Math Conjectures

Reinhardt's Smoothed-Octagon Conjecture

OPENMajorConjectureProposed 1934 · Standard version

Canonical statement

For a centrally symmetric convex disk K⊂R2K\subset\mathbb R^2, define
δL(K)=sup⁡Λarea⁡(K)det⁡Λ, \delta_L(K)=\sup_\Lambda\frac{\operatorname{area}(K)}{\det\Lambda},
where the supremum is over full-rank lattices Λ⊂R2\Lambda\subset\mathbb R^2 for which the interiors of the translates K+λK+\lambda, λ∈Λ\lambda\in\Lambda, are pairwise disjoint. Let OO be Reinhardt's smoothed octagon, obtained from a regular octagon by replacing each vertex by the hyperbola arc tangent to its two incident sides and asymptotic to the two adjacent nonincident sides. Then
δL(K)≥δL(O)=8−42−log⁡2 22−1 . \delta_L(K)\geq\delta_L(O) =\frac{8-4\sqrt2-\log2}{\,2\sqrt2-1\,}.
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For a centrally symmetric convex disk
\(K\subset\mathbb R^2\), define
\[
  \delta_L(K)=\sup_\Lambda\frac{\operatorname{area}(K)}{\det\Lambda},
\]
where the supremum is over full-rank lattices
\(\Lambda\subset\mathbb R^2\) for which the interiors of the translates
\(K+\lambda\), \(\lambda\in\Lambda\), are pairwise disjoint. Let \(O\)
be Reinhardt's smoothed octagon, obtained from a regular octagon by
replacing each vertex by the hyperbola arc tangent to its two incident
sides and asymptotic to the two adjacent nonincident sides. Then
\[
  \delta_L(K)\geq\delta_L(O)
  =\frac{8-4\sqrt2-\log2}{\,2\sqrt2-1\,}.
\]
DG-010:[Reinhardt1934][Hales2011Reinhardt][HalesVajjha2024Smoothed]
Discrete geometryGeometry and topology

Notes

Among centrally symmetric convex disks in the plane, which is worst at packing? Writing δL(K)\delta_L(K) for the largest density of a lattice packing by translates of KK, Reinhardt conjectured in 1934 that the minimum of δL\delta_L is attained by his smoothed octagon, a regular octagon whose corners are rounded by hyperbola arcs tangent to the incident sides [Reinhardt1934]. Its lattice packing density is (8−42−log⁡2)/(22−1)≈0.9024(8-4\sqrt2-\log 2)/(2\sqrt2-1)\approx 0.9024, slightly below the disk's π/12≈0.9069\pi/\sqrt{12}\approx 0.9069.

Substantial structure theory is now in place. A minimizing body is known to exist, and Hales developed a rigorous program for the problem, establishing regularity properties that any minimizer must satisfy [Hales2011Reinhardt]. Hales and Vajjha pushed this program much further with a detailed analysis of packings of smoothed polygons, proving strong local optimality results for the smoothed octagon [HalesVajjha2024Smoothed].

What is missing is the global statement: one must show that every global minimizer of δL\delta_L is affinely equivalent to the smoothed octagon. Until that step is completed the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.