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

$$\rho\equiv {V_{\rm sphere}\over V_{\rm local\ cell}}.$$

When the Local Cell is a regular Dodecahedron, then

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

See also Local Density Conjecture