The kissing number problem in geometry asks for the greatest number of non-overlapping unit spheres that can be arranged in a given space so that each one touches a single central unit sphere, a quantity also called the Newton number or contact number. In three dimensions the question is famous for a disagreement between Isaac Newton and David Gregory over whether the answer was 12 or 13; Newton's figure of 12 turned out to be correct, but the first rigorous proof was not published until 1953, by Kurt Schutte and Bartel van der Waerden. The exact kissing number remains unknown in most higher dimensions, making the general problem an active area of research.
Facts
StatementThe kissing number problem asks for the greatest number of non-overlapping unit spheres that can be arranged in a given dimension so that each one touches a common central unit sphere. 1 Progress Toward ResolutionThe kissing number is known exactly only in dimensions 1, 2, 3, 4, 8 and 24: it is 2, 6, 12, 24, 240 and 196560 respectively. The three-dimensional value of 12 was first proved by Schutte and van der Waerden in 1953, settling a historical dispute between Isaac Newton and David Gregory. The four-dimensional value of 24 was proved by Oleg Musin in 2003. The eight and twenty four dimensional values come from the E8 lattice and the Leech lattice. The value is unknown for every other dimension. 1 Prize Status
Prize Status (category)w-freetextdim2b-0926: category derived from a free-text property; original status/verification detail carried on the source property. Classification
Resolution Status Open Questions
Prize StatusNo prize is recorded.
No source consulted this pass names a monetary prize for this problem. Connections
Sources
1. Kissing Number (Wikipedia)
Wikimedia FoundationLead section
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres (i.e., of radius 1) that can be arranged in that space such that they each touch a common unit sphere.
Larger dimensions section
The existence of the highly symmetrical E8 lattice and Leech lattice has allowed to determine the kissing number for n = 8 (namely, 240) and for n = 24 (namely, 196,560).
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