5-polytope

5-polytope

Uniform prismatic forms

There are 6 categorical uniform prismatic families of polytopes based on the uniform 4-polytopes:

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 4-space:

There are three regular honeycomb of Euclidean 4-space:
#tesseractic honeycomb, with symbols {4,3,3,4}, . There are 19 uniform honeycombs in this family.
# Icositetrachoric honeycomb, with symbols {3,4,3,3}, . There are 31 uniform honeycombs in this family.
# Hexadecachoric honeycomb, with symbols {3,3,4,3},

Other families that generate uniform honeycombs:
* There are 23 uniform honeycombs, 4 unique in the demitesseractic honeycomb family. With symbols h{4,32,4} it is geometrically identical to the hexadecachoric honeycomb,
* There are 7 uniform honeycombs from the "A"~4, family, all unique.
* There are 7 uniform honeycombs in the "D"~4: [31,1,1,1] .

Pyramids

Pyramidal polyterons, or 5-pyramids, can be generated by a polychoron base in a 4-space hyperplane connected to a point off the hyperplane. The 5-simplex is the simplest example with a 4-simplex base.

A note on generality of terms for n-polytopes and elements

A 5-polytope, or polyteron, follows from the lower dimensional polytopes: 2: polygon, 3: polyhedron, and 4: polychoron.

Although there is no agreed upon standard terminology for higher polytopes, for dimensional clarity George Olshevsky advocates borrowing from the SI prefix sequencing, which can covers up to 10-polytopes with 9-dimensional facets:
* "Polyteron" for a 5-polytope (tera, a shortened on tetra-, for 4D faceted polytope), and "terons" for "4-face" element.
* "Polypeton" for a 6-polytope (peta, a shortened on penta-, for 5D faceted polytope), and "petons" for "5-face" elements.
* "Polyexon" for a 7-polytope (exa or "ecta", a shortened on hexa-, for 6D faceted polytope), and "exons" for "6-face" elements.
* "Polyzetton" for a 8-polytope (zetta, a variation on hepta-, for 7D faceted polytope), and "zettons" for "7-face" elements.
* "Polyyotton" for a 9-polytope (yotta, a variation on octa-, for 8D faceted polytope), and "yottons" for "8-face" elements.
* "Polyxennon" for a 10-polytope (xenna, a variation on ennea-, for 9D faceted polytope), and "xennons" for "9-face" elements.

For specific polytopes, like the lower dimensional polytopes, they can be named by their number of facets. For example a 5-simplex, with 6 facets can explicitly be called a "hexa-5-tope", representing a 6-faceted 5-polytope, and thus is named a "hexateron".

See also

* List of regular polytopes#Five Dimensions
* Polygon - 2-polytopes
** Regular polygon
* Polyhedron - 3-polytopes
** Uniform polyhedron
* Polychoron - 4-polytopes
** Uniform polychoron
* 6-polytope
* 7-polytope
* 8-polytope
* 9-polytope
* 10-polytope

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]
* N.W. Johnson: "The Theory of Uniform Polytopes and Honeycombs", Ph.D. Dissertation, University of Toronto, 1966
* Richard Klitzing 5D quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons

External links

* [http://www.steelpillow.com/polyhedra/ditela.html Polytope names] , Guy Inchbald
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions] , Jonathan Bowers
* [http://members.aol.com/Polycell/glossary.html#Polytope Glossary for hyperspace: Polytope] , George Olshevsky
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary] , Garrett Jones
* [http://www.geocities.com/os2fan2/gloss/pglossp.html#PGPOLYTOPE polytope names] , Wendy Krieger


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