Kepler's Conjecture

Theorem

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 $\dfrac \pi {3 \sqrt 2} = \dfrac \pi {\sqrt {18} }$:

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

This sequence is A093825 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

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Source of Name

This entry was named for Johannes Kepler.

Historical Note

This result was conjectured by Johannes Kepler in $1611$.

While it is in a certain sense obvious that the most efficient technique for packing spheres is the one traditionally used by greengrocer's to stack orange's, it proved challenging to actually prove it.

Many mathematicians believe, and all physicists know, that the density cannot exceed $\dfrac \pi {\sqrt {18} }$.

-- Claude Ambrose Rogers

The proof was finally provided in $2014$ by Thomas Callister Hales.

Sources

• 1958: C.A. Rogers: The packing of equal spheres (Proc. London Math. Soc. Ser. 3 Vol. 8: pp. 609 – 620)

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 7404 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 7404 \ldots$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Kepler's conjecture