What Is a Kissing Number?

Imagine placing identical billiard balls around one central ball. How many can touch it at once without any two outer balls overlapping? That maximum is the kissing number of the space, written k(n) in n-dimensional Euclidean space.

Three dimensions is not obvious

In two dimensions, six equal circles fit around a central circle. In three dimensions, the answer is 12. The picture looks familiar, but proving that 13 cannot fit is much harder than arranging 12. The three-dimensional case was settled in 1953 after a long history of competing constructions and arguments.

Four dimensions: exactly 24

The four-dimensional problem is a useful reminder that a construction is only half a result. It is easy to show that 24 spheres can touch a central sphere using a highly symmetric arrangement. The difficult part is proving that a 25th sphere is impossible. Oleg Musin’s proof established k(4) = 24 using a strengthened version of Delsarte’s method.

Why the Registry cares

A kissing number is a clean frontier: a precise object, a precise metric, and a number that can move only when a construction or an impossibility proof improves. Browse the Registry’s related discrete-geometry records for the same distinction between an achieved arrangement and a proven optimum.

Primary source

Musin, “The kissing number in four dimensions,” Annals of Mathematics ↗