List of uniform polychora (Meta, 11)

From Higher Dimensions Database

This page tabulates simple data for each of the (convex, Euclidean, non-prismatic) uniform polychora.

Families

Pyromorphs

Dx CD Variant Verf cell counts Element counts
CC (5×) CF (10×) CE (10×) CV (5×) |C| |F| |E| |V|
1xoooparent (33)4------ 510105
3xxootruncate (3·62)3----(33)1 10304020
2oxoorectate (34)3----(33)2 10303010
6oxxomesotruncate (3·62)2----(3·62)2 10406030
5xoxocantellate ({3·4}2)2--(3·42)2(34)1 20809030
7xxxocantitruncate (4·62)2--(3·42)1(3·62)1 208012060
9xooxruncinate (peritope) (33)1(3·42)3(3·42)3(33)1 30706020
11xxoxruncitruncate (3·62)1(6·42)2(3·42)1({3·4}2)1 3012015060
15xxxxomnitruncate (pantome) (4·62)1(6·42)1(6·42)1(4·62)1 30150240120

Stauromorphs

Dx CD Variant Verf cell counts Element counts
CC (8×) CF (24×) CE (32×) CV (16×) |C| |F| |E| |V|
1xoooparent (43)4------ 8243216
3xxootruncate (3·82)3----(33)1 248812864
2oxoorectate ({3·4}2)3----(33)2 24889632
6oxxomesotruncate (4·62)2----(3·62)2 2412019296
4ooxodual rectate (34)2----(34)4 24969624
12ooxxdual truncate (34)1----(3·62)4 249612048
8oooxdual ------(33)8 1632248
5xoxocantellate (3·43)2--(3·42)2(34)1 5624828896
10oxoxdual cantellate ({3·4}2)1(4·42)2--({3·4}2)2 4824028896
7xxxocantitruncate (4·6·8)2--(3·42)1(3·62)1 56248384192
14oxxxdual cantitruncate (4·62)1(4·42)1--(4·62)2 48240384192
9xooxruncinate (peritope) (43)1(4·42)3(3·42)3(33)1 8020819264
11xxoxruncitruncate (3·82)1(8·42)2(3·42)1({3·4}2)1 80368480192
13xoxxdual runcitruncate (3·43)1(4·42)1(6·42)2(3·62)1 80368480192
15xxxxomnitruncate (pantome) (4·6·8)1(8·42)1(6·42)1(4·62)1 80464768384

Xylomorphs

Dx CD Variant Verf cell counts Element counts
CC (24×) CF (96×) CE (96×) CV (24×) |C| |F| |E| |V|
1xoooparent (34)6------ 24969624
3xxootruncate (4·62)3----(43)1 48240384192
2oxoorectate ({3·4}2)3----(43)2 4824028896
6oxxomesotruncate (3·82)2----(3·82)2 48366576288
5xoxocantellate (3·43)2--(3·42)2({3·4}2)1 144720864288
7xxxocantitruncate (4·6·8)2--(3·42)1(3·82)1 1447201152576
9xooxruncinate (peritope) (34)1(3·42)3(3·42)3(34)1 240672576144
11xxoxruncitruncate (4·62)1(6·42)2(3·42)1(3·43)1 24011041440576
15xxxxomnitruncate (pantome) (4·6·8)1(6·42)1(6·42)1(4·6·8)1 240139223041152

Rhodomorphs

Dx CD Variant Verf cell counts Element counts
CC (120×) CF (720×) CE (1200×) CV (600×) |C| |F| |E| |V|
1xoooparent (53)4------ 1207201200600
3xxootruncate (3·102)3----(33)1 720312048002400
2oxoorectate ({3·5}2)3----(33)2 720312036001200
6oxxomesotruncate (5·62)2----(3·62)2 720432072003600
4ooxodual rectate (35)2----(34)5 72036003600720
12ooxxdual truncate (35)1----(3·62)5 720360043201440
8oooxdual ------(33)20 6001200720120
5xoxocantellate (3·4·5·4)2--(3·42)2(34)1 19209120108003600
10oxoxdual cantellate ({3·5}2)1(5·42)2--({3·4}2)2 14408640108003600
7xxxocantitruncate (4·6·10)2--(3·42)1(3·62)1 19209120144007200
14oxxxdual cantitruncate (5·62)1(5·42)1--(4·62)2 14408640144007200
9xooxruncinate (peritope) (53)1(5·42)3(3·42)3(33)1 2640744072002400
11xxoxruncitruncate (3·102)1(10·42)2(3·42)1({3·4}2)1 264013440180007200
13xoxxdual runcitruncate (3·4·5·4)1(5·42)1(6·42)2(3·62)1 264013440180007200
15xxxxomnitruncate (pantome) (4·6·10)1(10·42)1(6·42)1(4·62)1 2640170402880014400

Singularities

The snub demitesseract

The snub demitesseract, incorrectly referred to as the snub 24-cell, can be formed by the construction of alternated truncated 24-cell or alternated cantitruncated 16-cell - but in order to be a snub of one of these families, the alternation would need to be of the parent's omnitruncate, and it is not.

When correctly represented, the snub demitesseract's Coxeter-Dynkin symbol is that of a three-pronged star, with all four nodes as snub rings.

The snub demitesseract has 144 cells, 480 faces, 432 edges and 96 vertices.

  • Its cells are 24 icosahedra and 120 tetrahedra. They are joined with three icosahedra and five tetrahedra around each vertex.
  • Its faces are all triangles.

The grand antiprism

There is one (convex, Euclidean, non-prismatic) uniform polychoron not part of any of the above families, known as the grand antiprism. Because it doesn't exist within any family above, it is non-Wythoffian and has no Dx number, CD or Schlaefli symbols. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes.

Its elements are 320 cells, 720 faces, 500 edges and 100 vertices.

Uniform polyhedra usage statistics

#PolyhedronUnique uses
PyromorphsStauromorphsXylomorphsRhodomorphsTotal
Tetrahedral
1 Tetrahedron 54413
6 Tetrahedral truncate 54413
111 Triangular prism 545418
112 Hexagonal prism 323210
Octahedral
2 Cube 628
3 Octahedron 243211
7 Cuboctahedron 242210
8 Cubic truncate 235
9 Octahedral truncate 342211
10 Cuboctahedral rectate 224
11 Cuboctahedral truncate 235
113 Octagonal prism 22
Icosahedral
4 Dodecahedron 22
5 Icosahedron 22
13 Icosidodecahedron 22
14 Dodecahedral truncate 22
15 Icosahedral truncate 22
16 Icosidodecahedral rectate 22
17 Icosidodecahedral truncate 22
115 Pentagonal prism 44
116 Decagonal prism 22
Totals 25402540130