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Maximum density for binary sphere packings with radii 1 and √2 − 1

Determine whether the hexagonal compact packing of unit spheres with spheres of radius √2 − 1 inserted in every octahedral site achieves the maximum density among all three-dimensional packings of spheres of radii 1 and √2 − 1, with maximal density equal to (5/3 − √2)π ≈ 0.793.

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Background

The case of radii 1 and √2 − 1 is distinguished by the ability to place small spheres into octahedral sites of hexagonal compact packings of unit spheres, and by structural properties that make tetrahedralizations possible.

This configuration is widely conjectured to be globally density-optimal for this binary mixture. The authors’ upper bound (≈0.81254) is close to the general upper bound 0.813 from prior work, but still above the conjectured optimum 0.793, leaving the exact optimality status unresolved.

References

Indeed, this packing is conjectured (e.g. in ) to maximize the density among packings with spheres of size $1$ and $\sqrt{2}-1$, namely with the following value, slightly less than the above-mentioned upper bound of $0.813$:

\left(\tfrac{5}{3}-\sqrt{2}\right)\pi\approx 0.793.

— Bounding the density of binary sphere packing (2505.14110 - Fernique et al., 20 May 2025) in Introduction (after Figure 4)