Kissing numbers for surfaces

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...

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

New Upper Bounds for Kissing Numbers from Semidefinite Programming

In geometry, the kissing number problem asks for the maximum number τn of unit spheres that can simultaneously touch the unit sphere in n-dimensional Euclidean space without pairwise overlapping. The value of τn is only known for n = 1, 2, 3, 4, 8, 24. While its determination for n = 1, 2 is trivial, it is not the case for other values of n. The case n = 3 was the object of a famous discussion ...