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 using a double lattice.[3]

In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that the double lattice packing of the regular pentagon (which they call the "pentagonal ice-ray" packing, and which they trace to the work of Chinese artisans in 1900) has the optimal density among all packings of regular pentagons in the plane.[4] As of 2020, their proof has not yet been refereed and published.

References
  1. de Graaf, Joost; van Roij, René; Dijkstra, Marjolein (2011), "Dense regular packings of irregular nonconvex particles", Physical Review Letters, 107 (15): 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 & Computational Geometry, 5 (4): 389–397, doi:10.1007/BF02187800, MR 1043721
  4. Hales, Thomas; Kusner, Wöden (September 2016), Packings of regular pentagons in the plane, arXiv:1602.07220
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