Definition:Hilbert 23/18b

Hilbert $23$: Problem $18$b

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Kepler's Conjecture

The densest packing of identical spheres in space is obtained when the spheres are arranged with their centers at the points of a face-centered cubic lattice.

This obtains a density of $\dfrac \pi {3 \sqrt 2} = \dfrac \pi {\sqrt {18} }$:

$\dfrac \pi {\sqrt {18} } = 0 \cdotp 74048 \ldots$

Historical Note

The Hilbert 23 were delivered by David Hilbert in a famous address at Paris in $1900$.

He considered them to be the oustanding challenges to mathematicians in the future.

There was originally going to be a $24$th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers $4$, $6$, $16$ and $23$) too vague to ever be described as "solved".

Sources

• 1902: David Hilbert: Mathematical Problems (Bull. Amer. Math. Soc. Vol. 8, no. 10: pp. 437 – 479)

(translated by Mary Winston Newson from "Mathematische Probleme")