{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T09:52:46Z","timestamp":1775469166973,"version":"3.50.1"},"reference-count":10,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,7]],"date-time":"2018-08-07T00:00:00Z","timestamp":1533600000000},"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>In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter\u2013Dynkin diagram of H4.<\/jats:p>","DOI":"10.3390\/sym10080326","type":"journal-article","created":{"date-parts":[[2018,8,7]],"date-time":"2018-08-07T11:20:23Z","timestamp":1533640823000},"page":"326","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Binary Icosahedral Group and 600-Cell"],"prefix":"10.3390","volume":"10","author":[{"given":"Jihyun","family":"Choi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jae-Hyouk","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,7]]},"reference":[{"key":"ref_1","unstructured":"Van Hoboken, J. (2002). Platonic Solids, Binary Polyhedral Groups Kleinian Singularities and Lie Algebras of Type ADE. [Master\u2019s Thesis, University of Amsterdam]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1215\/S0012-7094-40-00722-0","article-title":"The binary polyhedral groups, and other generalizations of the quaternion group","volume":"7","author":"Coxeter","year":"1940","journal-title":"Duke Math. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7633","DOI":"10.1088\/1751-8113\/40\/27\/013","article-title":"Group theoretical analysis of 600-cell and 120-cell 4D polytopes with quaternions","volume":"40","author":"Koca","year":"2007","journal-title":"J. Phys. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"14047","DOI":"10.1088\/0305-4470\/39\/45\/013","article-title":"Quaternionic representation of the Coxeter group W(H4) and the polyhedra","volume":"39","author":"Koca","year":"2006","journal-title":"J. Phys. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1016\/j.laa.2010.10.005","article-title":"Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system","volume":"434","author":"Koca","year":"2011","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s00006-012-0371-3","article-title":"Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups","volume":"23","author":"Dechant","year":"2013","journal-title":"Adv. Appl. Clifford Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s10587-014-0126-5","article-title":"Gosset polytopes in integral octonions","volume":"64","author":"Chang","year":"2014","journal-title":"Czechoslov. Math. J."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Conway, J.H., and Smith, D.A. (2003). On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, AK Peters.","DOI":"10.1201\/9781439864180"},{"key":"ref_9","unstructured":"Coxeter, H.S.M. (1973). Regular Polytopes, Dover Publications, Inc.. [3rd ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"123","DOI":"10.4153\/CJM-2011-063-6","article-title":"Gosset polytopes in Picard groups of del Pezzo surfaces","volume":"64","author":"Lee","year":"2012","journal-title":"Can. J. 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