Domokos, Gábor and Lángi, Zsolt (2019) On some average properties of convex mosaics. Experimental Mathematics. ISSN 10586458 (In Press)

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Abstract
In a convex mosaic in Rd we denote the average number of vertices of a cell by v and the average number of cells meeting at a node by n. Except for the d = 2 planar case, there is no known formula prohibiting points in any range of the [n,v] plane (except for the unphysical n,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 PoissonVoronoi, PoissonDelaunay and random hyperplane mosaics, then these points appear to accumulate on a narrow strip of the [n,v] plane. To explore this phenomenon we introduce the harmonic degree h=nv/(n+v) of a ddimensional mosaic. We show that the observed narrow strip on the [n,v] plane corresponds to a narrow range of h. We prove that for every h∈(d,2^(d−1)] there exists a convex mosaic with harmonic degree h and we conjecture that there exist no ddimensional 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.
Item Type:  Article 

Subjects:  Q Science / természettudomány > QA Mathematics / matematika > QA73 Geometry / geometria 
Depositing User:  Dr. Zsolt Lángi 
Date Deposited:  22 Sep 2020 14:42 
Last Modified:  22 Sep 2020 14:42 
URI:  http://real.mtak.hu/id/eprint/114069 
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