page_title: Tetrahedron packing
section_title: Historical results
hierarchical_section_title: Tetrahedron packing / Historical results
caption_attribution_description: English: An illustration of a close-packing arrangement of tetrahedra, using tetrahedral milk cartons
context_page_description: 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% has been reported.
context_section_description: 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 approximant to a quasicrystal 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. 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.
caption: Tetrahedral packaging
exif: {"Image Make": "Canon", "Image Model": "Canon PowerShot A720 IS", "Image Orientation": "Horizontal (normal)", "Image XResolution": "180", "Image YResolution": "180", "Image ResolutionUnit": "Pixels/Inch", "Image DateTime": "2011:08:25 08:41:18", "Image YCbCrPositioning": "Centered", "Image ExifOffset": "196", "Thumbnail Compression": "JPEG (old-style)", "Thumbnail XResolution": "180", "Thumbnail YResolution": "180", "Thumbnail ResolutionUnit": "Pixels/Inch", "Thumbnail JPEGInterchangeFormat": "5108", "Thumbnail JPEGInterchangeFormatLength": "7495", "EXIF ExposureTime": "1/30", "EXIF FNumber": "14/5", "EXIF ISOSpeedRatings": "200", "EXIF ExifVersion": "0220", "EXIF DateTimeOriginal": "2011:08:25 08:41:18", "EXIF DateTimeDigitized": "2011:08:25 08:41:18", "EXIF ComponentsConfiguration": "YCbCr", "EXIF CompressedBitsPerPixel": "3", "EXIF ShutterSpeedValue": "157/32", "EXIF ApertureValue": "95/32", "EXIF ExposureBiasValue": "0", "EXIF MaxApertureValue": "95/32", "EXIF MeteringMode": "Pattern", "EXIF Flash": "Flash did not fire, compulsory flash mode", "EXIF FocalLength": "29/5", "EXIF FlashPixVersion": "0100", "EXIF ColorSpace": "sRGB", "EXIF ExifImageWidth": "3264", "EXIF ExifImageLength": "2448", "Interoperability InteroperabilityIndex": "R98", "Interoperability InteroperabilityVersion": "[48, 49, 48, 48]", "Interoperability RelatedImageWidth": "3264", "Interoperability RelatedImageLength": "2448", "EXIF InteroperabilityOffset": "3362", "EXIF FocalPlaneXResolution": "43520/3", "EXIF FocalPlaneYResolution": "2448000/169", "EXIF FocalPlaneResolutionUnit": "2", "EXIF SensingMethod": "One-chip color area", "EXIF FileSource": "Digital Camera", "EXIF CustomRendered": "Normal", "EXIF ExposureMode": "Auto Exposure", "EXIF WhiteBalance": "Auto", "EXIF DigitalZoomRatio": "1", "EXIF SceneCaptureType": "Standard"}