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Rhombic Dodecahedral Conjecture

The rhombic dodecahedral conjecture asserts that a regular rhombic dodecahedron has the least surface area among three-dimensional parallelohedra of a fixed inradius. Cesaroni and Novaga (2026) proved the sharp inequality

S(P)>=12sqrt(2)r(P)^2,

where S(P) is the surface area and r(P) is the inradius. Equality holds iff P is similar to a regular rhombic dodecahedron. This is a fixed-inradius problem, distinct from minimizing surface area at fixed volume.

Cesaroni and Novaga (2026) report using ChatGPT to check algebraic identities and explore proof arguments. The authors checked the final statements and proofs.

See also

Inradius, Parallelohedron, Rhombic Dodecahedron, Truncated Octahedron

Explore with Wolfram|Alpha

References

Cesaroni, A. and Novaga, M. "The Rhombic Dodecahedral Conjecture." 19 Sep 2026. https://arxiv.org/abs/2610.13975.

Cite this as:

Weisstein, Eric W. "Rhombic Dodecahedral Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RhombicDodecahedralConjecture.html

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