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Negative refraction and tiling billiards. (English) Zbl 1393.37048

The authors introduce a dynamical system, that they call tiling billiard, where the trajectories refract through planar tilings (a subdivision of the plane into regions).
They present examples of trajectories that obey the following conditions (from the established literature on inner and outer billiards): are periodic, cannot escape, fill a region densely.
Much of the literature on billiards is focused on the polygonal planar case. Several special situations where the tiling is obtained by dividing the plane in lines (for example into congruent triangles) are analyzed – cases where the tiling has even valence at any vertex (two-colorable).
The authors complete some previous results available in the literature. They conclude by giving some new interesting examples and providing several novel results on tiling billiards.
Some new open questions, based on their experimental results, are also conjectured.

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)

Software:

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Full Text: DOI arXiv

References:

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