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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}                  

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)
  • {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

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