User:Tetracube/Uniform polytera

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

Convex uniform polytera

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

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	$\left(0,\ 0,\ 0,\ 0,\ {\sqrt {2}}\right)$	Pentacross	o4o3o3o3x
2	$\left(0,\ 0,\ 0,\ {\sqrt {2}},\ {\sqrt {2}}\right)$	Rectified pentacross	o4o3o3x3o
3	$\left(0,\ 0,\ 0,\ {\sqrt {2}},\ 2{\sqrt {2}}\right)$	Truncated pentacross	o4o3o3x3x
4	$\left(0,\ 0,\ {\sqrt {2}},\ {\sqrt {2}},\ {\sqrt {2}}\right)$	Birectified penteract (Birectified pentacross)	o4o3x3o3o
5	$\left(0,\ 0,\ {\sqrt {2}},\ {\sqrt {2}},\ 2{\sqrt {2}}\right)$	Cantellated pentacross	o4o3x3o3x
6	$\left(0,\ 0,\ {\sqrt {2}},\ 2{\sqrt {2}},\ 2{\sqrt {2}}\right)$	Bitruncated pentacross	o4o3x3x3o
7	$\left(0,\ 0,\ {\sqrt {2}},\ 2{\sqrt {2}},\ 3{\sqrt {2}}\right)$	cantitruncated pentacross	o4o3x3x3x
8	$\left(0,\ {\sqrt {2}},\ {\sqrt {2}},\ {\sqrt {2}},\ {\sqrt {2}}\right)$	Rectified penteract	o4x3o3o3o
9	$\left(0,\ {\sqrt {2}},\ {\sqrt {2}},\ {\sqrt {2}},\ 2{\sqrt {2}}\right)$	Runcinated pentacross	o4x3o3o3x
10	$\left(0,\ {\sqrt {2}},\ {\sqrt {2}},\ 2{\sqrt {2}},\ 2{\sqrt {2}}\right)$	Bicantellated penteract (Bicantellated pentacross)	o4x3o3x3o
11	$\left(0,\ {\sqrt {2}},\ {\sqrt {2}},\ 2{\sqrt {2}},\ 3{\sqrt {2}}\right)$	Runcitruncated pentacross	o4x3o3x3x
12	$\left(0,\ {\sqrt {2}},\ 2{\sqrt {2}},\ 2{\sqrt {2}},\ 2{\sqrt {2}}\right)$	Bitruncated penteract	o4x3x3o3o
13	$\left(0,\ {\sqrt {2}},\ 2{\sqrt {2}},\ 2{\sqrt {2}},\ 3{\sqrt {2}}\right)$	runcicantellated pentacross	o4x3x3o3x
14	$\left(0,\ {\sqrt {2}},\ 2{\sqrt {2}},\ 3{\sqrt {2}},\ 3{\sqrt {2}}\right)$	Bicantitruncated penteract (Bicantitruncated pentacross)	o4x3x3x3o
15	$\left(0,\ {\sqrt {2}},\ 2{\sqrt {2}},\ 3{\sqrt {2}},\ 4{\sqrt {2}}\right)$	Runcicantitruncated pentacross	o4x3x3x3x
16	$\left(1,\ 1,\ 1,\ 1,\ 1\right)$	Penteract	x4o3o3o3o
17	$\left(1,\ 1,\ 1,\ 1,\ 1+{\sqrt {2}}\right)$	Stericated penteract (Stericated pentacross)	x4o3o3o3x
18	$\left(1,\ 1,\ 1,\ 1+{\sqrt {2}},\ 1+{\sqrt {2}}\right)$	Runcinated penteract	x4o3o3x3o
19	$\left(1,\ 1,\ 1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}}\right)$	Steritruncated pentacross	x4o3o3x3x
20	$\left(1,\ 1,\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+{\sqrt {2}}\right)$	Cantellated penteract	x4o3x3o3o
21	$\left(1,\ 1,\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}}\right)$	Stericantellated penteract (Stericantellated pentacross)	x4o3x3o3x
22	$\left(1,\ 1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}}\right)$	Runcicantellated penteract	x4o3x3x3o
23	$\left(1,\ 1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)$	Stericantitruncated pentacross	x4o3x3x3x
24	$\left(1,\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+{\sqrt {2}}\right)$	Truncated penteract	x4x3o3o3o
25	$\left(1,\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}}\right)$	Steritruncated penteract	x4x3o3o3x
26	$\left(1,\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}}\right)$	Runcitruncated penteract	x4x3o3x3o
27	$\left(1,\ 1+{\sqrt {2}},\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)$	Steriruncitruncated penteract (Steriruncitruncated pentacross)	x4x3o3x3x
28	$\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}}\right)$	cantitruncated penteract	x4x3x3o3o
29	$\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)$	Stericantitruncated penteract	x4x3x3o3x
30	$\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+3{\sqrt {2}}\right)$	Runcicantitruncated penteract	x4x3x3x3o
31	$\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+4{\sqrt {2}}\right)$	Omnitruncated penteract (omnitruncated pentacross)	x4x3x3x3x

Simplex coordinates

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