Uniform Polychoron - History of Discovery

History of Discovery

• Regular polytopes: (convex faces)
  • 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-polychora (star polyhedron cells and/or vertex figures)
  • 1852: Ludwig Schläfli also found 4 of the 10 regular star polychora, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
  • 1883: Edmund Hess completed the list of 10 of the nonconvex regular polychora, 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.
  • 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 polychora.
  • 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.
• Convex uniform polytopes:
  • 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
  • Convex uniform polychora:
    • 1965: The complete list of convex forms was finally done by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex polychoron, the grand antiprism.
    • 1966 N.W. 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
    • 1997: A complete enumeration of the names and elements of the convex uniform polychora is given online by George Olshevsky.
    • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope.
• Nonregular uniform star polychora: (similar to the nonconvex uniform polyhedra)
  • Ongoing: Thousands of nonconvex uniform polychora are known, but mostly unpublished. The list is presumed not to be complete, and there is no estimate of how long the complete list will be, although 1849 convex and nonconvex uniform polychora are currently known. Participating researchers include Jonathan Bowers, George Olshevsky and Norman Johnson.