On Some Average Properties of Convex Mosaics

Gábor Domokos, Zsolt Lángi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In a convex mosaic in (Formula presented.) we denote the average number of vertices of a cell by (Formula presented.) and the average number of cells meeting at a node by (Formula presented.) Except for the d = 2 planar case, there is no known formula prohibiting points in any range of the (Formula presented.) plane (except for the unphysical (Formula presented.) 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 (Formula presented.) plane. To explore this phenomenon we introduce the harmonic degree (Formula presented.) of a d-dimensional mosaic. We show that the observed narrow strip on the (Formula presented.) plane corresponds to a narrow range of (Formula presented.) We prove that for every (Formula presented.) there exists a convex mosaic with harmonic degree (Formula presented.) 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.

Original languageEnglish
JournalExperimental Mathematics
DOIs
Publication statusAccepted/In press - Jan 1 2019

Keywords

  • Convex mosaic
  • platonic solid
  • uniform mosaic

ASJC Scopus subject areas

  • Mathematics(all)

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