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In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum possible fraction of space. Currently, the best lower bound achieved on the optimal packing fraction of regular tetrahedra is 85.63%.〔 〕 Tetrahedra do not tile space, and an upper bound below 100% (namely 1-(2.6...)10−25) has been reported. ==Historical results== Aristotle claimed that tetrahedra could fill space completely. In 2006, Conway and Torquato showed that a packing fraction about 72% can be obtained by constructing a non-Bravais lattice packing of tetrahedra (with multiple particles with generally different orientations per repeating unit), and thus they showed that the best tetrahedron packing cannot be a lattice packing (with one particle per repeating unit such that each particle has a common orientation). These packing constructions almost doubled the optimal Bravais-lattice-packing fraction 36.73% obtained by Hoylman. In 2007 and 2010, Chaikin and coworkers experimentally showed that tetrahedron-like dice can randomly pack in a finite container up to a packing fraction between 75% and 76%. In 2008, Chen was the first to propose a packing of hard, regular tetrahedra that packed more densely than spheres, demonstrating numerically a packing fraction of 77.86%. A further improvement was made in 2009 by Torquato and Jiao, who compressed Chen's structure using a computer algorithm to a packing fraction of 78.2021%. In mid-2009 Haji-Akbari et al. showed, using MC simulations of initially random systems that at packing densities >50% an equilibrium fluid of hard tetrahedra spontaneously transforms to a dodecagonal quasicrystal, which can be compressed to 83.24%. They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic quasicrystal approximant with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%. In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel.〔 Available at (arXiv:0910.5226 )〕 These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%,〔 〕 and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%.〔 The Chen, Engel and Glotzer result currently stands as the densest known packing of hard, regular tetrahedra. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tetrahedron packing」の詳細全文を読む スポンサード リンク
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