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Local Density

Let each Sphere in a Sphere Packing expand uniformly until it touches its neighbors on flat faces. Call the resulting Polyhedron the Local Cell. Then the local density is given by

\begin{displaymath}
\rho\equiv {V_{\rm sphere}\over V_{\rm local\ cell}}.
\end{displaymath}

When the Local Cell is a regular Dodecahedron, then

\begin{displaymath}
\rho_{\rm dodecahedron}={\pi\sqrt{5+\sqrt{5}}\over 15\sqrt{10}\,(\sqrt{5}-2)}=0.7547\ldots.
\end{displaymath}

See also Local Density Conjecture




© 1996-9 Eric W. Weisstein
1999-05-25