Double lattice

In mathematics, especially in geometry, a double lattice in ℝn is a discrete subgroup of the group of Euclidean motions that consists only of translations and point reflections and such that the subgroup of translations is a lattice. The orbit of any point under the action of a double lattice is a union of two Bravais lattices, related to each other by a point reflection. A double lattice in two dimensions is a p2 wallpaper group. In three dimensions, a double lattice is a space group of the type 1, as denoted by international notation.

Double lattice packing

The best known packing of equal-sized regular pentagons on a plane is a double lattice structure which covers 92.131% of the plane.

A packing that can be described as the orbit of a body under the action of a double lattice is called a double lattice packing. In many cases the highest known packing density for a body is achieved by a double lattice. Examples include the regular pentagon, heptagon, and nonagon[1] and the equilateral triangular bipyramid.[2] Włodzimierz Kuperberg and Greg Kuperberg showed that all convex planar bodies can pack at a density of at least √3/2 by use a double lattice.[3]

References

  1. ↑ de Graaf, Joost; van Roij, René; Dijkstra, Marjolein (2011), "Dense Regular Packings of Irregular Nonconvex Particles", Physical Review Letters, 107: 155501, arXiv:1107.0603, Bibcode:2011PhRvL.107o5501D, doi:10.1103/PhysRevLett.107.155501, PMID 22107298
  2. ↑ Haji-Akbari, Amir; Engel, Michael; Glotzer, Sharon C. (2011), "Degenerate Quasicrystal of Hard Triangular Bipyramids", Phys. Rev. Lett., 107 (21): 215702, arXiv:1106.5561, Bibcode:2011PhRvL.107u5702H, doi:10.1103/PhysRevLett.107.215702, PMID 22181897
  3. ↑ Kuperberg, G.; Kuperberg, W. (1990), "Double-lattice packings of convex bodies in the plane", Discrete and Computational Geometry, 5 (4): 389–397, doi:10.1007/BF02187800, MR 1043721
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