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.
| Name | Extended Schläfli symbol |
Cell counts by location: {p,q,r,s} | Element counts | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| {p,q,r} (6) |
{}x{p,q} (15) |
{p}x{s} (20) |
{}x{r,s} (15) |
{q,r,s} (6) |
Facets | Cells | Faces | Edges | Vertices | ||
| t_0{3,3,3,3} | 5-simplex] | {3,3,3} | - | - | - | - | 6 | ? | ? | ? | ? |
| t_1{3,3,3,3} | Rectified 5-simplex | t_1{3,3,3} | - | - | - | {3,3,3} | 12 | ? | ? | ? | ? |
| t_2{3,3,3,3} | Birectified 5-simplex | t_2{3,3,3} | |||||||||
| t_0,1{3,3,3,3} | Truncated 5-simplex | t_0,1{3,3,3} | - | - | - | t{3,3,3} | 12 | ? | ? | ? | ? |
| t_1,2{3,3,3,3} | Bitruncated 5-simplex | t_1,2{3,3,3} | |||||||||
| t_0,2{3,3,3,3} | Cantellated 5-simplex | t_0,2{3,3,3} | |||||||||
| t_1,3{3,3,3,3} | Bicantellated 5-simplex | t_1,3{3,3,3} | |||||||||
| t_0,3{3,3,3,3} | Runcinated 5-simplex | t_0,3{3,3,3} | |||||||||
| t_0,4{3,3,3,3} | Stericated 5-simplex | {3,3,3} | |||||||||
| t_0,1,2{3,3,3,3} | Cantitruncated 5-simplex | t_0,1,2{3,3,3} | |||||||||
| t_1,2,3{3,3,3,3} | Bicantitruncated 5-simplex | t_1,2,3{3,3,3} | |||||||||
| t_0,1,3{3,3,3,3} | Runcitruncated 5-simplex | t_0,1,3{3,3,3} | |||||||||
| t_0,2,3{3,3,3,3} | Runcicantellated 5-simplex | t_0,1,3{3,3,3} | |||||||||
| t_0,1,4{3,3,3,3} | Steritruncated 5-simplex | t_0,1{3,3,3} | |||||||||
| t_0,2,4{3,3,3,3} | Stericantellated 5-simplex | t_0,2{3,3,3} | |||||||||
| t_0,1,2,3{3,3,3,3} | Runcicantitruncated 5-simplex | t_0,1,2,3{3,3,3} | |||||||||
| t_0,1,2,4{3,3,3,3} | Stericantitruncated 5-simplex | t_0,1,2,4{3,3,3} | |||||||||
| t_0,1,3,4{3,3,3,3} | Steriruncitruncated 5-simplex | t_0,1,3{3,3,3} | |||||||||
| t_0,1,2,3,4{3,3,3,3} | Omnitruncated 5-simplex | t_0,1,2,3{3,3,3} | |||||||||
- {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)
- {} 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