Periodic trajectories in the regular pentagon
英文摘要:
We study periodic linear trajectories in the double pentagon and periodic billiard trajectories in the regular pentagon.
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Diana Davis, Dmitry Fuchs, Serge Tabachnikov
We study periodic linear trajectories in the double pentagon and periodic billiard trajectories in the regular pentagon.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
In our recent paper, we studied periodic billiard trajectories in the regular pentagon and closed geodesic on the double pentagon, a translation surface of genus two. In particular, we made a number of conjectures concerning symbolic periodic trajectories. In this paper, we prove two of these conjectures.
In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the periodic trajectory through a given point is long, short, or a saddle connection. In this paper we present an algorithm resolving this question for trajectories emanating from the midpoints of the pentagon.
Periodic paths on the pentagon, double pentagon and golden L
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The study of polygonal billiards, particularly those in the regular pentagon, has been the subject of two recent papers. One of these papers approaches the problem of discovering the periodic trajectories on the pentagon by identifying slopes of periodic directions with points in the Poincaré disk generated by hyperbolic isometric transformations. The other approach, coming from the other paper, transforms the double pentagon into a rectilinear translation surface called the 'golden L', where periodic directions are generated by a set of matrices associated with this surface in a special way. …
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