# Kepler's Conjecture

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## 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

## 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 $\dfrac \pi {\sqrt {18} }$.*

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): Entry:**Kepler's conjecture**