The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve
Thomas C. Hales

TL;DR
This paper presents proofs of two conjectures in sphere packings in three-dimensional space, establishing lower bounds on Voronoi cell surface area and characterizing packings with twelve contacts per sphere, using computer-assisted methods.
Contribution
It provides the first complete proofs of the strong dodecahedral conjecture and Fejes Toth's contact conjecture in 3D sphere packings, employing computer-assisted techniques.
Findings
Voronoi cells have surface area at least that of a regular dodecahedron.
Packings with each sphere touched by twelve others form hexagonal layers.
Both theorems are proved with computer assistance, confirming longstanding conjectures.
Abstract
This article sketches the proofs of two theorems about sphere packings in Euclidean 3-space. The first is K. Bezdek's strong dodecahedral conjecture: the surface area of every bounded Voronoi cell in a packing of balls of radius 1 is at least that of a regular dodecahedron of inradius 1. The second theorem is L. Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. Both proofs are computer assisted. Complete proofs of these theorems appear in the author's book "Dense Sphere Packings" and a related preprint
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Theoretical and Applied Studies in Material Sciences and Geometry · Mathematics and Applications · Advanced Materials and Mechanics
