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