A proof of the dodecahedral conjecture

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 Pa…

菱形十二面体猜想

The rhombic dodecahedral conjecture

该研究证明三维平行多面体满足关于其边界二维豪斯多夫测度与内切半径平方的不等式，等号对应正菱形十二面体，证实了菱形十二面体猜想。

The Dodecahedron as a Voronoi Cell and its (minor) importance for the Kepler conjecture

The regular dodecahedron has a 2% smaller volume than the rhombic dodecahedron which is the Voronoi cell of a fcc packing. From this point of view it seems possible that the dodecahedral aspect which is the core of the so-called dodecahedral conjecture, will play a major part for an elementary proof of the Kepler conjecture. In this paper we will show that the icosahedral configuration caused by dodecahedron leads to tetrahedra with significantly larger volume than the fcc fundamental parallelotope tessellation tetrahedra. Therefore on the basis of a tetrahedral based point of view for sphere …

A Direct Proof of the 2nd Atiyah-Sutcliffe Conjecture for Convex Quadrilaterals

We present a direct proof of the second conjecture made by M. Atiyah and P. Sutcliffe for the case of convex quadrilaterals. Unlike previous work on this conjecture, our proof does not require any computer aided computations. The new proof relies on a new geometric inequality proved recently by the author.

另一个简单证明的1维平坦链猜想

Another simple proof of the 1-dimensional flat chain conjecture

本文提供了一个简短的证明，解决了1维平坦链猜想的核心问题。

A Proof of Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve

In 1969, Fejes Toth conjectured that in Euclidean 3-space any packing of equal balls such that each ball is touched by twelve others consists of hexagonal layers. This article verifies this conjecture.