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Periodic paths on the pentagon, double pentagon and golden L

Diana Davis, Samuel Lelièvre

Abstract

We give a tree structure on the set of all periodic directions on the golden L, which gives an associated tree structure on the set of periodic directions for the pentagon billiard table and double pentagon surface. We use this to give the periods of periodic directions on the pentagon and double pentagon. We also show examples of many periodic billiard trajectories on the pentagon, which are strikingly beautiful, and we describe some of their properties. Finally, we give conjectures and future directions based on experimental computer evidence.

Periodic paths on the pentagon, double pentagon and golden L

Abstract

We give a tree structure on the set of all periodic directions on the golden L, which gives an associated tree structure on the set of periodic directions for the pentagon billiard table and double pentagon surface. We use this to give the periods of periodic directions on the pentagon and double pentagon. We also show examples of many periodic billiard trajectories on the pentagon, which are strikingly beautiful, and we describe some of their properties. Finally, we give conjectures and future directions based on experimental computer evidence.

Paper Structure

This paper contains 46 sections, 82 theorems, 28 equations, 48 figures.

Table of Contents

  1. Main results, introduction and background
  2. Periodic trajectories in polygons: existence, abundance, enumeration
  3. Main results
  4. Analogous known results for the square
  5. Previous work on the pentagon, double pentagon and golden L
  6. Related work
  7. The double pentagon and the golden L
  8. From the pentagon to the necklace, double pentagon and golden L
  9. The tree of saddle connection vectors for the golden L
  10. Directions to saddle connection vectors on the golden L
  11. Combinatorial periods of pentagon and double pentagon trajectories
  12. Cutting sequences on the double pentagon
  13. Periods of periodic double pentagon trajectories
  14. The L-necklace, the Five Ls, and their Veech group
  15. Symmetries of trajectories via the group structure
  16. ...and 31 more sections

Key Result

Theorem 1.1

For the square torus and square billiard table:

Figures (48)

  • Figure 1: Some periodic paths on the regular pentagon. These are the trajectories 1000-short, 1231-long, 102-short, and 133-short, respectively (see §\ref{['sec:tree']}).
  • Figure 2: The golden L translation surface, with edge identifications indicated by color, and edge lengths as indicated in the figure
  • Figure 3: Left: the regular pentagon billiard table. Center: the unfolding of the pentagon billiard table into the necklace translation surface. The number 1-5 (along with the color) indicates which edge of the billiard table was unfolded to get that edge, and the letter indicates which of the copies are glued to each other. Right: The double pentagon translation surface.
  • Figure 4: The cut-and-paste, shear, and vertical compression between the golden L and the double pentagon. For an earlier paper with a similar figure, see curt.
  • Figure 5: The golden L showing saddle connection and cylinder vectors in the horizontal direction: the short saddle connection vector (blue) and the two long saddle connection vectors (orange and purple), and the short cylinder vector (white) and long cylinder vector (black). Notice that the short cylinder vector coincides with the long saddle connection vector, and the long cylinder vector is the sum of the short and long saddle connection vectors.
  • ...and 43 more figures

Theorems & Definitions (220)

  • Theorem 1.1
  • proof
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Definition 2.6
  • ...and 210 more