Kepler's Conjecture/Mistake

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Source Work

1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables:

Thème et variations
$0,77963 55700 \ldots$


  • $\sqrt {18} \paren {\operatorname {Arcos} 1/3 - \pi / 3}$
Le meilleur majorant connu pour la densité d'un empilement de sphères dans $R^3$.

That is, in English:

  • $\sqrt {18} \paren {\arccos 1/3 - \pi / 3}$
The best known upper bound for the density of a stack of spheres in $\R^3$.


This constant is in fact the packing density of a regular tetrahedron.

That is:

Let $S$ be a regular tetrahedron of edge length $2$.
Let $B$ be the part of $S$ that lies within distance $1$ of some vertex.
Then this constant is the ratio of the volume of $B$ to the volume of $S$.

This sequence is A267040 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see