# On some average properties of convex mosaics

## TL;DR

This paper investigates the average properties of convex mosaics in Euclidean space, introduces the harmonic degree as a key measure, and explores its range and geometric significance across dimensions.

## Contribution

It introduces the harmonic degree for convex mosaics, proves its range, and links it to geometric properties, providing new insights into mosaic structure.

## Key findings

- Convex mosaics' harmonic degree ranges between d and 2^{d-1} in d dimensions.
- Observed data points cluster within a narrow strip in the [ar n, ar v] plane.
- Harmonic degree relates to vertex angles and their polars, especially in 2D mosaics.

## Abstract

In a convex mosaic in $\mathbb{R} ^d$ we denote the average number of vertices of a cell by $\bar v$ and the average number of cells meeting at a node by $\bar n$. Except for the $d=2$ planar case, there is no known formula prohibiting points in any range of the $[\bar n, \bar v]$ plane (except for the unphysical $\bar n, \bar v < d+1$ 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 $[\bar n, \bar v]$ plane. To explore this phenomenon we introduce the harmonic degree $\bar h= \bar n\bar v/(\bar n + \bar v)$ of a $d$-dimensional mosaic. We show that the observed narrow strip on the $[\bar n, \bar v]$ plane corresponds to a narrow range of $\bar h$. We prove that for every $\bar h^{\star} \in (d, 2^{d-1}]$ there exists a convex mosaic with harmonic degree $\bar h^{\star}$ 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 2D mosaics, it is related to the average excess angle.