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User:Tetracube/Uniform polytera

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In geometry, a uniform polyteron is a 5-dimensional polytope which is vertex-transitive, and whose hypercells are uniform polychora.

Convex uniform polytera

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In 5 dimensions and higher, the only regular polytopes are the n-simplex, the n-cross polytope, and the n-hypercube. The hexateron (5-simplex) gives rise to the hexateric family of uniform polytera, while the pentacross (5-cross) and penteract (5-cube) give rise to the penteractic family of uniform polytera.

In addition, there is the family of demihypercubes which in 5D and above is distinct from the hypercube family of uniform polytopes. In 5D, this is the demipenteractic family.

Finally, there are the infinite families of prisms, derived from the prisms and duoprisms of the previous dimensions.

The penteractic family

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See Uniform_polyteron#The_B5_.5B4.2C3.2C3.2C3.5D_family_.28penteract.2Fpentacross.29

The penteractic family of polytera are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform polyteron. All coordinates correspond with uniform polytera of edge length 2.

# Base point Name CD symbol Coxeter-Dynkin
1 Pentacross o4o3o3o3x
2 Rectified pentacross o4o3o3x3o
3 Truncated pentacross o4o3o3x3x
4 Birectified penteract
(Birectified pentacross)
o4o3x3o3o
5 Cantellated pentacross o4o3x3o3x
6 Bitruncated pentacross o4o3x3x3o
7 cantitruncated pentacross o4o3x3x3x
8 Rectified penteract o4x3o3o3o
9 Runcinated pentacross o4x3o3o3x
10 Bicantellated penteract
(Bicantellated pentacross)
o4x3o3x3o
11 Runcitruncated pentacross o4x3o3x3x
12 Bitruncated penteract o4x3x3o3o
13 runcicantellated pentacross o4x3x3o3x
14 Bicantitruncated penteract
(Bicantitruncated pentacross)
o4x3x3x3o
15 Runcicantitruncated pentacross o4x3x3x3x
16 Penteract x4o3o3o3o
17 Stericated penteract
(Stericated pentacross)
x4o3o3o3x
18 Runcinated penteract x4o3o3x3o
19 Steritruncated pentacross x4o3o3x3x
20 Cantellated penteract x4o3x3o3o
21 Stericantellated penteract
(Stericantellated pentacross)
x4o3x3o3x
22 Runcicantellated penteract x4o3x3x3o
23 Stericantitruncated pentacross x4o3x3x3x
24 Truncated penteract x4x3o3o3o
25 Steritruncated penteract x4x3o3o3x
26 Runcitruncated penteract x4x3o3x3o
27 Steriruncitruncated penteract
(Steriruncitruncated pentacross)
x4x3o3x3x
28 cantitruncated penteract x4x3x3o3o
29 Stericantitruncated penteract x4x3x3o3x
30 Runcicantitruncated penteract x4x3x3x3o
31 Omnitruncated penteract
(omnitruncated pentacross)
x4x3x3x3x

Simplex coordinates

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The standard 2-simplex in R3
The permutohedron of order 3 (hexagon)

I found a useful extension - the n-simplex truncation coordinates can be found as facets of the (n+1)-orthoplexes in (n+1)-space. There's one "simplex truncation" facet in each coordinate orthant (just ignore the sign combinations) from each above polytopes with the end-node with the 4 edge unringed. Tom Ruen (talk) 02:43, 28 July 2010 (UTC)

Triangular truncations in 3-space:
# Base point Name Coxeter-Dynkin Vertices
1 (0, 0, 1) Triangle 3
2 (0, 1, 1) Rectified triangle 3
3 (0, 1, 2) Truncated triangle 6
Tetrahedra truncations in 4-space:
# Base point Name Coxeter-Dynkin Vertices
1 (0, 0, 0, 1) tetrahedron 4
2 (0, 0, 1, 1) Rectified tetrahedron 6
3 (0, 0, 1, 2) Truncated tetrahedron 12
4 (0, 1, 1, 1) Birectified tetrahedron 4
5 (0, 1, 1, 2) Cantellated tetrahedron 12
6 (0, 1, 2, 2) Bitruncated tetrahedron 12
7 (0, 1, 2, 3) Omnitruncated tetrahedron 24
Pentachora truncations in 5-space:
# Base point Name Coxeter-Dynkin Vertices
1 (0, 0, 0, 0, 1) Pentachoron 5
2 (0, 0, 0, 1, 1) Rectified pentachoron 10
3 (0, 0, 0, 1, 2) Truncated pentachoron 20
4 (0, 0, 1, 1, 1) Birectified pentachoron 10
5 (0, 0, 1, 1, 2) Cantellated pentachoron 30
6 (0, 0, 1, 2, 2) Bitruncated pentachoron 30
7 (0, 0, 1, 2, 3) Cantitruncated pentachoron 60
8 (0, 1, 1, 1, 1) Trirectified pentachoron 5
9 (0, 1, 1, 1, 2) Runcinated pentachoron 20
10 (0, 1, 1, 2, 2) Bicantellated pentachoron 30
11 (0, 1, 1, 2, 3) Runcitruncated pentachoron 60
12 (0, 1, 2, 2, 2) Tritruncated pentachoron 20
13 (0, 1, 2, 2, 3) Runcicantitruncated pentachoron 60
14 (0, 1, 2, 3, 3) Bicantitruncated pentachoron 60
15 (0, 1, 2, 3, 4) Omnitruncated pentachoron 120