Complementary bodies in sphere packing

TL;DR

This paper investigates the geometric properties of voids in sphere packings, specifically in cubic close packing, and explores how truncated polyhedra can tile these spaces, with implications for physical and biological modeling.

Contribution

It introduces the analysis of complementary bodies in sphere packings, deriving their surface area-to-volume ratios and demonstrating tiling of interstitial spaces by truncated polyhedra.

Findings

Derived surface area-to-volume ratios of complementary bodies.

Established tiling of interstitial space by truncated tetrahedra and octahedra.

Contributed to understanding packing density and geometric complements.

Abstract

Symbolic and graphical tools, such as Mathematica, enable precise visualization and analysis of void spaces in sphere packings. In the cubic close packing (CCP, or face-centred cubic packing; FCC) arrangement these voids can be partitioned into repeating geometric units we term spherically truncated polyhedra - bodies analogous to plane-truncated polyhedra but bounded by both planar and spherical surfaces. These structures are relevant in geometric studies and applications such as modelling diffusion in porous media and biological tissues. This work examines the properties of these complementary bodies, deriving their surface area-to-volume ratios, which are significant in physical contexts; and we establish a result concerning the packing density of truncated tetrahedra and octahedra, demonstrating how they tile the interstitial space surrounding packed spheres. These findings contribute to a deeper understanding of classical packing problems and their geometrical complements.