等边五边形的Bring六次曲线
The Bring sextic of equilateral pentagons
摘要：
考虑欧氏平面上的等边五边形$V_1,\ldots,V_5$。当我们识别平移、旋转和放大不同的五边形时，我们得到的可能形状的模空间是一个经常研究的多边形空间：一个拓扑上已知为四重环面（亏格4）的2流形$E_5$。我们研究$E_5$几何，我们的目标是一个保形地图的地形可能的形状。微分几何，我们使用的都是由于高斯，虽然它的大部分是命名后，他的学生黎曼。流形$E_5$从豪斯曼和克努森的格拉斯曼方法中继承了一个黎曼度量，在这个度量下$e_5$有240个等距：一个可选的反射与任何排列的组合，其中五个边向量$V_{k+1}-V_k$被组装成一个五边形。赋予$E_5$由$e_5$施加的共形结构，得到亏格为4的紧致黎曼曲面，具有120个自同构：120个保持方向的等距。但只有一个黎曼曲面具有这些性质：Bring六次曲线。因此，$(E_5, e_5)$像Bring六次曲线一样，作为240个三角形的重复模式共形嵌入双曲平面，每个三角形的顶角为$\fracπ{2}$、$\fracπ{4}$和$\fracπ{5}$。这个共形映射实现了我们的目标。为了在我们的地图上绘制五边形，我们通过求解高斯贝尔特拉米方程来计算$E_5$的初始等温坐标对。然后，我们使用保角映射将其中一个等温三角形区域转换为$(\fracπ{2},\fracπ{4},\fracπ{5})$双曲三角形的庞加莱投影。
英文摘要：
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.
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