contact number | coordination number | kissing number problem | ligancy | Newton number

The number of equivalent hyperspheres in n dimensions which can touch an equivalent hypersphere without any intersections, also sometimes called the Newton number, contact number, coordination number, or ligancy. Newton correctly believed that the kissing number in three dimensions was 12, but the first proofs were not produced until the 19th century by Bender, Hoppe, and Günther. More concise proofs were published by Schütte and van der Waerden and Leech. After packing 12 spheres around the central one (which can be done, for example, by arranging the spheres so that their points of tangency with the central sphere correspond to the vertices of an icosahedron), there is a significant amount of free space left (above figure), although not enough to fit a 13th sphere.

Coxeter-Todd lattice | dodecahedral conjecture | Hermite constants | hypersphere packing | Kepler conjecture | Leech lattice | Minkowski-Hlawka theorem | sphere packing