{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T00:08:01Z","timestamp":1773360481106,"version":"3.50.1"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2010,10,14]],"date-time":"2010-10-14T00:00:00Z","timestamp":1287014400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2011,12]]},"DOI":"10.1007\/s00454-010-9304-x","type":"journal-article","created":{"date-parts":[[2010,10,13]],"date-time":"2010-10-13T12:47:57Z","timestamp":1286974077000},"page":"799-818","source":"Crossref","is-referenced-by-count":18,"title":["Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra"],"prefix":"10.1007","volume":"46","author":[{"given":"Simon","family":"Gravel","sequence":"first","affiliation":[]},{"given":"Veit","family":"Elser","sequence":"additional","affiliation":[]},{"given":"Yoav","family":"Kallus","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2010,10,14]]},"reference":[{"key":"9304_CR1","unstructured":"Aristotle: On the Heavens, vol. III. ebooks@adelaide.edu. Translated by J.L. Stocks"},{"key":"9304_CR2","unstructured":"Bezdek, A., Kuperberg, W.: Dense packing of space with various convex solids. Preprint arXiv:1008.2398v1 (2010)"},{"key":"9304_CR3","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1007\/s00454-008-9101-y","volume":"40","author":"E.R. Chen","year":"2008","unstructured":"Chen, E.R.: A dense packing of regular tetrahedra. Discrete Comput. Geom. 40, 214 (2008)","journal-title":"Discrete Comput. Geom."},{"key":"9304_CR4","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s00454-010-9273-0","volume":"44","author":"E.R. Chen","year":"2010","unstructured":"Chen, E.R., Engel, M., Glotzer, S.: Dense crystalline dimer packings of regular tetrahedra. Discrete Comput. Geom. 44, 253 (2010)","journal-title":"Discrete Comput. Geom."},{"issue":"28","key":"9304_CR5","doi-asserted-by":"crossref","first-page":"10612","DOI":"10.1073\/pnas.0601389103","volume":"103","author":"J.H. Conway","year":"2006","unstructured":"Conway, J.H., Torquato, S.: Packing, tiling, and covering with tetrahedra. Proc. Natl. Acad. Sci. USA 103(28), 10612 (2006)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"9304_CR6","volume-title":"The Thirteen Books of Euclid\u2019s Elements: Books X\u2013XIII and Appendix","author":"Euclid","year":"1908","unstructured":"Euclid, Heath, T.L., Heiberg, J.L.: The Thirteen Books of Euclid\u2019s Elements: Books X\u2013XIII and Appendix. Cambridge University Press, Cambridge (1908)"},{"key":"9304_CR7","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1038\/nature08641","volume":"462","author":"A. Haji-Akbari","year":"2009","unstructured":"Haji-Akbari, A., Engel, M., Keys, A., Zheng, X.: Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra. Nature 462, 773 (2009)","journal-title":"Nature"},{"key":"9304_CR8","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.4007\/annals.2005.162.1065","volume":"162","author":"T.C. Hales","year":"2005","unstructured":"Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. Second Ser. 162, 1065 (2005)","journal-title":"Ann. Math. Second Ser."},{"key":"9304_CR9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00454-009-9148-4","volume":"44","author":"T.C. Hales","year":"2010","unstructured":"Hales, T.C., Harrison, J., McLaughlin, S., Nipkow, T., Obua, S., Zumkeller, R.: A revision of the proof of the Kepler conjecture. Discrete Comput. Geom. 44, 1 (2010)","journal-title":"Discrete Comput. Geom."},{"key":"9304_CR10","first-page":"253","volume":"3","author":"D.C. Hilbert","year":"1900","unstructured":"Hilbert, D.C.: Mathematische probleme. Nachr. Ges. Wiss. G\u00f6tt., Math. Phys. Kl. 3, 253 (1900)","journal-title":"Nachr. Ges. Wiss. G\u00f6tt., Math. Phys. Kl."},{"key":"9304_CR11","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1090\/S0002-9904-1970-12400-4","volume":"76","author":"D.J. Hoylman","year":"1970","unstructured":"Hoylman, D.J.: The densest lattice packing of tetrahedra. Bull. Am. Math. Soc. 76, 135 (1970)","journal-title":"Bull. Am. Math. Soc."},{"key":"9304_CR12","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.104.185501","volume":"104","author":"A. Jaoshvili","year":"2010","unstructured":"Jaoshvili, A., Esakia, A., Porrati, M., Chaikin, P.M.: Experiments on the random packing of tetrahedral dice. Phys. Rev. Lett. 104, 185501 (2010)","journal-title":"Phys. Rev. Lett."},{"key":"9304_CR13","doi-asserted-by":"crossref","unstructured":"Kallus, Y., Elser, V., Gravel, S.: A method for dense packing discovery. Preprint arXiv:1003.3301 (2010)","DOI":"10.1103\/PhysRevE.82.056707"},{"key":"9304_CR14","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s00454-010-9254-3","volume":"44","author":"Y. Kallus","year":"2010","unstructured":"Kallus, Y., Elser, V., Gravel, S.: Dense periodic packings of tetrahedra with small repeating units. Discrete Comput. Geom. 44, 245 (2010)","journal-title":"Discrete Comput. Geom."},{"issue":"5","key":"9304_CR15","doi-asserted-by":"crossref","first-page":"227","DOI":"10.2307\/2689983","volume":"54","author":"M. Senechal","year":"1981","unstructured":"Senechal, M.: Which tetrahedra fill space? Math. Mag. 54(5), 227 (1981)","journal-title":"Math. Mag."},{"key":"9304_CR16","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.80.041104","volume":"80","author":"S. Torquato","year":"2009","unstructured":"Torquato, S., Jiao, Y.: Dense packings of polyhedra: Platonic and archimedean solids. Phys. Rev. E 80, 041104 (2009)","journal-title":"Phys. Rev. E"},{"key":"9304_CR17","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1038\/nature08239","volume":"460","author":"S. Torquato","year":"2009","unstructured":"Torquato, S., Jiao, Y.: Dense packings of the platonic and archimedean solids. Nature 460, 876 (2009)","journal-title":"Nature"},{"key":"9304_CR18","volume":"81","author":"S. Torquato","year":"2010","unstructured":"Torquato, S., Jiao, Y.: Exact constructions of a family of dense periodic packings of tetrahedra. Phys. Rev. E 81, 041310 (2010)","journal-title":"Phys. Rev. E"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-010-9304-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-010-9304-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-010-9304-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,5]],"date-time":"2019-06-05T08:36:24Z","timestamp":1559723784000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-010-9304-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,10,14]]},"references-count":18,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2011,12]]}},"alternative-id":["9304"],"URL":"https:\/\/doi.org\/10.1007\/s00454-010-9304-x","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,10,14]]}}}