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On Dense Tetrahedra in Binary Sphere Packings

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Pith's one-line read A computer-assisted geometric proof shows that packings of spheres of radii 1 and sqrt(2)-1 have density at most about 0.812542, slightly improving the previous best bound of 0.813.

arxiv 2505.14110 v2 pith:5CLGWX7F submitted 2025-05-20 math.MG cs.CG

Thomas Fernique , Daria Pchelina This is my paper · ORCID
classification math.MGcs.CG
keywords packingsspheresbounddensedensitypackingsizessphere
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies packings of spheres of two sizes in three-dimensional space: large spheres of radius 1 and small spheres of radius about 0.414. This particular size ratio is special because each hole in a hexagonal close packing of large spheres can hold exactly one small sphere. The authors want to prove an upper bound on how much space any such packing can occupy. They use a decomposition of space into tetrahedra whose vertices are sphere centers, called FM-tetrahedra. The density of the whole packing is at most the largest density of any single tetrahedron in this decomposition. The main work is to show that the densest possible tetrahedron has a specific shape, with one small sphere and three large spheres or four large spheres, etc. The proof combines a local analysis near the suspected best tetrahedra with a global computer check that divides the space of all possible tetrahedra into over 800 million small blocks and verifies, using interval arithmetic, that every block has density below the claimed bound. Interval arithmetic gives guaranteed bounds: each calculation returns an interval that is guaranteed to contain the true value. The result is an upper bound of about 0.81254, slightly better than the previous 0.813, but still above the conjectured optimal density of about 0.793. The interest lies partly in the method: using interval arithmetic and dimension-reduction tricks to turn a continuous optimization problem in six variables into a finite computer check.
Extended reading notes

Core claim

Theorem 1 and Corollary 1: For r = sqrt(2)-1, the densest FM-tetrahedra of types 1111, 11rr, 1rrr have only tight edges; the densest FM-tetrahedron of type rrrr has two equal stretched edges; the densest FM-tetrahedron of type 111r has one stretched edge between two large spheres; and consequently any packing of spheres of radii 1 and r has density at most delta*_111r = 0.812542027810834866943600528883352220338748559263354479...

Load-bearing premise

The correctness of the global computer-assisted verification. The proof's conclusion depends on the recursive interval-arithmetic block check (check.cpp, Section 6.3) correctly implementing the FM-tetrahedron domain, the density upper bound via solid angles, the support-sphere radius computation via the quadratic polynomial, and the dimension reduction via sliding. In particular, degenerate cases in the quadratic solver (where both coefficients A and B vanish, Section 6.1) are argued to be safe only because the recursive search terminated; no explicit proof is given that every such degenerate block is correctly discarded.

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Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The proof relies on standard geometric results (Cayley-Menger, solid angle formulas) and domain-specific assumptions (FM-decomposition, saturation, support sphere bounds). The free parameters listed are proof certificates (neighborhood radii and heuristic exponent vectors), not physical parameters; they do not enter the theorem statement. No new speculative physical entities are introduced.

free parameters (7)
  • epsilon_1111 = 1/46
    Neighborhood radius in Section 4.1 for type 1111; chosen by hand to make the mean-value-theorem check succeed.
  • epsilon_11rr = 1/203
    Neighborhood radius in Section 4.1 for type 11rr; chosen by hand.
  • epsilon_1rrr = 1/148
    Neighborhood radius in Section 4.1 for type 1rrr; chosen by hand.
  • epsilon_rrrr = 1/173
    Neighborhood radius in Section 4.2 for type rrrr; chosen by hand.
  • epsilon_111r = 1/445
    Neighborhood radius in Section 4.2 for type 111r; chosen by hand.
  • k_rrrr = (35,35,58,40,35,35)
    Exponents in the certificate function alpha for type rrrr (Section 4.2), found heuristically.
  • k_111r = (20,80,80,40,40,160)
    Exponents in the certificate function alpha for type 111r (Section 4.2), found heuristically.
assumptions (6)
  • domain assumption The packing can be assumed saturated without loss of generality.
    Introduction and Section 2.1; adding spheres to holes cannot decrease density.
  • domain assumption FM-decomposition covers space and the packing density is at most the maximum density of its FM-tetrahedra.
    Section 2.1, from [FTM58]; the density is a weighted average of tetrahedron densities, each bounded by the max.
  • domain assumption Every face of an FM-tetrahedron is an FM-triangle (Proposition 3).
    Proven in Appendix B using computer algebra; used in Lemma 1 to bound |AH'| below by min radius.
  • domain assumption The support sphere of an FM-tetrahedron has radius < r and satisfies the quadratic polynomial of Proposition 4.
    Propositions 1 and 4; the radius formula is derived by computer algebra (radius.sage).
  • domain assumption The interval arithmetic libraries used (SageMath and the C++ interval type) compute outward-rounded results.
    Section 3.1; necessary for the certified inequalities in Sections 4-6.
  • standard math The Cayley-Menger determinant and Lagrange's formula for solid angles are correct.
    Section 2.2; standard results.

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Pith. "Pith review of On Dense Tetrahedra in Binary Sphere Packings." pith.science (2026). https://pith.science/paper/5CLGWX7F

@misc{pith2026250514110,
  author       = {Pith},
  title        = {Pith review of: On Dense Tetrahedra in Binary Sphere Packings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5CLGWX7F}},
  note         = {Machine review of arXiv:2505.14110}
}
abstract

This paper considers the density of tetrahedra arising in a specific decomposition of packings of unequal spheres in $\mathbb{R}^3$. It aims to extend a bound obtained in 2D in the 1960s by Florian. The focus is on packings of spheres of sizes $1$ and $\sqrt{2}-1$: the small sphere fits exactly into each octahedral hole of a hexagonal close packing of large spheres, yielding a conjecturally maximally dense packing (for these sizes). The paper slightly improves, by completely different means, the previous best upper bound on the density of such packings. The proof combines geometric insight with challenging interval arithmetic computations, which may be of independent interest.

Figures

Figures reproduced from arXiv: 2505.14110 by the authors.

Figure 1
Figure 1. Left: a packing of unit disks in the Euclidean plane and a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Left: a layer of a hexagonal compact packing (yellow spheres) and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Left: the densest triangle that can appear in a packing of disks of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: A cannonball packing of unit spheres (yellow) with spheres of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: The densest FM-tetrahedra of each type, with a numerical ap [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Left: consider a triangle ABC with disks centered on its vertices. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Left: the densest Vorono¨ı cell in a packing of unit disks. Center: [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: The part of the box below the blue veil is a 3-dim. cut of [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Density of some extremal FM-tetrahedra as a function of [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Zoom on the black frame in Fig. 9. The extreme tetrahedra for [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: The set of FM-tetrahedra is delimited by the surface [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Consider the tetrahedron of type 11rr with edge lengths [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]
Figure 13
Figure 13. Figure 13: Consider a tetrahedron T = ABCD, where the sphere centered in A is slided towards D and becomes centered in A′ . Denote by dT the tetrahedron AA′BC, by H the orthogonal projection of A onto A′BC, by H′ the orthogonal projection of H onto BC and by h the length |AH| of…
Figure 14
Figure 14. Figure 14: Sliding spheres towards the support sphere (in red) until a [PITH_FULL_IMAGE:figures/full_fig_p031_14.png]
Figure 15
Figure 15. Figure 15: There are 5 types of tetrahedra (one per column). The radii of [PITH_FULL_IMAGE:figures/full_fig_p032_15.png]

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