Semiregular k 21 polytope

Semiregular k 21 polytope

In geometry, a semiregular k21 polytope is a polytope in (k+4) dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by Coxeter as k21 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the k-node sequence.

Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polyopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the "5-ic semiregular figure".

The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation in of hyperbolic 9-space constructed of (∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.)

The family starts uniquely as 6-polytopes. The "triangular prism" and "rectified 5-cell" are included at the beginning for completeness. The "demipenteract" also exists in the demihypercube family.

They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the E6 symmetry.

The complete family of Gosset semiregular polytopes are:
# triangular prism: -121 (2 Triangles and 3 square faces)
# rectified 5-cell: 021, "Tetroctahedric" (5 tetrahedra and 5 octahedra cells)
# demipenteract: 121, "5-ic semiregular figure" (16 5-cell and 10 16-cell facets)
# Gosset 2 21 polytope: 221, "6-ic semiregular figure" (72 5-simplex and 27 5-orthoplex facets)
# Gosset 3 21 polytope: 321, "7-ic semiregular figure" (567 6-simplex and 126 6-orthoplex facets)
# Gosset 4 21 polytope: 421, "8-ic semiregular figure" (17280 7-simplex and 2160 7-orthoplex facets)
# E8 lattice: 521, "9-ic semiregular check" tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets)

Each polytope is constructed from (n-1)-simplex and (n-1)-orthoplex facets.

The orthoplex faces are constructed from the Coxeter group Dn-1 and have a Schlafli symbol of {31,n-1,1} rather than the regular {3n-2,4}. This construction is an implication of two "facet types". Half the facets around each orthoplex ridge are attached to another orthoplex, and the others are attached to a simplex. In contrast, every simplex ridge is attached to an orthoplex.

Each has a vertex figure as the previous form. For example the "rectified 5-cell" has a vertex figure as a "triangular prism".

Elements

See also

* Uniform 2k1 polytope family
* Uniform 1k2 polytope family

References

* T. Gosset: "On the Regular and Semi-Regular Figures in Space of n Dimensions", Messenger of Mathematics, Macmillan, 1900
* 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
* G.Blind and R.Blind, "The semi-regular polyhedra", Commentari Mathematici Helvetici 66 (1991) 150--154
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, "The Symmetry of Things" 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 411-413: The Gosset Series: n21)

External links

* [http://www.geocities.com/os2fan2/gloss.htm#gossetfig PolyGloss v0.05: Gosset figures (Gossetoicosatope)]
* [http://www.liga.ens.fr/~dutour/Regular Regular, SemiRegular, Regular faced and Archimedean polytopes]


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Semiregular 4-polytope — In geometry, a semiregular 4 polytope (or polychoron) is a 4 dimensional polytope which is vertex transitive (i.e. the symmetry group of the polytope acts transitively on the vertices) and whose cells are regular polyhedra. These represent a… …   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

  • Semiregular polyhedron — A semiregular polyhedron is a polyhedron with regular faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset s 1900 definition of the more general semiregular polytope. [Thorold… …   Wikipedia

  • Gosset 3 21 polytope — In 7 dimensional geometry, the 321 is a semiregular polytope, enumerated by Thorold Gosset in his 1900 paper. He called it an 7 ic semi regular figure . It is called the Hess polytope for Edmund Hess who first discovered it.Its construction is… …   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

  • Gosset 4 21 polytope — The Gosset 421 polytope is an 8 dimensional semiregular uniform polytope composed of 17,280 7 simplex and 2,160 7 orthoplex facets.It was discovered by Thorold Gosset, who described it in his 1900 paper as an 8 ic semi regular figure. It is the… …   Wikipedia

  • 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… …   Wikipedia

  • 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

  • Gosset 2 21 polytope — In 6 dimensional geometry, 221 is a semiregular polytope, discovered by Thorold Gosset, published in his 1900 paper. He called it an 6 ic semi regular figure .Its construction is based on the E6 group. Coxeter named it 221 by its bifurcating… …   Wikipedia

  • Cross-polytope — In geometry, a cross polytope,[1] orthoplex,[2] hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross polytope are all the permutations of (±1, 0, 0, …, 0). The cross polytope is …   Wikipedia

Share the article and excerpts

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