Dense packing of space with various convex solids

TL;DR

This paper surveys recent progress in the dense packing of convex solids, focusing on lattice packings in two and three dimensions, and discusses open problems and future research directions.

Contribution

It provides a comprehensive overview of current results in convex packing density and introduces new open problems for advancing three-dimensional packing theory.

Findings

Significant results on lattice packing densities in 2D and 3D

Identification of key open problems in convex packing

Discussion of connections with number theory and crystallography

Abstract

One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of the problem arise depending on the type of isometries allowed for the packing: packing by translates, lattice packing, translates and point reflections, or all isometries. Due to its connections with number theory, crystallography, etc., lattice packing has been studied most extensively. In two dimensions the theory is fairly well developed, and there are several significant results on lattice packing in three dimensions as well. This article surveys the known results, focusing on the most recent progress. Also, many new problems are stated, indicating directions in which future development of the general packing theory in three dimensions seems feasible.