The kissing numbers of tetrahedra

Packings with large minimum kissing numbers

For each proper power of 4, n, we describe a simple explicit construction of a finite collection of pairwise disjoint open unit balls in R in which each ball touches more than 2 √ n others. A packing of balls in the Euclidean space is a finite or infinite collection of pairwise disjoint open unit balls in Rn. It is called a lattice packing if the centers of the balls form a lattice in Rn. The m...

A Lower Bound for the Translative Kissing Numbers of Simplices

First we recall some standard definitions. By a d-dimensional convex body we mean a compact convex subset of Rd with non-empty interior. Two subsets of Rd with non-empty interiors are non-overlapping if they have no common interior point, and we say that they touch each other if they are non-overlapping and their intersection is non-empty. Denote by H(K) the translative kissing number of a d-di...

High-Accuracy Semidefinite Programming Bounds for Kissing Numbers

The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for n ≤ 24. The bound for n = 16 implies a ...

Quick asymptotic upper bounds for lattice kissing numbers

General upper bounds for lattice kissing numbers are derived using Hurwitz zeta functions and new inequalities for Mellin transforms. 1 Statement of results Let τn be the kissing number in dimension n, i.e. the maximal number of balls of equal size in Euclidean space of dimension n which can touch another one of the same radius without any two overlapping. Similarly let λn be the maximal lattic...

Average Kissing Numbers for Non-congruent Sphere Packings