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Bounding the density of binary sphere packing

Thomas Fernique, Daria Pchelina

TL;DR

The paper proves an explicit upper bound on the density of packings with two sphere sizes in $\mathbb{R}^3$, focusing on radii $1$ and $r=\sqrt{2}-1$ so that a small sphere can occupy octahedral sites. It introduces FM-tetrahedra, a restricted additively-weighted Delaunay decomposition, and computes the density bound by bounding the density of each FM-tetrahedron via interval arithmetic and computer-assisted proof. The main result identifies the densest FM-tetrahedra types (including $1111$, $11rr$, $1rrr$, $rrrr$, $111r$) and establishes a global bound $\delta^*_{111r}=0.8125420...$, achieved within explicit local neighborhoods, with a dimension-reduction and sliding-compression strategy facilitating a full certification. This work demonstrates a concrete, rigorous approach to tight density bounds for binary sphere packings and provides a framework potentially adaptable to other size ratios or higher dimensions.

Abstract

This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each octahedral hole of a hexagonal compact packing of large spheres. This upper bound is obtained by bounding from above the density of the tetrahedra which can appear in the additively-weighted Delaunay decomposition of the sphere centers of such packings. The proof relies on challenging computer calculations in interval arithmetic and may be of interest by their own.

Bounding the density of binary sphere packing

TL;DR

The paper proves an explicit upper bound on the density of packings with two sphere sizes in , focusing on radii and so that a small sphere can occupy octahedral sites. It introduces FM-tetrahedra, a restricted additively-weighted Delaunay decomposition, and computes the density bound by bounding the density of each FM-tetrahedron via interval arithmetic and computer-assisted proof. The main result identifies the densest FM-tetrahedra types (including , , , , ) and establishes a global bound , achieved within explicit local neighborhoods, with a dimension-reduction and sliding-compression strategy facilitating a full certification. This work demonstrates a concrete, rigorous approach to tight density bounds for binary sphere packings and provides a framework potentially adaptable to other size ratios or higher dimensions.

Abstract

This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each octahedral hole of a hexagonal compact packing of large spheres. This upper bound is obtained by bounding from above the density of the tetrahedra which can appear in the additively-weighted Delaunay decomposition of the sphere centers of such packings. The proof relies on challenging computer calculations in interval arithmetic and may be of interest by their own.

Paper Structure

This paper contains 26 sections, 11 theorems, 51 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. FM-tetrahedra
  3. Definitions and properties
  4. Formulas
  5. An optimization problem
  6. Towards a 3D Florian's bound?
  7. Proof strategy
  8. Interval arithmetic
  9. Checking a strict inequality
  10. Optimal points
  11. Near optimal tetrahedra
  12. Tight tetrahedra
  13. Stretched tetrahedra
  14. Away from optimal tetrahedra
  15. Dimension reduction
  16. ...and 11 more sections

Key Result

Theorem 1

Let $r:=\sqrt{2}-1$. The densest FM-tetrahedra of types $1111$, $11rr$ and $1rrr$ have only tight edges. The densest FM-tetrahedron of type $rrrr$ has $4$ tight edges and two incident edges of length The densest FM-tetrahedron of type $111r$ has $5$ tight edges and one edge between two large spheres of length

Figures (15)

  • Figure 1: Left: a packing of unit disks in the Euclidean plane and a Delaunay triangulation of the disk centers, with the densest possible triangle emphasized. Right: a packing made of densest possible triangles only.
  • Figure 2: Left: a layer of a hexagonal compact packing (yellow spheres) and some spheres (oranges) of the next layer. Right: how spheres of consecutive layers form tetrahedra or octahedra.
  • Figure 3: Left: the densest triangle that can appear in a packing of disks of sizes in $[r,1]$. Right: there is no packing with only such triangles. Consider indeed the $k$ triangles which share a common small disk: since these triangles must always be paired along their shortest edge, $k$ is even, but $\alpha<\pi/2$ yields $k>4$ while $\alpha>\pi/3$ (because $r<1$) yields $k<6$.
  • Figure 4: A cannonball packing of unit spheres (yellow) with spheres of size $\sqrt{2}-1$ filling octaedral sites.
  • Figure 5: The densest FM-tetrahedra of each type, with a numerical approximation of their density. Blue spheres have radius $\sqrt{2}-1$, yellow ones radius $1$, and red ones are the support spheres defined in Section \ref{['sec:settings']}: they are tangent to the four other spheres). In the two bottom cases, the upper blue sphere is tangent to the opposite face and the red sphere has the same size.
  • ...and 10 more figures

Theorems & Definitions (16)

  • Theorem 1
  • Corollary 1
  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Conjecture 1
  • Proposition 3
  • Proposition 4
  • Definition 3
  • ...and 6 more