8-polytope

Three are based on the uniform 6-polytopes and regular polygons:

Uniform duoprismatic forms

There are 28 categorical uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.

There are 4 based on the uniform 6-polytopes and regular polygons:

Uniform triprismatic forms

There are 11 categorical uniform triprismatic forms based on Cartesian products of lower dimensional uniform polytopes, for example these regular products:

Six are based on products of the uniform 4-polytopes and uniform duoprisms:

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 7-space:

Regular and uniform tessellations include:
* B^~₇ or [4,3⁵,4] :
**regular honeycomb hepteractic honeycomb, with symbols {4,3⁵,4},
* C^~₇ or [3^1,1,3⁴,4] :
**Uniform demihepteractic honeycomb, with symbols h{4,3⁵,4} = {3^1,1,3⁴,4},
* E^~₇ or [3^3,3,1] :
**Uniform 3₃₁ honeycomb with 3₂₁ polytope vertex figure and 2₃₁ polytope and 7-simplex facets with symbols {3^3,3,1} and
** Uniform 1_3,3 honeycomb,

See also

*List of regular polytopes#Higher dimensions
* polygon
* polyhedron
* polychoron
* 5-polytope
* 6-polytope
* 7-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

External links

* [http://www.steelpillow.com/polyhedra/ditela.html Polytope names]
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
* [http://members.aol.com/Polycell/glossary.html Glossary for hyperspace]