标题： 正五边形在平面中的排列 显示英文标题

标题： Packings of Regular Pentagons in the Plane

摘要： 我们证明了欧几里得平面上全等正五边形的任何排列密度最多为$(5-\sqrt5)/3$，这大约是0.92。 更具体地说，本文证明了Henley（1986）和Kuperberg与Kuperberg（1990）提出的正五边形冰射线猜想，该猜想断言平面中全等正五边形的最佳排列是一个双格子，由对齐的垂直列向上指的五边形和对齐的垂直列向下指的五边形交替组成。 该策略基于Delaunay三角形面积的估计。 我们的策略将正五边形冰射线猜想简化为最多涉及四个Delaunay三角形的面积最小化问题。 这些最小化问题通过计算机解决。 证明中的计算机辅助部分使用了区间算术、自动微分和中间相遇算法等技术。 显示英文摘要

摘要： We show that every packing of congruent regular pentagons in the Euclidean plane has density at most $(5-\sqrt5)/3$, which is about 0.92. More specifically, this article proves the pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice, formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons. The strategy is based on estimates of the areas of Delaunay triangles. Our strategy reduces the pentagonal ice-ray conjecture to area minimization problems that involve at most four Delaunay triangles. These minimization problems are solved by computer. The computer-assisted portions of the proof use techniques such as interval arithmetic, automatic differentiation, and a meet-in-the-middle algorithm.