---
title: "Symplectic Tiling Billiards on Complete Affine Tori"
authors: ["Charles Daly", "Fabian Lander"]
year: 2026
venue: "arXiv:math.DS"
arxiv: "2608.28894"
doi: null
source: "https://arxiv.org/abs/2608.28894"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1583
seed_batch: "curiosity-2026-09-05"
reviewed: "2026-09-05"
pool: "maths-foundations"
relevance_score: 10
lineage: billiards
cites:
  - title: "Symplectic Tiling Billiards on Complete Affine Tori"
    url: "https://arxiv.org/abs/2608.28894"
    year: 2026
    arxiv: "2608.28894"
    doi: null
see: []
---

# Symplectic Tiling Billiards on Complete Affine Tori

## One-sentence takeaway

Schwartz’s symplectic×tiling billiards played on complete affine tori — dynamics on non-Euclidean flat structures.

## Why it's lovely

Why you might love this: billiards squared (tiling + symplectic) on affine tori. In 2023 Richard Schwartz married tiling billiards to symplectic billiards; Daly–Lander put that system on tilings that come from non-Euclidean geometries on the torus — affine analogues of flat conformal structures, via Baues–Goldman. Visual dynamics meets symplectic geometry on a deformation space you can actually mark.

## Key ideas

- Schwartz 2023: a dynamical system combining tiling billiards and symplectic billiards.
- Played on tilings of the plane arising from non-Euclidean (complete affine) geometries on the torus.
- Affine analogue of flat conformal structures (Baues–Goldman); an open subset of the deformation space for marked complete affine structures, with algebraic conditions on tiling symmetries.
- Long-term dynamics: a stability result for divergent trajectories, with divergence measured by geometric invariants of the tiling symmetry group.
- Heuristic: non-Euclidean tiles become “thin” far away; a thinness notion adapted to affine tilings.

## Caveats

Complete affine tori, not Euclidean tiling billiards in the plane. Divergence/stability is the dynamical headline, not a full classification of periodic orbits. Relies on the Schwartz symplectic-tiling setup as given.

## Implications for Broadside

Dynamics delight on the billiards shelf. Affine-torus tilings are a reminder that “flat” can still be non-Euclidean — the same kind of surprise as a mesh that is locally Euclidean and globally twisted.

## Links

- arXiv abs: https://arxiv.org/abs/2608.28894
- PDF: https://arxiv.org/pdf/2608.28894.pdf