The Lonely Vertex Problem

TL;DR

This paper investigates a geometric problem involving tilings of Euclidean space with convex polytopes, focusing on the distribution of vertices and their connectivity within the tiling.

Contribution

It introduces the 'Lonely Vertex Problem,' analyzing conditions under which vertices are either shared by multiple tiles or absent, advancing understanding of tiling vertex configurations.

Findings

Characterization of vertex distribution in convex tilings

Conditions for the existence of isolated vertices

Implications for tiling symmetry and structure

Abstract

In a locally finite tiling of n-dim Euclidean space by convex polytopes, each point of the space is either a vertex of at least two tiles, or no vertex at all.