# A5 polytope

 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 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 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}
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}