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September 26, 2025Open Access

Billiards in regular polygons

Key Points

Billiards in regular polygons exhibit complex dynamics dependent on polygon sides.
The introduction of chaotic behavior occurs beginning at n=12, suggesting a critical threshold.
Control over billiard trajectories is linked to combinatorial patterns projected onto a finite line.
The study connects arithmetic perspectives to the unpredictability observed in polygonal billiards.

Abstract

In this paper we study billiards in regular n -sided polygons and monodromy over Teichmüller curves from an arithmetic perspective. Our main results show that the combinatorics of billiards in a periodic direction s is controlled by the projection of s to a finite projective line.This arithmetic control coexists, experimentally, with chaotic behavior that first emerges when n=12 .

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Cite This Study

Curtis T. McMullen (Wed,) studied this question.

synapsesocial.com/papers/68d6c68eb1249cec298b2d32 https://doi.org/https://doi.org/10.4171/lem/1099

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