论文标题
等边五边形的六六元
The Bring sextic of equilateral pentagons
论文作者
论文摘要
考虑欧几里得平面中的等边五角大$ v_1，\ ldots，v_5 $。当我们确定五角星因翻译，旋转和放大倍率而有所不同时，我们得到的可能形状的模量空间是一个经常研究的多边形空间：一个2个manifold $ e_5 $在拓扑位上已知是四倍体圆环（属4）。我们从几何上研究$ e_5 $，我们的目标是那个可能形状的地形的保形图。我们使用的差异几何形状全部归功于高斯，尽管其中大部分是以他的学生Riemann命名的。 The manifold $E_5$ inherits a Riemannian metric from the Grassmannian approach of Hausmann and Knutson, a metric $e_5$ under which $E_5$ has 240 isometries: an optional reflection combined with any permutation of the order in which the five edge vectors $V_{k+1}-V_k$ get assembled into a pentagon.给出$ e_5 $ $ e_5 $施加的保形结构，产生了一个紧凑的riemann属，具有120个自动形态的属：保留方向的120个同镜。但是只有一个带有这些特性的Riemann表面：Bring Sextic。因此，$（e_5，e_5）$ $将同型嵌入双曲线平面（如Bring Sextic），作为240个三角形的重复模式，每个模式为$ \fracπ{2} $，$ \fracπ{4} $，以及$ \fracπ{5} $。那个共形地图实现了我们的目标。 为了在我们的地图上绘制五角大州，我们通过求解Beltrami方程式àlaGauss来计算以$ e_5 $的初始等温坐标。然后，我们使用共形映射将其中一个等温三角区域转换为$（\fracπ{2}，\fracπ{4}，\fracπ{5}）$双曲三角形的庞加莱投影。
Consider equilateral pentagons $V_1,\ldots,V_5$ in the Euclidean plane. When we identify pentagons that differ by translation, rotation, and magnification, the moduli space of possible shapes that we get is an oft-studied polygon space: a 2-manifold $E_5$ known topologically to be a quadruple torus (genus 4). We study $E_5$ geometrically, our goal being a conformal map of that terrain of possible shapes. The differential geometry that we use is all due to Gauss, though much of it is named after his student Riemann. The manifold $E_5$ inherits a Riemannian metric from the Grassmannian approach of Hausmann and Knutson, a metric $e_5$ under which $E_5$ has 240 isometries: an optional reflection combined with any permutation of the order in which the five edge vectors $V_{k+1}-V_k$ get assembled into a pentagon. Giving $E_5$ the conformal structure imposed by $e_5$ yields a compact Riemann surface of genus 4 with 120 automorphisms: the 120 isometries that preserve orientation. But there is only one Riemann surface with those properties: the Bring sextic. So $(E_5, e_5)$ conformally embeds in the hyperbolic plane, like the Bring sextic, as a repeating pattern of 240 triangles, each with vertex angles of $\fracπ{2}$, $\fracπ{4}$, and $\fracπ{5}$. That conformal map realizes our goal. To plot pentagons on our map, we compute an initial pair of isothermal coordinates for $E_5$ by solving the Beltrami equation à la Gauss. We then use a conformal mapping to convert one of those isothermal triangular regions into a Poincaré projection of a $(\fracπ{2},\fracπ{4},\fracπ{5})$ hyperbolic triangle.
