On the uniqueness of collections of pennies and marbles

TL;DR

This paper investigates the conditions under which collections of touching unit spheres, like pennies and marbles, are uniquely determined by their contact graphs, with a focus on rigidity in dimensions 2 and 3.

Contribution

It provides a precise characterization of global rigidity for contact graphs of spheres in 2D and 3D when the graphs are chordal, using graph rigidity theory.

Findings

Characterization of global rigidity for chordal contact graphs in 2D and 3D.

Examples of flexible, locally rigid, and globally rigid sphere collections.

Contrast between special chordal cases and generic rigidity settings.

Abstract

In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit $d$-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity.