正多边形的边缘几何 —— 第1部分
The Edge Geometry of Regular Polygons -- Part 1
AI总结：
本文研究正N边形的边缘几何，通过外弹球等三种映射证明其共享相同的局部网格几何，提出正多边形可分为8类的“八重道”，并扩展了局部不变区域的代数判据。
摘要：
有多种映射可用于生成我们称之为正 N 边形的“边缘几何”的结构，但它们均基于作用于 N 的扩展边以形成“奇异”集合 W 的分段等距。该奇异集合也被称为“网格”，因为它是连通的并由射线或线段组成，且在极限情况下可能存在聚点。我们将在此使用三种此类映射，它们似乎都共享 W 的相同局部几何。这些映射分别是外弹球映射 Tau、Chou 和 Lin 的“数字滤波器”映射以及 Arek Goetz 的“双中心”映射。在文献（arXiv:1206.5223）中，我们证明了这些映射分别等价于环面空间和复平面中的“剪切与旋转”，而在主论文 [H5]《正多边形的第一族及其突变》（arXiv:1612.09295）中，我们证明了 Tau 网格 W 也可以简化为剪切与旋转。映射间的这种等价性支持了该网格几何是 N 边形固有的这一前提。就边缘几何而言，似乎只有 8 类正 N 边形构成了一个“八重道”。由于 W 的拓扑非常复杂，我们希望通过研究 N 的局部区域来取得一些进展。每个正 N 边形的边都是 Tau 不变区域的一部分，该区域应至少包含 N 的 1/4 的第一族 S[k] 瓦片，因此边缘几何与全局网格重叠。在 [H5] 的附录 II 中，我们基于相邻瓦片的“共享边”提出了这些 N 局部区域不变性的可能充分条件。在本文中，我们扩展了这一猜想并提供了代数判据及大量示例。在附录中，我们基于 R.Schwartz 在 [S2] 中的工作给出了轨道“投影”的示例。
英文摘要：
There are multiple mappings that can be used to generate what we call the 'edge geometry' of a regular N-gon, but they are all based on piecewise isometries acting on the extended edges of N to form a 'singularity' set W. This singularity set is also known as the 'web' because it is connected and consists of rays or line segments, with possible accumulation points in the limit. We will use three such maps here, all of which appear to share the same local geometry of W. These mappings are the outer-billiards map Tau, the 'digital-filter' map of Chou and Lin and a 'dual-center' map of Arek Goetz. In (arXiv:1206.5223) we show that these maps are equivalent to a 'shear and rotation' in a toral space and the complex plane respectively, and in the main paper [H5] 'First Families of Regular Polygons and their Mutations' (arXiv:1612.09295) we show that the Tau-web W can also be reduced to a shear and rotation. This equivalence of maps supports the premise that this web geometry is inherent in the N-gon. In terms of Edge Geometry there appears to be just 8 classes of regular N-gons which make up an '8-Fold Way'. Since the topology of W is very complex, we hope to make some progress by studying the region local to N. The edges of every regular N-gon are part of a Tau-invariant region that should include at least 1/4 of the First Family S[k] tiles of N, so the edge geometry overlaps the global web. In Appendix II to [H5] we formulated what may be sufficient conditions for invariance of these regions local to N, based on 'edge-sharing' of adjacent tiles. In this paper we extend this conjecture and provide algebraic criteria with many examples. In the appendix we give examples of 'projections' of orbits based on the work of R.Schwartz in [S2].
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