正多边形的第一类
First Families of Regular Polygons
摘要：
每一个正多边形都可以被看作是相关的正多边形的一个定义明确的“家族”的成员。这些家庭自然出现在分段旋转的研究，如外部台球。在某些情况下，它们存在于所有尺度上，可以用来定义“奇点集”的分形维数。这对于规则的N-边形（例如五边形、八边形和十二边形）是有据可查的，它们的代数复杂度是“二次”（EulerPhi[N]/2 = 2）。最近的证据表明，这些家庭的几何形状是固有的“父”多边形，可以独立于任何映射。这是本文的目的，以显示如何第一家庭的任何正多边形自然产生的几何形状的“星多边形”第一次研究托马斯布拉德沃丁（1290年至1349年）。第一家庭的核心是S[k]“瓷砖”。每个S[k]图块具有对应的“星点”sk = tan（kPi/N）和匹配尺度。基于C.Siegel和S.Chowla在1949年的一个结果，gcd（k，N）= 1的本原标度构成N的分圆域的最大实子域的单位基.传统上，这个子字段一直是仿射分段有理旋转的缩放和旋转参数的来源，我们的研究结果表明，第一家庭缩放可以作为一个“自然”的基础，这样的调查-其中包括外部台球地图。
英文摘要：
Every regular polygon can be regarded as a member of a well-defined 'family' of related regular polygons. These families arise naturally in the study of piecewise rotations such as outer billiards. In some cases they exist on all scales and can be used to define the fractal dimension of the 'singularity set'. This is well-documented for regular N-gons such as the pentagon, octagon and dodecagon, whose algebraic complexity is 'quadratic' (EulerPhi[N]/2 = 2). Recent evidence suggests that the geometry of these families is intrinsic to the 'parent' polygon and can be derived independently of any mapping. It is the purpose of this paper to show how the First Family for any regular polygon arises naturally from the geometry of the 'star polygons' first studied by Thomas Bradwardine (1290-1349). The nucleus of the First Family are the S[k] 'tiles'. Each S[k] tile has a corresponding 'star-point' sk = tan(kPi/N) and a matching scale. Based on a 1949 result by C.Siegel and S.Chowla, the primitive scales with gcd(k,N) = 1 form a unit basis for the maximal real subfield of the cyclotomic field of N. Traditionally, this subfield has been the source of scaling and rotational parameters for affine piecewise rational rotations, and our results show that the First Family scaling can serve as a 'natural' basis for such investigations - which include the outer billiards map.
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