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Tiling Billards on Triangle Tilings, and Interval Exchange Transformations

Paul Baird-Smith, Diana Davis, Elijah Fromm, Sumun Iyer

Abstract

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing three-interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We show that, for a particular choice of triangle tiling, certain trajectories approach the Rauzy fractal, under rescaling.

Tiling Billards on Triangle Tilings, and Interval Exchange Transformations

Abstract

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation-reversing three-interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We show that, for a particular choice of triangle tiling, certain trajectories approach the Rauzy fractal, under rescaling.

Paper Structure

This paper contains 21 sections, 33 theorems, 15 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on tiling billiards
  3. Acknowledgments
  4. Tiling billiards on triangle tilings, and folding
  5. Triangle tilings
  6. Folding
  7. A trajectory crosses a triangle at most once
  8. From trajectories to interval and polygon exchange transformations
  9. Parameters in the circle, to set up the IET
  10. From the circle to an interval exchange transformation
  11. From IETs to a polygon exchange transformation
  12. Results about tiling billiards from the IET and PET
  13. Results from the IET
  14. Results from the PET
  15. The Arnoux-Yoccoz IET and its tiling billiards trajectory
  16. ...and 6 more sections

Key Result

Theorem 1

trihex For almost every initial point and direction, a trajectory with this initial position and direction is dense in all of the plane except for a periodic family of triangular open sets.

Figures (19)

  • Figure 1: (a) A tiling billiards trajectory refracts across an edge of a triangle tiling. (b) When folded across an edge, the pieces of trajectory lie on the same line. (c) In the folded position, all of the triangles are inscribed in the same circle, and the folded trajectory lies on a chord.
  • Figure 2: A periodic trajectory with period 6 (left) and a drift-periodic trajectory with period 4 (right) on the isosceles right triangle tiling
  • Figure 3: After folding the three triangles on the left, all of the triangles share a circumcenter (Lemma \ref{['circumcenters']}), and the blue triangles are rotations of one another about the circumcenter (Lemma \ref{['lem:rotations']}).
  • Figure 4: Folding up the example trajectory in the Appendix: (a) Folding along every edge of the tiling. (b) In the folded position, all the pieces of trajectory lie on the same line, and (c) all of the triangles are inscribed in the same circle, with the blue triangles on one side (shown) and red on the other.
  • Figure 5: From a given positively-oriented triangle (center), there are six adjacent positively-oriented triangles (dark blue), which are reached via the six two-edge moves described in Definition \ref{['def:moves']}.
  • ...and 14 more figures

Theorems & Definitions (89)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 79 more