Compact packings of space with three sizes of spheres

TL;DR

This paper classifies all compact packings of three-dimensional space using three sphere sizes, revealing specific configurations where tetrahedral holes are filled with smaller spheres, advancing understanding of dense sphere arrangements.

Contribution

It provides a complete classification of all compact packings with three different sphere sizes, detailing the configurations involving tetrahedral holes filled with smaller spheres.

Findings

All such compact packings are identified and described.

Four distinct ways of filling tetrahedral holes with smaller spheres are characterized.

The work extends the understanding of dense sphere packings with multiple sizes.

Abstract

A sphere packing of the three-dimensional Euclidean space is compact if it has only tetrahedral holes, that is, any local maximum of the distance to the spheres is at equal distance to exactly four spheres. This papers describes all the compact packings with spheres of three different sizes. They are close-compact packings of unit spheres with holes filled in four different ways by smaller spheres.