View source for Uniform k 21 polytope

{{DISPLAYTITLE:Uniform ''k''&thinsp;<sub>21</sub> polytope}}
In [[Geometry|geometry]], a '''uniform ''k''<sub>21</sub> polytope''' is a [[Polytope|polytope]] in ''k''&nbsp;+&nbsp;4 dimensions constructed from the [[En (Lie algebra)|''E''<sub>''n''</sub>]] [[Coxeter group]], and having only [[Regular polytope|regular polytope]] facets. The family was named by their Coxeter symbol '''''k''<sub>21</sub>''' 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 polytope|regular]] and [[Semiregular polytope|semiregular polytope]]s, 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''.

== Family members ==
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''': 6<sub>21</sub>. It is a tessellation of hyperbolic 9-space constructed of ∞ 9-[[Simplex|simplex]] and ∞ 9-orthoplex facets with all vertices at infinity.)

The family starts uniquely as [[6-polytope|6-polytope]]s. The ''triangular prism'' and ''rectified 5-cell'' are included at the beginning for completeness. The ''demipenteract'' also exists in the [[Demihypercube|demihypercube]] family.

They are also sometimes named by their symmetry group, like '''E6 polytope''', although there are many uniform polytopes within the [[E6 (mathematics)|''E''<sub>6</sub>]] symmetry.

The complete family of Gosset semiregular polytopes are:
# [[Triangular prism|triangular prism]]: −1<sub>21</sub> (2 [[Triangle|triangle]]s and 3 square faces)
# [[Rectified 5-cell|rectified 5-cell]]: 0<sub>21</sub>, ''Tetroctahedric'' (5 [[Tetrahedron|tetrahedra]] and 5 [[Octahedron|octahedra]] cells)
# demipenteract: 1<sub>21</sub>, ''5-ic semiregular figure'' (16 [[5-cell|5-cell]] and 10 [[16-cell|16-cell]] facets)
# [[2 21 polytope|2 21 polytope]]: 2<sub>21</sub>, ''6-ic semiregular figure'' (72 5-[[Simplex|simplex]] and 27 5-orthoplex facets)
# [[3 21 polytope|3 21 polytope]]: 3<sub>21</sub>, ''7-ic semiregular figure'' (576 6-[[Simplex|simplex]] and 126 6-orthoplex facets)
# 4 21 polytope: 4<sub>21</sub>, ''8-ic semiregular figure'' (17280 7-[[Simplex|simplex]] and 2160 7-orthoplex facets)
# [[5 21 honeycomb|5 21 honeycomb]]: 5<sub>21</sub>, ''9-ic semiregular check'' tessellates Euclidean 8-space (∞ 8-[[Simplex|simplex]] and ∞ 8-orthoplex facets)
# 6 21 honeycomb: 6<sub>21</sub>, tessellates hyperbolic 9-space (∞ 9-[[Simplex|simplex]] and ∞ 9-orthoplex facets)

Each polytope is constructed from (''n''&nbsp;&minus;&nbsp;1)-[[Simplex|simplex]] and (''n''&nbsp;&minus;&nbsp;1)-orthoplex facets.

The orthoplex faces are constructed from the [[Coxeter group]] ''D''<sub>''n''&minus;1</sub> and have a [[Schläfli symbol]] of {3<sup>1,''n''&minus;1,1</sup>} rather than the regular {3<sup>''n''&minus;2</sup>,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|vertex figure]] as the previous form. For example, the ''rectified 5-cell'' has a vertex figure as a ''triangular prism''.

