Alternative names: \(A_2\) hexagonal lattice.
Description
Two-dimensional lattice that corresponds to the triangular tiling and that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. As a tiling, its dual (whose points lie at the centers of each triangle) is the honeycomb tiling.
It's generator matrix is \begin{align} V=\begin{pmatrix}1 & 0\\ -1/2 & \sqrt{3}/2 \end{pmatrix}~. \tag*{(1)}\end{align} All possible sublattices are characterized in Refs. [1,2] from the point of view of information transmission over the AGWN channel.
Protection
The triangular lattice exhibits the densest packing with density \(\Delta = \pi/\sqrt{12} \approx 0.9069\) [3; Sec. 1.4], the highest kissing number of 6, and the thinnest covering with thickness \(\Theta = 2\pi/3\sqrt{3}\approx 1.2092\) [4] in two dimensions. It solves the quantizer problem in two dimensions with \(G_2 = \frac{5}{36\sqrt{3}}\) [5–8]. It also solves the Gaussian channel coding problem [6].Cousins
- Polygon code— The Voronoi cell of the triangular lattice is the hexagon.
- Honeycomb tiling— The Voronoi cell of the triangular lattice (honeycomb tiling) is a hexagon (triangle). Triangular and hexagonal tilings are dual to each other as tilings, i.e., the vertices of one tiling lie at the centers of faces of the other. Points of the honeycomb tiling form two triangular lattices. The ruby tiling is a fattened honeycomb tiling interpolating between the honeycomb tiling and triangular lattice.
- Universally optimal sphere packing— The triangular lattice is universally optimal among all lattices, but has not been proven to be optimal over all periodic packings [11].
- Hexagonal GKP code— The hexagonal GKP code is based on the triangular lattice.
Member of code lists
Primary Hierarchy
Parents
\(A_2\) triangular lattice
References
- [1]
- M. Bernstein, N. J. A. Sloane, and P. E. Wright, “On sublattices of the hexagonal lattice”, Discrete Mathematics 170, 29 (1997) DOI
- [2]
- M. Baake and P. A. B. Pleasants, “Algebraic Solution of the Coincidence Problem in Two and Three Dimensions”, Zeitschrift für Naturforschung A 50, 711 (1995) DOI
- [3]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [4]
- R. Kershner, “The Number of Circles Covering a Set”, American Journal of Mathematics 61, 665 (1939) DOI
- [5]
- L. Fejes Toth, Sur Ia representation d' une population infinie par une nombre fini d'elements, AMAH 10 (1959), 299-304
- [6]
- L. F. Tóth, Lagerungen in Der Ebene Auf Der Kugel Und Im Raum (Springer Berlin Heidelberg, 1972) DOI
- [7]
- A. Gersho, “Asymptotically optimal block quantization”, IEEE Transactions on Information Theory 25, 373 (1979) DOI
- [8]
- D. Newman, “The hexagon theorem”, IEEE Transactions on Information Theory 28, 137 (1982) DOI
- [9]
- C. Campopiano and B. Glazer, “A Coherent Digital Amplitude and Phase Modulation Scheme”, IEEE Transactions on Communications 10, 90 (1962) DOI
- [10]
- G. Foschini, R. Gitlin, and S. Weinstein, “Optimization of Two-Dimensional Signal Constellations in the Presence of Gaussian Noise”, IEEE Transactions on Communications 22, 28 (1974) DOI
- [11]
- H. L. Montgomery, “Minimal theta functions”, Glasgow Mathematical Journal 30, 75 (1988) DOI
Page edit log
- Victor V. Albert (2022-11-23) — most recent
Cite as:
“\(A_2\) triangular lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hexagonal