User:Tomruen/uniform polyteron

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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:

The hexateron family {3,3,3,3}

There are 19 forms:

# Schläfli symbol Name Cell counts by location: {3,3,3,3} Element counts
{3,3,3}
(6) 		{}x{3,3}
(15) 		{3}x{3}
(20) 		{}x{3,3}
(15) 		{3,3,3}
(6) 		Facets Cells Faces Edges Vertices
1
t0{3,3,3,3} 		hexateron
Hix 		{3,3,3} - - - - 6 15 20 15 6
2
t1{3,3,3,3} 		Rectified hexateron
Rix 		t1{3,3,3} - - - {3,3,3} 12 45 80 60 15
3
t2{3,3,3,3} 		Birectified hexateron
Dot 		t2{3,3,3}         12 60 120 90 20
4
t0,1{3,3,3,3} 		Truncated hexateron
Tix 		t0,1{3,3,3} - - - {3,3,3} 12 45 80 75 30
5
t1,2{3,3,3,3} 		Bitruncated hexateron
Bittix 		t1,2{3,3,3}         12 60 140 150 60
6
t0,2{3,3,3,3} 		Cantellated hexateron
Sarx 		t0,2{3,3,3}         27 135 290 240 60
7
t1,3{3,3,3,3} 		Bicantellated hexateron
Sibrid 		t1,3{3,3,3}                  
8
t0,3{3,3,3,3} 		Runcinated hexateron
Spix 		t0,3{3,3,3}                  
9
t0,4{3,3,3,3} 		Stericated hexateron
Scad 		{3,3,3}                  
10
t0,1,2{3,3,3,3} 		Cantitruncated hexateron
Garx 		t0,1,2{3,3,3}                  
11
t1,2,3{3,3,3,3} 		Bicantitruncated hexateron
Gibrid 		t1,2,3{3,3,3}                  
12
t0,1,3{3,3,3,3} 		Runcitruncated hexateron
Pattix 		t0,1,3{3,3,3}                  
13
t0,2,3{3,3,3,3} 		Runcicantellated hexateron
Pirx 		t0,1,3{3,3,3}                  
14
t0,1,4{3,3,3,3} 		Steritruncated hexateron
Cappix 		t0,1{3,3,3}                  
15
t0,2,4{3,3,3,3} 		Stericantellated hexateron
Card 		t0,2{3,3,3}                  
16
t0,1,2,3{3,3,3,3} 		Runcicantitruncated hexateron
Gippix 		t0,1,2,3{3,3,3}                  
17
t0,1,2,4{3,3,3,3} 		Stericantitruncated hexateron
Cograx 		t0,1,2,4{3,3,3}                  
18
t0,1,3,4{3,3,3,3} 		Steriruncitruncated hexateron
Captid 		t0,1,3{3,3,3}                  
19
t0,1,2,3,4{3,3,3,3} 		Omnitruncated hexateron
Gocard 		t0,1,2,3{3,3,3}                  

The penteract family {4,3,3,3}

There are 20 forms, 7 shared with the pentacross family. Four are shared with the demipenteract family.

