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Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere intersection is forbidden, the maximum non-intersecting radius of each sphere is $r_c = \frac{1}{\sqrt{2}}$.

Let $R_v > \frac{r_c\sqrt{18}}{\pi}$ be the minimum radius of each sphere in the fcc packing arrangment s.t., allowing for sphere-sphere overlaps, no point in the lattice is outside of a sphere. Here, $\frac{\pi}{\sqrt{18}}=\frac{\pi}{3\sqrt{2}}$ is the maximum possible packing density of non-intersecting spheres as per Hales' proof of the Kepler conjecture.

What is the exact value of $R_v$? Can this value be easily calculated for other sphere packing geometries provided the lattice basis?

Update: Noam D. Elkies answered by question in the comments below. What I'm looking for is the "covering radius" for the lattice. I'm currently searching for this value for the fcc/$A_3$ lattice, but if anyone happens to know it, it would be great if you could help me out!

Update 2: The covering radius for the fcc/$A_3$ lattice appears to simply be $R_v=r_c*\sqrt{2}=1$.

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That's called the covering radius of the lattice. See Conway and Sloane's Sphere Packings, Lattices and Groups for more information, including the covering radius of this lattice (which they call $A_3$) and many others. – Noam D. Elkies May 29 '13 at 2:57
@Noam D. Elkies Thanks so much for your response, that more or less answers my question. – RMoser May 29 '13 at 3:05
A caution, SPLAG mostly write the covering radius as a multiple of the packing radius, which you have already scaled as $1/\sqrt 2.$ – Will Jagy May 29 '13 at 4:40

1 Answer 1

Although not directly answering the question, the discussion here highlights connections between the question of $R_v$ and the Voronoi cell of this lattice. In short, it turns out that $R_v$ is the maximal distance from a lattice point to any of its Voronoi vertices. Computing Voronoi vertices can be done relatively quickly, certainly in 3D.

If the points are not arranged on a lattice, then a "covering radius" of the set of points can be defined as the maximum/supremum over the $R_v$, where $R_v$ can be different for each point. Of course this reduces to the above in the case of lattices, where $R_v$ is identical for all points.

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