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A5 polytope

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Orthographic projections
A5 Coxeter plane

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

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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 graphs Coxeter-Dynkin diagram
Schläfli symbol
Name
[6] [5] [4] [3]
A5 A4 A3 A2
1
{3,3,3,3}
5-simplex (hix)
2
t1{3,3,3,3} or r{3,3,3,3}
Rectified 5-simplex (rix)
3
t2{3,3,3,3} or 2r{3,3,3,3}
Birectified 5-simplex (dot)
4
t0,1{3,3,3,3} or t{3,3,3,3}
Truncated 5-simplex (tix)
5
t1,2{3,3,3,3} or 2t{3,3,3,3}
Bitruncated 5-simplex (bittix)
6
t0,2{3,3,3,3} or rr{3,3,3,3}
Cantellated 5-simplex (sarx)
7
t1,3{3,3,3,3} or 2rr{3,3,3,3}
Bicantellated 5-simplex (sibrid)
8
t0,3{3,3,3,3}
Runcinated 5-simplex (spix)
9
t0,4{3,3,3,3} or 2r2r{3,3,3,3}
Stericated 5-simplex (scad)
10
t0,1,2{3,3,3,3} or tr{3,3,3,3}
Cantitruncated 5-simplex (garx)
11
t1,2,3{3,3,3,3} or 2tr{3,3,3,3}
Bicantitruncated 5-simplex (gibrid)
12
t0,1,3{3,3,3,3}
Runcitruncated 5-simplex (pattix)
13
t0,2,3{3,3,3,3}
Runcicantellated 5-simplex (pirx)
14
t0,1,4{3,3,3,3}
Steritruncated 5-simplex (cappix)
15
t0,2,4{3,3,3,3}
Stericantellated 5-simplex (card)
16
t0,1,2,3{3,3,3,3}
Runcicantitruncated 5-simplex (gippix)
17
t0,1,2,4{3,3,3,3}
Stericantitruncated 5-simplex (cograx)
18
t0,1,3,4{3,3,3,3}
Steriruncitruncated 5-simplex (captid)
19
t0,1,2,3,4{3,3,3,3}
Omnitruncated 5-simplex (gocad)


A5 polytopes

t0

t1

t2

t0,1

t0,2

t1,2

t0,3

t1,3

t0,4

t0,1,2

t0,1,3

t0,2,3

t1,2,3

t0,1,4

t0,2,4

t0,1,2,3

t0,1,2,4

t0,1,3,4

t0,1,2,3,4

References

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  • 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, ISBN 978-0-471-01003-6 [1]
    • (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
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Notes

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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds