# Compact packings of the space with two sizes of spheres

## TL;DR

This paper characterizes all compact packings of Euclidean space using two sphere sizes, showing they are formed by filling octahedral holes of close packings with smaller spheres, which helps understand dense arrangements.

## Contribution

It provides a complete classification of two-size sphere packings as fillings of octahedral holes in close packings, a novel geometric insight.

## Key findings

- All compact packings with two sphere sizes are obtained by filling octahedral holes.
- This characterization helps identify densest sphere arrangements.
- The results link packing structures to geometric tilings.

## Abstract

Compact packings are specific packings of spheres which can be seen as tilings and are good candidates to maximize the density. We show that the compact packings of the Euclidean space with two sizes of spheres are exactly those obtained by filling with smaller spheres the octahedral holes of a close-packing of equal spheres.