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A5 polytope

[[File:5-simplex t0.svg160px]][5-simplex](5-simplex)

In 5-dimensional geometry, there are 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 vertices.

Each can be visualized as symmetric orthographic projections in the Coxeter planes of the A5 Coxeter group and other subgroups.

Graphs

Symmetric orthographic projections of these 19 polytopes can be made in the A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetrically nodea_1ed-diagrams, symmetry doubles to [2(k+1)].

These 19 polytopes are each shown in these 4 symmetry planes, with vertices and edges drawn and vertices colored by the number of overlapping vertices in each projective position.

#Coxeter plane graphsCoxeter-Dynkin diagramSchläfli symbolName[6][5][4][3]A5A4A3A212345678910111213141516171819
[[File:5-simplex t0.svg80px]][[File:5-simplex t0 A4.svg80px]][[File:5-simplex t0 A3.svg80px]][[File:5-simplex t0 A2.svg80px]]{3,3,3,3}[5-simplex](5-simplex) (hix)
[[File:5-simplex t1.svg80px]][[File:5-simplex t1 A4.svg80px]][[File:5-simplex t1 A3.svg80px]][[File:5-simplex t1 A2.svg80px]]t1{3,3,3,3} or r{3,3,3,3}Rectified 5-simplex (rix)
[[File:5-simplex t2.svg80px]][[File:5-simplex t2 A4.svg80px]][[File:5-simplex t2 A3.svg80px]][[File:5-simplex t2 A2.svg80px]]t2{3,3,3,3} or 2r{3,3,3,3}Birectified 5-simplex (dot)
[[File:5-simplex t01.svg80px]][[File:5-simplex t01 A4.svg80px]][[File:5-simplex t01 A3.svg80px]][[File:5-simplex t01 A2.svg80px]]t0,1{3,3,3,3} or t{3,3,3,3}Truncated 5-simplex (tix)
[[File:5-simplex t12.svg80px]][[File:5-simplex t12 A4.svg80px]][[File:5-simplex t12 A3.svg80px]][[File:5-simplex t12 A2.svg80px]]t1,2{3,3,3,3} or 2t{3,3,3,3}Bitruncated 5-simplex (bittix)
[[File:5-simplex t02.svg80px]][[File:5-simplex t02 A4.svg80px]][[File:5-simplex t02 A3.svg80px]][[File:5-simplex t02 A2.svg80px]]t0,2{3,3,3,3} or rr{3,3,3,3}Cantellated 5-simplex (sarx)
[[File:5-simplex t13.svg80px]][[File:5-simplex t13 A4.svg80px]][[File:5-simplex t13 A3.svg80px]][[File:5-simplex t13 A2.svg80px]]t1,3{3,3,3,3} or 2rr{3,3,3,3}Bicantellated 5-simplex (sibrid)
[[File:5-simplex t03.svg80px]][[File:5-simplex t03 A4.svg80px]][[File:5-simplex t03 A3.svg80px]][[File:5-simplex t03 A2.svg80px]]t0,3{3,3,3,3}Runcinated 5-simplex (spix)
[[File:5-simplex t04.svg80px]][[File:5-simplex t04 A4.svg80px]][[File:5-simplex t04 A3.svg80px]][[File:5-simplex t04 A2.svg80px]]t0,4{3,3,3,3} or 2r2r{3,3,3,3}Stericated 5-simplex (scad)
[[File:5-simplex t012.svg80px]][[File:5-simplex t012 A4.svg80px]][[File:5-simplex t012 A3.svg80px]][[File:5-simplex t012 A2.svg80px]]t0,1,2{3,3,3,3} or tr{3,3,3,3}Cantitruncated 5-simplex (garx)
[[File:5-simplex t123.svg80px]][[File:5-simplex t123 A4.svg80px]][[File:5-simplex t123 A3.svg80px]][[File:5-simplex t123 A2.svg80px]]t1,2,3{3,3,3,3} or 2tr{3,3,3,3}Bicantitruncated 5-simplex (gibrid)
[[File:5-simplex t013.svg80px]][[File:5-simplex t013 A4.svg80px]][[File:5-simplex t013 A3.svg80px]][[File:5-simplex t013 A2.svg80px]]t0,1,3{3,3,3,3}Runcitruncated 5-simplex (pattix)
[[File:5-simplex t023.svg80px]][[File:5-simplex t023 A4.svg80px]][[File:5-simplex t023 A3.svg80px]][[File:5-simplex t023 A2.svg80px]]t0,2,3{3,3,3,3}Runcicantellated 5-simplex (pirx)
[[File:5-simplex t014.svg80px]][[File:5-simplex t014 A4.svg80px]][[File:5-simplex t014 A3.svg80px]][[File:5-simplex t014 A2.svg80px]]t0,1,4{3,3,3,3}Steritruncated 5-simplex (cappix)
[[File:5-simplex t024.svg80px]][[File:5-simplex t024 A4.svg80px]][[File:5-simplex t024 A3.svg80px]][[File:5-simplex t024 A2.svg80px]]t0,2,4{3,3,3,3}Stericantellated 5-simplex (card)
[[File:5-simplex t0123.svg80px]][[File:5-simplex t0123 A4.svg80px]][[File:5-simplex t0123 A3.svg80px]][[File:5-simplex t0123 A2.svg80px]]t0,1,2,3{3,3,3,3}Runcicantitruncated 5-simplex (gippix)
[[File:5-simplex t0124.svg80px]][[File:5-simplex t0124 A4.svg80px]][[File:5-simplex t0124 A3.svg80px]][[File:5-simplex t0124 A2.svg80px]]t0,1,2,4{3,3,3,3}Stericantitruncated 5-simplex (cograx)
[[File:5-simplex t0134.svg80px]][[File:5-simplex t0134 A4.svg80px]][[File:5-simplex t0134 A3.svg80px]][[File:5-simplex t0134 A2.svg80px]]t0,1,3,4{3,3,3,3}Steriruncitruncated 5-simplex (captid)
[[File:5-simplex t01234.svg80px]][[File:5-simplex t01234 A4.svg80px]][[File:5-simplex t01234 A3.svg80px]][[File:5-simplex t01234 A2.svg80px]]t0,1,2,3,4{3,3,3,3}Omnitruncated 5-simplex (gocad)

References

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover, New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, and Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
    • (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

Notes

References

  1. "Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter".
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