# Uniform polyteron

*1,016*pages on

this wiki

A **uniform 5-polytope** is a uniform polytope that exists in 5-dimensional Euclidean space. Using a Wythoff construction, the set of uniform 5-polytopes are enumerated below, grouped with the generation symmetry, although there are overlaps as different generators can create the same forms.

** Regulars and truncations **
The three regular 5-polytopes above create 2 families of uniform 5-polytopes. Using a naming scheme proposed by Norman Johnson, these are:

**{3,3,3,3} Family ** - There are 19 forms.

- {4,3,3,3} - 31 truncated forms
- t_0{4,3,3,3} regular penteract
- t_1{4,3,3,3} rectified penteract
- t_2{4,3,3,3} birectified penteract
- t_3{4,3,3,3} trirectified penteract
- t_4{4,3,3,3} quadrirectified penteract
- t_0,1{4,3,3,3} truncated penteract
- t_1,2{4,3,3,3} bitruncated penteract
- t_2,3{4,3,3,3} tritruncated penteract
- t_3,4{4,3,3,3} quadritruncated penteract
- t_0,2{4,3,3,3} cantellated penteract
- t_1,3{4,3,3,3} bicantellated penteract
- t_2,4{4,3,3,3} tricantellated penteract
- t_0,3{4,3,3,3} runcinated penteract
- t_1,4{4,3,3,3} biruncinated penteract
- t_0,4{4,3,3,3} stericated penteract
- t_0,1,2{4,3,3,3} cantitruncated penteract
- t_1,2,3{4,3,3,3} bicantitruncated penteract
- t_2,3,4{4,3,3,3} tricantitruncated penteract
- t_0,1,3{4,3,3,3} runcitruncated penteract
- t_1,2,4{4,3,3,3} biruncitruncated penteract
- t_0,2,3{4,3,3,3} runcicantellated penteract
- t_1,3,4{4,3,3,3} biruncicantellated penteract
- t_0,1,4{4,3,3,3} steritruncated penteract
- t_0,2,4{4,3,3,3} stericantellated penteract
- t_0,3,4{4,3,3,3} steriruncinated penteract
- t_0,1,2,3{4,3,3,3} runcicantitruncated penteract
- t_1,2,3,4{4,3,3,3} biruncicantitruncated penteract
- t_0,1,2,4{4,3,3,3} stericantitruncated penteract
- t_0,1,3,4{4,3,3,3} steriruncitruncated penteract
- t_0,2,3,4{4,3,3,3} steriruncicantellated penteract
- t_0,1,2,3,4{4,3,3,3} Omnitruncated penteract

## Uniform alternate truncations

There is a one semiregular polytope from a set of semiregular n-polytopes called a half measure polytope, discovered by Thorold Gosset in his complete enumeration of semiregular polytopes. They are all formed by half the vertices of a measure polytope (alternatingly truncated).

This one is called a demipenteract. It has 16 vertices, with *10* 16-cells, and *16* 5-cells.

The semiregular demipenteract can also be used to create 7 truncated forms. There are up to 3 types of hypercells, truncations of the 16-cell, truncations of the 5-cell, and truncations of the tetrahedral hyperprism.

** Prismatic forms **

There are 3 categorical uniform prismatic forms:

- {} x {p,q,r} - uniform polychoron prisms (Each
*uniform polychoron*forms one uniform prism)- {} x {3,3,3} - 9 forms
- {} x {3,3,4} - 15 forms (Three shared with {}x{3,4,3} family)
- {} x {3,4,3} - 10 forms
- {} x {3,3,5} - 15 forms
- Grand antiprism prism

- {p} x {q,r} - Regular polygon - uniform polyhedron duoprisms
- {p} x {3,3} - 5 forms for each (
*p*>=3) (Three shared with {p}x{3,4} family) - {p} x {3,4} - 7 forms for each (
*p*>=3) - {p} x {3,5} - 7 forms for each (
*p*>=3)

- {p} x {3,3} - 5 forms for each (
- {} x {p} x {q} - Uniform duoprism prisms - 1 form for each
*p*and*q*, (each >=3).

## References

- T. Gosset:
*On the Regular and Semi-Regular Figures in Space of n Dimensions*, Messenger of Mathematics, Macmillan, 1900 - Norman Johnson
*Uniform Polytopes*, Manuscript (1991) - Richard Klitzing 5D quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons