Kissing number

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Kissing number

Summary

The kissing number is a classic problem in geometry that asks for the maximum number of identical, non-overlapping spheres that can simultaneously touch a single sphere of the same size. Although the idea is easy to visualize, finding the exact kissing number becomes increasingly difficult as the number of dimensions grows. The problem originated in a famous 1694 debate between Isaac Newton and David Gregory over whether 12 or 13 spheres could touch one central sphere in three dimensions; Newton was correct, but a rigorous proof was not discovered until 1953. 

The exact kissing numbers are known only for a few dimensions, including 1 (2), 2 (6), 3 (12), 4 (24), 8 (240), and 24 (196,560), while for most higher dimensions mathematicians have established only upper and lower bounds. The kissing number problem is closely related to sphere packing, coding theory, lattice geometry, and optimisation, making it important in both pure and applied mathematics. Research continues to improve bounds and discover new methods, including semidefinite programming and modern AI-assisted techniques, highlighting the problem as one of the most fascinating and enduring challenges in discrete geometry.


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