28 research outputs found
The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve
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
preprintComment: The citations and title have been update
A formal proof of the Kepler conjecture
This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project
A formal proof of the Kepler conjecture
This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project
On a strong version of the Kepler conjecture
We raise and investigate the following problem that one can regard as a very
close relative of the densest sphere packing problem. If the Euclidean 3-space
is partitioned into convex cells each containing a unit ball, how should the
shapes of the cells be designed to minimize the average surface area of the
cells? In particular, we prove that the average surface area in question is
always at least 13.8564... .Comment: 9 page
A Dense Packing of Regular Tetrahedra
We construct a dense packing of regular tetrahedra, with packing density .Comment: full color versio
A glimpse into Thurston's work
We present an overview of some significant results of Thurston and their
impact on mathematics. The final version of this paper will appear as Chapter 1
of the book "In the tradition of Thurston: Geometry and topology", edited by K.
Ohshika and A. Papadopoulos (Springer, 2020)