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In geometry, a uniform 4-polytope is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. 47 non-prismatic convex uniform 4-polytopes, one finite set of convex prismatic forms, and two infinite sets of convex prismatic forms have been described. There are also an unknown number of non-convex star forms. == History of discovery == * Convex Regular polytopes: * * 1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions. * Regular star 4-polytopes (star polyhedron cells and/or vertex figures) * * 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures and . * * 1883: Edmund Hess completed the list of 10 of the nonconvex regular 4-polytopes, in his book (in German) ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'' (). * Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) * * 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimensions''.〔T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900〕 * * 1910: Alicia Boole Stott, in her publication ''Geometrical deduction of semiregular from regular polytopes and space fillings'', expanded the definition by also allowing Archimedean solid and prism cells. This construction enumerated 45 semiregular 4-polytopes.〔http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf〕 * * 1911: Pieter Hendrik Schoute published ''Analytic treatment of the polytopes regularly derived from the regular polytopes'', followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell. * * 1912: E. L. Elte independently expanded on Gosset's list with the publication ''The Semiregular Polytopes of the Hyperspaces'', polytopes with one or two types of semiregular facets.〔Elte (1912)〕 * Convex uniform polytopes: * * 1940: The search was expanded systematically by H.S.M. Coxeter in his publication ''Regular and Semi-Regular Polytopes''. * * Convex uniform 4-polytopes: * * * 1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication ''Four-Dimensional Archimedean Polytopes'', established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism. * * * 1966 Norman Johnson completes his Ph.D. dissertation ''The Theory of Uniform Polytopes and Honeycombs'' under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher. * * * 1986 Coxeter published a paper ''Regular and Semi-Regular Polytopes II'' which included analysis of the unique snub 24-cell structure, and the symmetry of the anomalous grand antiprism. * * * 1998〔https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html December 6, 1998 oldest archive〕-2000: The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration (used as a basis for this listing). Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots ''poly'' ("many") and ''choros'' ("room" or "space").〔(The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes ) By David Darling, (2004) ASIN: B00SB4TU58〕 The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms called rectified, and bi,tri-prefixes added when the first ring was on the second or third nodes.〔Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 ''full polychoric groups''〕 * * * 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, ''Vierdimensionale Archimedische Polytope''. Möller reproduced Johnson's naming system in his listing.〔(2004 Dissertation (German): VierdimensionaleArhimedishe Polytope ) 〕 * * * 2008: ''The Symmetries of Things''〔Conway (2008)〕 was published by John H. Conway contains the first print-published listing of the convex uniform 4-polytopes and higher dimensions by coxeter group family, with general vertex figure diagrams for each ringed Coxeter diagram permutation, snub, grand antiprism, and duoprisms which he called proprisms for product prisms. He used his own ''ijk''-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, with all of Johnson's names were included in the book index. * Nonregular uniform star 4-polytopes: (similar to the nonconvex uniform polyhedra) * * 2000-2005: In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky.〔() ''Convex and Abstract Polytopes'' workshop (2005), N.Johnson — "Uniform Polychora" abstract〕 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「Uniform 4-polytope」の詳細全文を読む スポンサード リンク
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