*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 ...
More

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.

*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/9780691181387.003.0015
- Subject:
- Mathematics, Educational Mathematics

This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, ...
More

This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ denote the second iterate of the outer billiards map defined on R2 − P. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the X-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.Less

This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon *P* without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ denote the second iterate of the outer billiards map defined on *R*^{2} − *P*. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the *X*-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.