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421 polytope
E8Petrie.svg
Skew orthogonal projection
inside Petrie polygon
Type Uniform 8-polytope
Family k21 polytope
Schläfli symbol {34,2,1}
Coxeter symbol 421
Coxeter-Dynkin diagram CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png
7-faces 19440 total:
2160 {34,1,1}Cross graph 7 Nodes highlighted.svg
17280 {36}Complete graph K8.svg
6-faces 207360:
138240 {34,0,1}Complete graph K7.svg
68120 {35}Complete graph K7.svg
5-faces 483840 {34}Complete graph K6.svg
4-faces 483840 {33}Complete graph K5.svg
Cells 241920 {3,3}Complete graph K4.svg
Faces 60480 {3}
Edges 6720
Vertices 240
Vertex figure 321 polytope
Petrie polygon 30-gon
Coxeter group E8, [34,2,1]
Properties convex

In geometry, 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 last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.

Donald Coxeter called it 421 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 4, 2, and 1, and having a single ring on the final node of the 4 branch. As this graph is a representation of the simple Lie group E8, the polytope is sometimes referred to as the E8 polytope.

For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a 30-gonal regular polygon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc) can also be extracted and drawn on this projection.

Contents

Construction

The facet information can be extracted from its Coxeter-Dynkin diagram.

  • CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png

Removing the node on the short branch leaves the 7-simplex:

  • CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png

Removing the node on the end of the 2-length branch leaves the 7-orthoplex in its alternated form: (411)

  • CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png

Every simplex facet touches an 7-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 321 polytope.

  • CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png

Coordinates

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The 240 vertices of the 421 polytope can be constructed in two set, 112 (22C28) with integer coordinates obtained from (\pm 1,\pm 1,0,0,0,0,0,0)\, by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with half-integer coordinates obtained from \left(\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12\right) \, by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).

projections

E8 roots zome.jpg
The 421 polytope can be projected into 3-space as a physical vertex-edge model as shown using Zome tools[1] (Only a subset of edges are included)
E8 3D.png
The E8 421 polytope projected into 3-space with all 6720 edges[2]
Gosset 4 21 polytope petrie.svg
Projection into Petrie polygon for all unique vertex positions, 30-fold symmetry
Gosset 4 21 polytope Coxeter plane.svg
Projection into Coxeter plane, 12-fold symmetry. Red,orange,and yellow vertices have multiplicity of 1,8, and 24

Related polytopes

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k21 family

The 421 polytope is last in a family called the k21 polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2-orthoplexes) and two triangles (2-simplexes).

Regular complex polytopes

As a 4-dimensional regular complex polytope 3{3}3{3}3{3}3, Coxeter called the Witting polytope, after Alexander Witting.[3]

Uniform 8-polytopes

It is one of a family of 255 (28 − 1) convex uniform polytopes in eight dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram:

CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png

Among these 255, two other special polytopes in this family are: 141 and 241, with a ring at the end of the other branches.

Tessellations

This polytope, along with the 8-simplex, can create a uniform tessellaton of hyperbolic 8-dimensional space, represented by symbol 521 and Coxeter-Dynkin diagram:

CD ring.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png

Notes

  1. ^ http://homepages.wmich.edu/~drichter/gossetzome.htm
  2. ^ e8Flyer.nb
  3. ^ Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, (1974)., 12.5 The Witting 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
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p347 (figure 3.8c) by Peter mcMullen: (30-gonal node-edge graph of 421)
  • Richard Klitzing 8D quasiregular polyzetta o3o3o3o *c3o3o3o3x - fy

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