| 词条 | Double lattice |
| 释义 |
In mathematics, especially in geometry, a double lattice in {{math|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 {{overline|1}}, as denoted by international notation. Double lattice packingA 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 {{math|{{sqrt|3}}/2}} by using a double lattice.[3]References1. ^{{citation | last = de Graaf | first = Joost | last2 = van Roij | first2 = René | last3 = Dijkstra | first3 = Marjolein | author3-link = Marjolein Dijkstra | journal = Physical Review Letters | pages = 155501 | title = Dense Regular Packings of Irregular Nonconvex Particles | doi = 10.1103/PhysRevLett.107.155501 | volume = 107 | issue = 15 | year = 2011 | bibcode=2011PhRvL.107o5501D | pmid=22107298| arxiv = 1107.0603}} 2. ^{{Citation | title = Degenerate Quasicrystal of Hard Triangular Bipyramids | last1 = Haji-Akbari | first1 = Amir | last2 = Engel | first2 = Michael | last3 = Glotzer | first3 = Sharon C. | journal = Phys. Rev. Lett. | volume = 107 | issue = 21 | pages = 215702 | year = 2011 | doi = 10.1103/PhysRevLett.107.215702 | bibcode=2011PhRvL.107u5702H | pmid=22181897| arxiv = 1106.5561}} 3. ^{{Citation | last1 = Kuperberg | first1 = G. | last2 = Kuperberg | first2 = W. | title = Double-lattice packings of convex bodies in the plane | journal = Discrete and Computational Geometry | volume = 5 | year = 1990 | issue = 4 | pages = 389–397 | doi = 10.1007/BF02187800 |mr=1043721}} 2 : Crystallography|Lattice points |
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