Uniform 2 k1 polytope


Uniform 2 k1 polytope

In geometry, 2k1 polytope or {32,k,1} is a uniform polytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by Coxeter as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.

The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex (pentacross) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions.

Each polytope is constructed from (n-1)-simplex and 2k-1,1 (n-1)-polytope facets, each has a vertex figure as an (n-1)-demicube, "{31,n-2,1}".

The sequence ends with k=5 (n=9), as an infinite tessellation of 8-space.

The complete family of 2k1 polytope polytopes are:
# 5-cell: 201, (5 tetrahedra cells)
# Pentacross: 211, (32 5-cell (201) facets)
# Gosset 2 21 polytope: 221, (72 5-simplex and 27 5-orthoplex (211) facets)
# Gosset 2 31 polytope: 231, (576 6-simplex and 56 221 facets)
# Gosset 2 41 polytope: 241, (17280 7-simplex and 240 231 facets)
# Gosset 2_51 lattice: 251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets)

Elements

See also

* k21 polytope family
* 1k2 polytope family

References

* Alicia 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
** Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
** Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1-24 plus 3 plates, 1910.
** Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
* Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, "Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam" (eerstie sectie), vol 11.5, 1913.
* H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
* N.W. Johnson: "The Theory of Uniform Polytopes and Honeycombs", Ph.D. Dissertation, University of Toronto, 1966
* H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
* H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988

External links

* [http://www.geocities.com/os2fan2/gloss.htm#gossetfig PolyGloss v0.05: Gosset figures (Gossetooctotope)]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Uniform 1 k2 polytope — In geometry, 1k2 polytope or {31,k,2} is a uniform polytope in n dimensions (n = k+4) constructed from the En Coxeter group. The family was named by Coxeter as 1k2 by its bifurcating Coxeter Dynkin diagram, with a single ring on the end of the 1… …   Wikipedia

  • Uniform polytope — A uniform polytope is a vertex transitive polytope made from uniform polytope facets. A uniform polytope must also have only regular polygon faces.Uniformity is a generalization of the older category semiregular, but also includes the regular… …   Wikipedia

  • Polytope — Not to be confused with polytrope. In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on… …   Wikipedia

  • 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 …   Wikipedia

  • Uniform Polychora Project — The Uniform Polychora Project is a collaborative effort in geometry to recognize and standardize terms used to describe objects in higher dimensional spaces. The project aims to:# Collect information about uniform polychora as well as information …   Wikipedia

  • Uniform star polyhedron — A display of uniform polyhedra at the Science Museum in London …   Wikipedia

  • Uniform tiling — In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the… …   Wikipedia

  • 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… …   Wikipedia

  • Convex regular 4-polytope — The tesseract is one of 6 convex regular 4 polytopes In mathematics, a convex regular 4 polytope is a 4 dimensional polytope that is both regular and convex. These are the four dimensional analogs of the Platonic solids (in three dimensions) and… …   Wikipedia

  • 6-polytope — Uniform duoprismatic forms There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower dimensional uniform polytopes. Five are formed as the product of a uniform polychoron with a regular polygon, and… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.