Uniform polychoron

In geometry, a uniform polychoron (plural: uniform polychora) is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra.

This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms.

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" [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001] .
* Semiregular polytopes: (convex polytopes)
** 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".
** 1912: E. L. Elte expanded on Gosset's work with the publication "The Semiregular Polytopes of the Hyperspaces", including a special subset of polytopes with semiregular facets (Those constructible by a single ringed node of a Coxeter-Dynkin diagram.)
* Convex uniform polytopes:
** 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.
** 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 J. T. Guy, in their publication "Four-Dimensional Archimedean Polytopes", established by computer analysis, adding only one non-Wythoffian convex polychoron, the grand antiprism.
*** 1997: A complete enumeration of the names and elements of the convex uniform polychora is given online by George Olshevsky. [http://members.aol.com/Polycell/uniform.html]
*** 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, "Vierdimensionale Archimedische Polytope" (in German).
* 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. Participating researchers include Jonathan Bowers, George Olshevsky and Norman Johnson.

Regular polychora

The uniform polychora include two special subsets, which satisfy additional requirements:
* The 16 regular polychora, with the property that all cells, faces, edges, and vertices are congruent:
** 6 convex regular 4-polytopes;
** 10 Schläfli-Hess polychora.

Convex uniform polychora

There are 64 convex uniform polychora, including the 6 regular convex polychora, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms.
* 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
* 13 are polyhedral prisms based on the Archimedean solids
* 9 are in the self-dual regular A₄ [3,3,3] group (5-cell) family.
* 9 are in the self-dual regular F₄ [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
* 15 are in the regular B₄ [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
* 15 are in the regular H₄ [3,3,5] group (120-cell/600-cell) family.
* 1 special snub form in the [3,4,3] group (24-cell) family.
* 1 special non-Wythoffian polychoron, the grand antiprism.
* TOTAL: 68 − 4 = 64

These 64 uniform polychora are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.

In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:
* Set of uniform antiprismatic prisms - s{p,2}x{} - Polyhedral prisms of two antiprisms.
* Set of uniform duoprisms - {p}x{q} - A product of two polygons.

The A₄ [3,3,3] family - (5-cell)

The 5-cell has "diploid pentachoric" symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

The pictures are draw as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

:(*) Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family.

The "snub 24-cell" is repeat to this family for completeness. It is an alternation of the "cantitruncated 16-cell" or "truncated 24-cell". The truncated octahedral cells become icosahedra. The cube becomes a tetrahedron, and 96 new tetrahedra are created in the gaps from the removed vertices.

The F₄ [3,4,3] family - (24-cell)

This family has "diploid icositetrachoric" symmetry, of order 24*48=1152: the 48 symmetries of the octahedron for each of the 24 cells.

The D₄ [3^1,1,1] group family (Demitesseract)

This demitesseract family introduces no new uniform polychora, but it is worthy to repeat these alternative constructions.

Octahedral prisms: B₃×A₁ - [4,3] × [ ]

See also convex uniform honeycombs, some of which illustrate these operations as applied to the regular cubic honeycomb.

If two polytopes are duals of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent.

ee also

*Polychoron
*Semiregular 4-polytopes
*Duoprism

References

* T. Gosset: "On the Regular and Semi-Regular Figures in Space of n Dimensions", Messenger of Mathematics, Macmillan, 1900
* A. Boole Stott: "Geometrical deduction of semiregular from regular polytopes and space fillings", Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
*
* H.S.M. Coxeter:
** H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: "Uniform Polyhedra", Philosophical Transactions of the Royal Society of London, Londne, 1954
** H.S.M. Coxeter, "Regular Polytopes", 3rd Edition, Dover New York, 1973
* Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 22) H.S.M. Coxeter, "Regular and Semi Regular Polytopes I", [Math. Zeit. 46 (1940) 380-407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, "Regular and Semi-Regular Polytopes II", [Math. Zeit. 188 (1985) 559-591]
** (Paper 24) H.S.M. Coxeter, "Regular and Semi-Regular Polytopes III", [Math. Zeit. 200 (1988) 3-45]
* J.H. Conway and M.J.T. Guy: "Four-Dimensional Archimedean Polytopes", Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
* N.W. Johnson: "The Theory of Uniform Polytopes and Honeycombs", Ph.D. Dissertation, University of Toronto, 1966
* M. Möller: "Definitions and computations to the Platonic and Archimedean polyhedrons", thesis (diploma), University of Hamburg, 2001
* B. Grünbaum "Convex polytopes", New York ; London : Springer, c2003. ISBN 0-387-00424-6.
Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.

External links

*
* Convex uniform polychora
** [http://www.polytope.de/ Polytope in R⁴] , Marco Möller
*** [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope]
** [http://members.aol.com/Polycell/uniform.html Uniform Polytopes in Four Dimensions] , George Olshevsky
**# [http://members.aol.com/Polycell/section1.html Convex uniform polychora based on the pentachoron (5-cell)]
**# [http://members.aol.com/Polycell/section2.html Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell)]
**# [http://members.aol.com/Polycell/section3.html Convex uniform polychora based on the icositetrachoron (24-cell)]
**# [http://members.aol.com/Polycell/section4.html Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell)]
**# [http://members.aol.com/Polycell/section5.html Anomalous convex uniform polychoron: (grand antiprism)]
**# [http://members.aol.com/Polycell/section6.html Convex uniform prismatic polychora]
**# [http://members.aol.com/Polycell/section7.html Uniform polychora derived from glomeric tetrahedron B4]
** [http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html Regular and semi-regular convex polytopes a short historical overview]
* Nonconvex uniform polychora
** [http://www.polytope.net/hedrondude/polychora.html Uniform polychora] by Jonathan Bowers
** [http://www.software3d.com/Stella.php#stella4D Stella4D] Stella (software) produces interactive views of all 1849 known uniform polychora including the 64 convex forms and the infinite prismatic families. Was used to create most images on this page.