正多边形的边几何——第二部分
Edge Geometry of Regular Polygons -- Part 2
AI总结：
本文研究正N边形边几何中奇点集W的局部结构，通过外台球映射、数字滤波器映射和双中心映射的剪切与旋转等价性，分析S[1]和S[2]第一族瓦片及其演化，并依据边猜想提出正多边形的“八重方式”。
摘要：
有多种映射可用于生成我们所称的正N边形的“边几何”，但它们都基于作用于N的延长边上的分段等距映射，以形成一个“奇点”集W。该奇点集也被称为“网”，因为它是连通的，并且由射线或线段组成，在极限情况下可能存在聚点。这里我们将使用三种这样的映射，它们似乎都具有相同的W的局部几何。这些映射是外台球映射Tau、数字滤波器映射Df和“双中心”映射Dc。在《外台球、数字滤波器与踢动哈密顿量》（arXiv:1206.5223）中，我们证明了Df和Dc映射分别等价于环面空间和复平面中的“剪切与旋转”，而在《正多边形的第一族及其变异》（arXiv:1612.09295）中，我们证明了Tau网W也可以归结为剪切与旋转。这种映射等价性支持了该网几何内在于N边形这一前提。由于W的拓扑结构复杂，我们希望通过研究N的局部区域取得一些进展。每个正N边形的边都是N的局部Tau不变区域的一部分。这里的重点是紧邻N的S[1]和S[2]第一族瓦片，但我们也将研究它们与相邻瓦片的相互作用。由于所有S[k]瓦片都以多步方式演化，因此可以对网中存活下来的“下一代”瓦片做出预测。边猜想只定义了8类N边形，因此正多边形存在一条“八重方式”。
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 Df and the 'dual-center' map Dc. In 'Outer-billiards, digital filters and kicked Hamiltonians' (arXiv:1206.5223) we show that the Df and Dc maps are equivalent to a 'shear and rotation' in a toral space and the complex plane respectively, and in '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. Since the topology of W is 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 local to N. The emphasis here are the S[1] and S[2] First Family tiles adjacent to N, but we will also study their interaction with neighboring tiles. Since all S[k] tiles evolve in a multi-step fashion, it is possible to make predictions about the 'next-generation' tiles which survive in the web. The Edge Conjecture defines just 8 classes of N-gons so there is an 'Eightfold Way' for regular polygons.
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