# Extended
Schläfli
symbol 		Name Cell counts by location: {4,3,3,3} Element counts
{4,3,3}
(32) 		{}x{4,3}
(80) 		{4}x{3}
(80) 		{}x{3,3}
(40) 		{3,3,3}
(10) 		Facets Cells Faces Edges Vertices
20
t0{4,3,3,3} 		penteract
Pent 		{4,3,3} - - - - ? ? ? ? ?
21
t1{4,3,3,3} 		Rectified penteract
Rin 		t1{4,3,3} - - - {3,3,3} ? ? ? ? ?
22
t2{4,3,3,3} 		Birectified penteract
Nit 		t2{4,3,3}                  
23
t0,1{4,3,3,3} 		Truncated penteract
Tan 		t0,1{4,3,3} - - - {3,3,3} ? ? ? ? ?
24
t1,2{4,3,3,3} 		Bitruncated penteract
Bittin 		t1,2{4,3,3}                  
25
t0,2{4,3,3,3} 		Cantellated penteract
Sirn 		t0,2{4,3,3}                  
26
t1,3{4,3,3,3} 		Bicantellated penteract
Sibrant 		t1,3{4,3,3}                  
27
t0,3{4,3,3,3} 		Runcinated penteract
Span 		t0,3{4,3,3}                  
28
t0,4{4,3,3,3} 		Stericated penteract
Scant 		{4,3,3}                  
29
t0,1,2{4,3,3,3} 		Cantitruncated penteract
Girn 		t0,1,2{4,3,3}                  
30
t1,2,3{4,3,3,3} 		Bicantitruncated penteract
Gibrant 		t1,2,3{4,3,3}                  
31
t0,1,3{4,3,3,3} 		Runcitruncated penteract
Pattin 		t0,1,3{4,3,3}                  
32
t0,2,3{4,3,3,3} 		Runcicantellated penteract
Prin 		t0,1,3{4,3,3}                  
33
t0,1,4{4,3,3,3} 		Steritruncated penteract
Capt 		t0,1{4,3,3}                  
34
t0,2,4{4,3,3,3} 		Stericantellated penteract
Carnit 		t0,2{4,3,3}                  
35
t0,1,2,3{4,3,3,3} 		Runcicantitruncated penteract
Gippin 		t0,1,2,3{4,3,3}                  
36
t0,1,2,4{4,3,3,3} 		Stericantitruncated penteract
Cogrin 		t0,1,2,4{4,3,3}                  
37
t0,1,3,4{4,3,3,3} 		Steriruncitruncated penteract
Captint 		t0,1,3{4,3,3}                  
38
t0,1,2,3,4{4,3,3,3} 		Omnitruncated penteract
Gacnet 		t0,1,2,3{4,3,3}                  
39
h_0{4,3,3,3} 		demipenteract
Hin 		{3,3,3} - - - - ? ? ? ? ?

Pentacross family {3,3,3,4}

There are 19 forms, 7 shared with the penteract family.

# Extended
Schläfli
symbol 		Name Cell counts by location: {3,3,3,4} Element counts
{3,3,3}
(10) 		{}x{3,3}
(40) 		{3}x{4}
(80) 		{}x{3,4}
(80) 		{3,3,4}
(32) 		Facets Cells Faces Edges Vertices
40
t0{3,3,3,4} 		pentacross
Tac 		{3,3,3} - - - - ? ? ? ? ?
41
t1{3,3,3,4} 		Rectified pentacross
Rat 		t1{3,3,3} - - - ? ? ? ? ? ?
[22]
t2{3,3,3,4} 		Birectified pentacross
Nit 		t2{3,3,3}                  
42
t0,1{3,3,3,4} 		Truncated pentacross
Tot 		t0,1{3,3,3} - - - ? ? ? ? ? ?
43
t1,2{3,3,3,4} 		Bitruncated pentacross
Bittit 		t1,2{3,3,3}                  
44
t0,2{3,3,3,4} 		Cantellated pentacross
Sart 		t0,2{3,3,3}                  
[26]
t1,3{3,3,3,4} 		Bicantellated pentacross
Sibrant 		t1,3{3,3,3}                  
45
t0,3{3,3,3,4} 		Runcinated pentacross
Spat 		t0,3{3,3,3}                  
[28]
t0,4{3,3,3,4} 		Stericated pentacross
Scant 		{3,3,3}                  
46
t0,1,2{3,3,3,4} 		Cantitruncated pentacross
Gart 		t0,1,2{3,3,3}                  
47
t1,2,3{3,3,3,4} 		Bicantitruncated pentacross
Gibrant 		t1,2,3{3,3,3}                  
48
t0,1,3{3,3,3,4} 		Runcitruncated pentacross
Pattit 		t0,1,3{3,3,3}                  
49
t0,2,3{3,3,3,4} 		Runcicantellated pentacross
Pirt 		t0,1,3{3,3,3}                  
50
t0,1,4{3,3,3,4} 		Steritruncated pentacross
Cappin 		t0,1{3,3,3}                  
[34]
t0,2,4{3,3,3,4} 		Stericantellated pentacross
Carnit 		t0,2{3,3,3}                  
51
t0,1,2,3{3,3,3,4} 		Runcicantitruncated pentacross
Gippit 		t0,1,2,3{3,3,3}                  
52
t0,1,2,4{3,3,3,4} 		Stericantitruncated pentacross
Cogart 		t0,1,2,4{3,3,3}                  
53
t0,1,3,4{3,3,3,4} 		Steriruncitruncated pentacross
Captint 		t0,1,3{3,3,3}                  
54
t0,1,2,3,4{3,3,3,4} 		Omnitruncated pentacross
Gacnet 		t0,1,2,3{3,3,3}                  

