Hermite constant
Constant relating to close packing of spheres
title: "Hermite constant" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["systolic-geometry", "geometry-of-numbers", "mathematical-constants"] description: "Constant relating to close packing of spheres" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Hermite_constant" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Constant relating to close packing of spheres ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/63/Hermite_constant.svg" caption="A hexagonal lattice with unit covolume (the area of the quadrilateral is 1). Both arrows are minimum non-zero elements for n-1 with length \lambda_n=\sqrt{\gamma_n}=\sqrt{2/\sqrt{3}}."] ::
In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.
The constant \gamma_n for integers n0 is defined as follows. For a lattice L in Euclidean space \R^n with unit covolume, i.e. \operatorname{vol}(\R^n/L)=1, let \lambda_1(L) denote the least length of a nonzero element of L. Then \sqrt{\gamma_n} is the maximum of \lambda_1(L) over all such lattices L.
The square root in the definition of the Hermite constant is a matter of historical convention.
Alternatively, the Hermite constant \gamma_n can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.
Examples
The Hermite constant is known in dimensions 1–8 and 24.
::data[format=table]
| n | \gamma_n^n |
|---|---|
| 1 | 2 |
| 1 | \frac 4 3 |
| :: |
For n=2, one has \gamma_2=2/\sqrt{3}. This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area.
Estimates
It is known that
\gamma_n \le \left( \frac 4 3 \right)^\frac{n-1}{2}.
A stronger estimate due to Hans Frederick Blichfeldt is
\gamma_n \le \left( \frac 2 \pi \right)\Gamma\left(2 + \frac n 2\right)^\frac{2}{n}, where \Gamma(x) is the gamma function.
References
References
- Cassels (1971) p. 36
- Kitaoka (1993) p. 36
- Blichfeldt, H. F.. (1929). "The minimum value of quadratic forms, and the closest packing of spheres". Math. Ann..
- Kitaoka (1993) p. 42
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::