{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:45:24Z","timestamp":1760784324098,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2017,9,11]],"date-time":"2017-09-11T00:00:00Z","timestamp":1505088000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Where are the atoms in complex crystals such as quasicrystals or periodic crystals with one hundred or more atoms per unit cell? How did they get there? The first of these questions has been gradually answered for many materials over the quarter-century since quasicrystals were discovered; in this paper we address the second. We briefly review a history of proposed models for describing atomic positions in crystal structures. We then present a revised description and possible growth model for one particular system of alloys, those containing Tsai-type clusters, that includes an important class of quasicrystals.<\/jats:p>","DOI":"10.3390\/sym9090188","type":"journal-article","created":{"date-parts":[[2017,9,11]],"date-time":"2017-09-11T10:28:46Z","timestamp":1505125726000},"page":"188","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters"],"prefix":"10.3390","volume":"9","author":[{"given":"Jean","family":"Taylor","sequence":"first","affiliation":[{"name":"Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erin","family":"Teich","sequence":"additional","affiliation":[{"name":"Applied Physics Program, University of Michigan, Ann Arbor, MI 48109, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pablo","family":"Damasceno","sequence":"additional","affiliation":[{"name":"Department of Cellular and Molecular Pharmacology, University of California, San Francisco, CA 94143, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yoav","family":"Kallus","sequence":"additional","affiliation":[{"name":"Santa Fe Institute, Santa Fe, NM 87501, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marjorie","family":"Senechal","sequence":"additional","affiliation":[{"name":"Mathematics and History of Science and Technology, Smith College, Northampton, MA 01063, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,9,11]]},"reference":[{"key":"ref_1","first-page":"551","article-title":"The Open Mathematics of Crystallization","volume":"64","author":"Radin","year":"2017","journal-title":"Not. AMS"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shechtman, D., Blech, I., Gratias, D., and Cahn, J.W. (1984). Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Phys. Rev. Lett., 53.","DOI":"10.1103\/PhysRevLett.53.1951"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00283-013-9381-9","article-title":"Quasicrystals: The view from Stockholm","volume":"35","author":"Senechal","year":"2013","journal-title":"Math. Intell."},{"key":"ref_4","unstructured":"Kepler, J. (1611). Strena, seu de Nive Sexangula, Godefridum Tampach."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hooke, R. (1665). Micrographia: Or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses, with Observations and Inquiries Thereupon, The Royal Society.","DOI":"10.5962\/bhl.title.105738"},{"key":"ref_6","unstructured":"H\u00e4uy, R.J. (1784). Essai D\u2019une Th\u00e9orie sur la Structure des Crystaux, Appliq\u00e9e a Plusiers Genres de Substances Crystallis\u00e9es, Gogu\u00e9 & N\u00e9e de la Rochelle, Libraries, MDCCLXXIV."},{"key":"ref_7","unstructured":"(2017, September 03). Nobel Foundation Website. Available online: https:\/\/www.nobelprize.org\/nobel_prizes\/physics\/laureates\/1914\/laue-facts.html."},{"key":"ref_8","first-page":"1","article-title":"M\u00e9moire sur les syst\u00e8mes form\u00e9s par les points distribu\u00e9s sur un plan ou dans l\u2019espace","volume":"33","author":"Bravais","year":"1850","journal-title":"J. l\u2019\u00c9cole Polytech. Cah."},{"key":"ref_9","unstructured":"Senechal, M. (1996). Quasicrystals and Geometry, Cambridge University Press."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Baake, M., and Grimm, U. (2013). Aperiodic Order, Cambridge University Press.","DOI":"10.1017\/CBO9781139025256"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/PL00009413","article-title":"Geometric Models for Quasicrystals I. Delone Sets of Finite Type","volume":"21","author":"Lagarias","year":"1999","journal-title":"Discret. Comput. Geom."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7354","DOI":"10.1021\/acs.inorgchem.7b00245","article-title":"Three-Dimensional Modeling of Quasicrystal Structures from X-ray Diffraction: An Icosahedral Al-Cu-Fe Alloy","volume":"56","author":"Hong","year":"2017","journal-title":"Inorg. Chem."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1038\/nmat1799","article-title":"Atomic structure of the binary icosahedral Yb\u2013Cd quasicrystal","volume":"6","author":"Takakura","year":"2007","journal-title":"Nat. Mater."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1098\/rspa.1952.0194","article-title":"Supercooling of Liquids","volume":"215","author":"Frank","year":"1952","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1038\/1691057a0","article-title":"Crystal structure of the intermetallic compound Mg32 (Al, Zn)49 and related phases","volume":"169","author":"Bergman","year":"1952","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1107\/S0365110X6200239X","article-title":"A dense non-crystallographic packing of equal spheres","volume":"15","author":"Mackay","year":"1962","journal-title":"Acta Crystallogr."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3576","DOI":"10.1021\/cg500498k","article-title":"Applied topological analysis of crystal structures with the program packageTopos","volume":"14","author":"Blatov","year":"2014","journal-title":"Proc. Cryst. Growth Des."},{"key":"ref_18","unstructured":"Fujiwara, T., and Ogawa, T. (1989, January 14\u201319). Progress on the atomic structure of quasicrystals in quasicrystals. Proceedings of the 12th Taniguchi Symposium, Shima Mie Prefecture, Japan. Available online: http:\/\/www.lassp.cornell.edu\/clh\/PUBS\/tani90.pdf."},{"key":"ref_19","unstructured":"Taylor, J., Teich, E., Damasceno, P., Kallus, Y., and Senechal, M. (2017). The role of geometric frustration in the self-assembly of complex crystals, Unpublished Work."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"E669","DOI":"10.1073\/pnas.1524875113","article-title":"Clusters of polyhedra in spherical confinement","volume":"113","author":"Teich","year":"2016","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_21","unstructured":"Pfoertner, H. (2015, March 27). Densest Packing of Spheres in a Sphere. Available online: www.randomwalk.de\/sphere\/insphr\/spheresinsphr.html."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"415403","DOI":"10.1088\/0953-8984\/24\/41\/415403","article-title":"Ordering and dynamics of the central tetrahedron in the 1\/1 Zn6Sc periodic approximant to quasicrystal","volume":"24","author":"Euchner","year":"2012","journal-title":"J. Phys. Condens. Matter"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/9\/188\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:44:36Z","timestamp":1760208276000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/9\/188"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,11]]},"references-count":22,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["sym9090188"],"URL":"https:\/\/doi.org\/10.3390\/sym9090188","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2017,9,11]]}}}