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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.12259 |
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| _version_ | 1866916718163525632 |
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| author | Schwartz, Richard Evan |
| author_facet | Schwartz, Richard Evan |
| contents | In this paper I will unite two games,
symplectic billiards and tiling billiards. The new
game is called symplectic tiling billiards.
I will prove a result about periodic orbits of
symplectic tiling billiards in a
very special case and then show how this result
combines with the construction in
Thurston's paper {\it Shapes of Polyhedra\/} to
give hyperbolic structures on moduli spaces
of planar equilateral polygons. One corollary
is that the configuration space of the hexagonal
planar linkage with unit-length rods (modulo
isometry) has an
algebraically defined hyperbolic structure in which
it is a $10$-cusped hyperbolic $3$-manifold
that is tiled by $15$ regular ideal octahedra.
The $10$ cusps correspond to the $10$ maximally degenerate
configurations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12259 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry Schwartz, Richard Evan Dynamical Systems In this paper I will unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of symplectic tiling billiards in a very special case and then show how this result combines with the construction in Thurston's paper {\it Shapes of Polyhedra\/} to give hyperbolic structures on moduli spaces of planar equilateral polygons. One corollary is that the configuration space of the hexagonal planar linkage with unit-length rods (modulo isometry) has an algebraically defined hyperbolic structure in which it is a $10$-cusped hyperbolic $3$-manifold that is tiled by $15$ regular ideal octahedra. The $10$ cusps correspond to the $10$ maximally degenerate configurations. |
| title | Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2307.12259 |