Demipenteract family h{4,3,3,3}

There are 23 forms.

# Extended
Schläfli
symbol 		Name Cell counts by location: h{4,3,3,3} Element counts
Facets Cells Faces Edges Vertices
[39]
h_0{3,3,3,4} 		Demipenteract
Hin 		- - - - - 26 120 160 80 16
[22]
h_{3,3,3,4} 		Birectified penteract
Nit 		- - - - - ? ? ? ? ?
[41]
h_{3,3,3,4} 		Rectified pentacross
Rat 		- - - - - ? ? ? ? ?
[40]
h_{3,3,3,4} 		Pentacross
Tac 		- - - - - ? ? ? ? ?
55
h_{3,3,3,4} 		Thin - - - - - ? ? ? ? ?
[21]
h_{3,3,3,4} 		Rectified penteract
Rin 		- - - - - ? ? ? ? ?
56
h_{3,3,3,4} 		Sirhin - - - - - ? ? ? ? ?
57
h_{3,3,3,4} 		Siphin - - - - - ? ? ? ? ?
[24]
h_{3,3,3,4} 		Bitruncated pentacross
Bittit 		- - - - - ? ? ? ? ?
[44]
h_{3,3,3,4} 		Cantellated pentacross
Sart 		- - - - - ? ? ? ? ?
[42]
h_{3,3,3,4} 		Truncated pentacross
Tot 		- - - - - ? ? ? ? ?
[24]
h_{3,3,3,4} 		Bitruncated penteract
Bittin 		- - - - - ? ? ? ? ?
58
h_{3,3,3,4} 		Girhin - - - - - ? ? ? ? ?
59
h_{3,3,3,4} 		Pithin - - - - - ? ? ? ? ?
[26]
h_{3,3,3,4} 		Bicantellated penteract
Sibrant 		- - - - - ? ? ? ? ?
[45]
h_{3,3,3,4} 		Runcinated pentacross
Spat 		- - - - - ? ? ? ? ?
60
h_{3,3,3,4} 		Pirhin - - - - - ? ? ? ? ?
[46]
h_{3,3,3,4} 		Cantitruncated pentacross
Gart 		- - - - - ? ? ? ? ?
[47]
h_{3,3,3,4} 		Bicantitruncated pentacross
Gibrant 		- - - - - ? ? ? ? ?
[49]
h_{3,3,3,4} 		Runcicantellated pentacross
Pirt 		- - - - - ? ? ? ? ?

h_{3,3,3,4} 		Giphin - - - - - ? ? ? ? ?

h_{3,3,3,4} 		Runcitruncated pentacross
Pattit 		- - - - - ? ? ? ? ?

h_{3,3,3,4} 		Runcicantitruncated pentacross
Gippit 		- - - - - ? ? ? ? ?

Uniform alternate truncations

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