Abstract

In a convex mosaic in ${\mathbb{R}}^d$ we denote the average number of vertices of a cell by $\overline{v}$ and the average number of cells meeting at a node by $\overline{n}.$ Except for the d = 2 planar case, there is no known formula prohibiting points in any range of the $[\overline{n},\overline{v}]$ plane (except for the unphysical $\overline{n},\overline{v}<d+1$ strips). Nevertheless, in d = 3 dimensions if we plot the 28 points corresponding to convex uniform honeycombs, the 28 points corresponding to their duals and the 3 points corresponding to Poisson-Voronoi, Poisson-Delaunay and random hyperplane mosaics, then these points appear to accumulate on a narrow strip of the $[\overline{n},\overline{v}]$ plane. To explore this phenomenon we introduce the harmonic degree $\overline{h}=\overline{n}\overline{v}/(\overline{n}+\overline{v})$ of a d-dimensional mosaic. We show that the observed narrow strip on the $[\overline{n},\overline{v}]$ plane corresponds to a narrow range of $\overline{h}.$ We prove that for every ${\overline{h}}^{\star }\in (d,2^{d-1}]$ there exists a convex mosaic with harmonic degree ${\overline{h}}^{\star }$ and we conjecture that there exist no d-dimensional mosaic outside this range. We also show that the harmonic degree has deeper geometric interpretations. In particular, in case of Euclidean mosaics it is related to the average of the sum of vertex angles and their polars, and in case of 2 D mosaics, it is related to the average excess angle.

摘要

在一个凸马赛克中${\mathbb{R}}^d$我们将一个单元格的平均顶点数表示为$\overline{v}$和在一个节点相遇的平均细胞数$\overline{n}.$除了d  = 2 平面情况外，没有已知的公式禁止在任何范围内的点$[\overline{n},\overline{v}]$平面（非物质的除外$\overline{n},\overline{v}<d+1$带子）。然而，在d  = 3 维中，如果我们绘制对应于凸均匀蜂窝的 28 个点、对应于它们的对偶的 28 个点以及对应于 Poisson-Voronoi、Poisson-Delaunay 和随机超平面马赛克的 3 个点，那么这些点似乎是累积的在一条狭窄的地带$[\overline{n},\overline{v}]$飞机。为了探索这种现象，我们引入调和度$\overline{H}=\overline{n}\overline{v}/(\overline{n}+\overline{v})$d维马赛克。我们展示了观察到的窄带$[\overline{n},\overline{v}]$平面对应的范围很窄$\overline{H}.$我们证明对于每个${\overline{H}}^{\star }\in (d,2^{d-1}]$存在调和度的凸马赛克${\overline{H}}^{\star }$我们推测在这个范围之外不存在d维镶嵌。我们还表明，谐波度具有更深层次的几何解释。特别是，在欧几里得马赛克的情况下，它与顶点角和它们的极角之和的平均值有关，而在二维马赛克的情况下，它与平均超角有关。