The raspberries in three dimensions with at most two sizes of berry

TL;DR

This paper classifies all possible arrangements of tangent spheres, called raspberries, with at most two sizes, around a central sphere in three-dimensional space, providing a comprehensive catalog of configurations.

Contribution

It introduces a complete catalog of raspberry configurations with up to two sphere sizes, advancing understanding of sphere arrangements in 3D space.

Findings

Catalog of all feasible radius configurations for raspberries.

Identification of configurations with at most two sphere sizes.

Discussion of construction methods for these configurations.

Abstract

In three dimensional Euclidean space, a raspberry is defined to be an arrangement of spheres with pairwise disjoint interiors, where all spheres are tangent to a central unit sphere and such that the contact graph of the non-central spheres triangulates the central sphere. We discuss the relevance of these structures in related work. We present a catalog of all configurations of radii that permit the formation of raspberries that have at most two sizes of non-central spheres. Throughout, we discuss the construction of this catalog.