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\(D_3\) face-centered cubic (fcc) lattice

Alternative names: Cannonball lattice.

Description

Laminated three-dimensional lattice consisting of layers of triangular 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].

Cousins

  • \(\Lambda_{24}\) Leech lattice— The Leech lattice can be constructed via the Turyn construction and the holy construction using the octacode as the glue code; one of these constructions uses eight copies of the \(D_3\) fcc lattice [6,7].
  • Body-centered cubic (bcc) lattice— 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.

Member of code lists

Primary Hierarchy

Classical Domain
Analog Kingdom
Analog codeECC
Sphere packing
Lattice-based codeECC
Root lattice
\(D_n\) checkerboard latticeECC
Parents
\(D_n\) checkerboard lattice
The \(D_3\) root lattice is equivalent to the \(A_3\) root lattice [8; Ch. 10].
\(D_3\) 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]
Feng-Wen Sun and H. C. A. van Tilborg, “The Leech lattice, the octacode, and decoding algorithms”, IEEE Transactions on Information Theory 41, 1097 (1995) DOI
[7]
A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
[8]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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Zoo Code ID: dthree

Cite as:
“\(D_3\) face-centered cubic (fcc) lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dthree
BibTeX:
@incollection{eczoo_dthree, title={\(D_3\) face-centered cubic (fcc) lattice}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dthree} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/sphere_packing/lattice/root/dthree.yml.