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 has the optimal density among all packings of regular pentagons in the plane.[4] This packing has been used as a decorative pattern in China since at least 1900, and in this context has been called the "pentagonal ice-ray".[5] As of 2021[update], the proof of its optimality has not yet been refereed and published.
It has been conjectured that, among all convex shapes, the regular heptagon has the lowest packing density for its optimal double lattice packing, but this remains unproven.[6]
double, lattice, mathematics, especially, geometry, double, lattice, ℝn, discrete, subgroup, group, euclidean, motions, that, consists, only, translations, point, reflections, such, that, subgroup, translations, lattice, orbit, point, under, action, double, la. 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 edit nbsp 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 Wlodzimierz 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 Woden Kusner announced a proof that the double lattice packing of the regular pentagon has the optimal density among all packings of regular pentagons in the plane 4 This packing has been used as a decorative pattern in China since at least 1900 and in this context has been called the pentagonal ice ray 5 As of 2021 update the proof of its optimality has not yet been refereed and published It has been conjectured that among all convex shapes the regular heptagon has the lowest packing density for its optimal double lattice packing but this remains unproven 6 References edit de Graaf Joost van Roij Rene 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 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 Kuperberg G Kuperberg W 1990 Double lattice packings of convex bodies in the plane Discrete amp Computational Geometry 5 4 389 397 doi 10 1007 BF02187800 MR 1043721 Hales Thomas Kusner Woden September 2016 Packings of regular pentagons in the plane arXiv 1602 07220 Dye Daniel Sheets 2012 Chinese Lattice Designs Dover pp 307 309 ISBN 9780486146225 Kallus Yoav 2015 Pessimal packing shapes Geometry amp Topology 19 1 343 363 arXiv 1305 0289 doi 10 2140 gt 2015 19 343 MR 3318753 Retrieved from https en wikipedia org w index php title Double lattice amp oldid 1101448570, wikipedia, wiki, book, books, library,