== Elements ==

{| class="wikitable"
|+
Gosset semiregular figures
|-
!rowspan=2|''n''-ic
!rowspan=2|''k''<sub>21</sub>
!rowspan=2| [[Graph (discrete mathematics)|Graph]]
!rowspan=2| Name<BR>Coxeter<BR>diagram
!colspan=2|Facets
!colspan=8|Elements
|-
! (''n''&nbsp;&minus;&nbsp;1)-[[Simplex|simplex]]<BR>{3<sup>''n''&minus;2</sup>}
! (''n''&nbsp;&minus;&nbsp;1)-orthoplex<BR>{3<sup>''n''&minus;4,1,1</sup>}
! [[Vertex (geometry)|Vertices]]
! [[Edge (geometry)|Edges]]
! [[Face (geometry)|Faces]]
! Cells
! 4-faces
! 5-faces
! 6-faces
! 7-faces
|- align=center
|3-ic
| &minus;1<sub>21</sub>
| [[File:Triangular prism graphs.png|80px]]
|[[Triangular prism]]<BR>{{CDD|node|3|node_1|2|node_1}}
|2 [[Triangle|triangle]]s<BR>[[File:2-simplex t0.svg|40px]][[File:Triangular prism simplex.png|40px]]<BR>{{CDD|node_1|3|node}}
|3 squares<BR>[[File:2-orthoplex.svg|40px]][[File:Triangular prism orthoplex.png|40px]]<BR>{{CDD|node_1|2|node_1}}
|6
|9
|5
|&nbsp;
|&nbsp;
|&nbsp;
|&nbsp;
|&nbsp;
|- align=center
|4-ic
|0<sub>21</sub>
|[[File:E4 graph ortho.png|80px]]
|[[Rectified 5-cell]]<BR>{{CDD|nodea|3a|nodea|3a|branch_10}}
|5 [[Tetrahedron|tetrahedron]]<BR>[[File:3-simplex t0.svg|40px]][[File:Uniform polyhedron-33-t0.png|40px]]<BR>{{CDD|node_1|3|node|3|node}}
|5 [[Octahedron|octahedron]]<BR>[[File:3-orthoplex.svg|40px]][[File:Uniform polyhedron-33-t1.png|40px]]<BR>{{CDD|branch_10|3a|nodea}}
|10
|30
|30
|10
|&nbsp;
|&nbsp;
|&nbsp;
|&nbsp;
|- align=center
|5-ic
|1<sub>21</sub>
|[[File:Demipenteract graph ortho.svg|80px]]
|Demipenteract<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea_1}}
|16 [[5-cell|5-cell]]<BR>[[File:4-simplex t0.svg|40px]][[File:Schlegel wireframe 5-cell.png|40px]]<BR>{{CDD|node_1|3|node|3|node|3|node}}
|10 [[16-cell|16-cell]]<BR>[[File:4-orthoplex.svg|40px]] [[File:Schlegel wireframe 16-cell.png|40px]]<BR>{{CDD|nodea_1|3a|branch|3a|nodea}}
|16
|80
|160
|120
|26
|&nbsp;
|&nbsp;
|&nbsp;
|- align=center
|6-ic
|2<sub>21</sub>
|[[File:E6 graph.svg|80px]]
|[[2 21 polytope|2<sub>21</sub> polytope]]<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}
|72 [[5-simplex|5-simplex]]es<BR>[[File:5-simplex t0.svg|40px]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}
|27 [[5-orthoplex|5-orthoplex]]es<BR>[[File:5-orthoplex.svg|40px]]<BR>{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea}}
|27
|216
|720
|1080
|648
|99
|&nbsp;
|&nbsp;
|- align=center
|7-ic
|3<sub>21</sub>
|[[File:E7 graph.svg|80px]]
|[[3 21 polytope|3<sub>21</sub> polytope]]<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}
|576 [[6-simplex|6-simplex]]es<BR>[[File:6-simplex t0.svg|40px]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
|126 [[6-orthoplex|6-orthoplex]]es<BR>[[File:6-orthoplex.svg|40px]]<BR>{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea}}
|56
|756
|4032
|10080
|12096
|6048
|702
|&nbsp;
|- align=center
|8-ic
|4<sub>21</sub>
|[[File:E8 graph.svg|80px]]
|4<sub>21</sub> polytope<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}
|17280 [[7-simplex|7-simplex]]es<BR>[[File:7-simplex t0.svg|40px]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
|2160 [[7-orthoplex|7-orthoplex]]es<BR>[[File:7-orthoplex.svg|40px]]<BR>{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea}}
|240
|6720
|60480
|241920
|483840
|483840
|207360
|19440
|- align=center
|9-ic
|5<sub>21</sub>
|
|[[5 21 honeycomb|5<sub>21</sub> honeycomb]]<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea_1}}
|∞ 8-simplexes<BR>[[File:8-simplex t0.svg|40px]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|∞ 8-orthoplexes<BR>[[File:8-orthoplex.svg|40px]]<BR>{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea}}
|colspan=8|∞
|- align=center
|10-ic
|6<sub>21</sub>
|
|6<sub>21</sub> honeycomb<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a||nodea|3a|nodea_1}}
|∞ [[9-simplex|9-simplex]]es<BR>[[File:9-simplex t0.svg|40px]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|∞ [[9-orthoplex|9-orthoplex]]es<BR>[[File:9-orthoplex.svg|40px]]<BR>{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|colspan=8|∞
|}

== See also ==
* [[Uniform 2 k1 polytope|Uniform 2<sub>k1</sub> polytope]] family
* [[Uniform 1 k2 polytope|Uniform 1<sub>k2</sub> polytope]] family

== References ==
* T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
* [[Biography:Alicia Boole Stott|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.&nbsp;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.
* [[Biography:Harold Scott MacDonald Coxeter|H. S. M. Coxeter]]: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
* [[Biography:Norman Johnson (mathematician)|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
* [[Biography:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;411–413: The Gosset Series: n<sub>21</sub>)

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

{{Polytopes}}
{{Honeycombs}}

[[Category:Polytopes]]
[[Category:Multi-dimensional geometry]]

{{Sourceattribution|Uniform k 21 polytope}}