On Some Average Properties of Convex Mosaics

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.

Keywords and phrases:

• Convex mosaic
• uniform mosaic
• platonic solid

2010 Mathematics Subject Classification:

• 52C22
• 52B11
• 52A38

Acknowledgments

The authors are very grateful to Rolf Schneider for his repeated encouragement which contributed to shape this manuscript. We also thank Egon Schulte and Frank Morgan for their positive comments, and an anonymous referee for careful reading and valuable comments.

Declaration of Interest

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the NKFIH Hungarian Research Fund grant 119245 and of grant BME FIKP-VÍZ by EMMI is kindly acknowledged. ZL has been supported by grant UNKP-19-4 New National Excellence Program of the Ministry of Innovation and Technology and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.