet Comparison Theorem to get more information about the surface Σ(p/q) from Corollary 6.6. Section 7.5 proves a curious result from elementary number theory which underlies the Truchet Comparison Theorem. Section 7.6 puts together the ingredients and proves the Truchet Comparison Theorem.Less
This chapter fixes some even rational parameter p/q as usual. It shows that the pixelated spacetime slices of capacity 2p are combinatorially equivalent to certain of the tilings from P. Hooper's Truchet tile system [H]. Section 7.2 describes the Truchet tile system. Section 7.3 states the main result, the Truchet Comparison Theorem. One can view the Truchet Comparison Theorem as a computational tool for understanding some of the pixelated spacetime diagrams. Section 7.4 uses the Truchet Comparison Theorem to get more information about the surface Σ(p/q) from Corollary 6.6. Section 7.5 proves a curious result from elementary number theory which underlies the Truchet Comparison Theorem. Section 7.6 puts together the ingredients and proves the Truchet Comparison Theorem.
Richard Evan Schwartz
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691181387
- eISBN:
- 9780691188997
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691181387.001.0001
- Subject:
- Mathematics, Educational Mathematics
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 ...
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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. This book provides a combinatorial model for orbits of outer billiards on kites. The book 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.Less
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. This book provides a combinatorial model for orbits of outer billiards on kites. The book 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.
