Also known as Cannonball lattice.

## Description

Laminated three-dimensional lattice consisting of layers of hexagonal lattices.

## Protection

The \(D_3\) fcc lattice exhibits the densest packing, highest kissing number of 12, and the thinnest lattice covering in three dimensions. The Kepler conjecture [1] states that the \(D_3\) fcc lattice has the densest packing in 3D.

It was first proven that this lattice was the densest 3D lattice packing [2]. Determining the maximum density of any sphere packing in 3D was then reduced to a computationally tractable problem [3], which was solved [4] and formalized in automated proof checking software [5].

## Parent

- \(D_n\) checkerboard lattice code — The \(D_3\) root lattice is equivalent to the \(A_3\) root lattice [6; Ch. 10].

## Cousins

- Body-centered cubic (bcc) lattice code — The bcc and fcc lattices are dual to each other.
- Cubeoctahedron code — Cubeoctahedron codewords form the minimal shell of the \(D_3\) face-centered cubic (fcc) lattice.

## References

- [1]
- Kepler, Johannes. The six-cornered snowflake. Paul Dry Books, 2010.
- [2]
- C.F. Gauss, Besprechung des Buchs von L.A. Seeber, Untersuchungen uber die Eigenschaften der positiven terndren quadratischen Formen usw. Gottingsche Gelehrte Anzeigen, July 9, 1831 = Werke, II, pp. 18g8-196, 1876.
- [3]
- L. F. Tóth, Lagerungen in Der Ebene Auf Der Kugel Und Im Raum (Springer Berlin Heidelberg, 1972) DOI
- [4]
- T. C. Hales, “An overview of the Kepler conjecture”, (2002) arXiv:math/9811071
- [5]
- T. Hales et al., “A formal proof of the Kepler conjecture”, (2015) arXiv:1501.02155
- [6]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.

## Page edit log

- Victor V. Albert (2022-11-23) — most recent

## Cite as:

“\(D_3\) face-centered cubic (fcc) lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dthree