Uniform polychoron

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Schlegel diagram for the truncated 120-cell with tetrahedral cells visible.

In geometry, a uniform polychoron (plural: uniform polychora) is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra.

This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms.

[edit] History of discovery

Regular polytopes: (convex faces)
- 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität" that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
Regular star-polychora (star polyhedron cells and/or vertex figures)
- 1852: Ludwig Schläfli also found 4 of the 10 regular star polychora, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
- 1883: Edmund Hess completed the list of 10 of the nonconvex regular polychora, in his book (in German) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [1].
Convex semiregular polytopes:
- 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.
- 1912: E. L. Elte expanded on Gosset's work with the publication The Semiregular Polytopes of the Hyperspaces, including a special subset of polytopes with semiregular facets (those constructible by a single ringed node of a Coxeter-Dynkin diagram).
Convex uniform polytopes:
- 1910: Alicia Boole Stott, in her publication Geometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowing Archimedean solid and prism cells.
- 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
- Convex uniform polychora:
  - 1965: The complete list of convex forms was finally done by John Horton Conway and Michael J. T. Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex polychoron, the grand antiprism.
  - 1997: A complete enumeration of the names and elements of the convex uniform polychora is given online by George Olshevsky. (Uniform Polytopes in Four Dimensions, George Olshevsky.)
  - 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope.^[1]
Nonregular uniform star polychora: (similar to the nonconvex uniform polyhedra)
- Ongoing: Thousands of nonconvex uniform polychora are known, but mostly unpublished. The list is presumed not to be complete, and there is no estimate of how long the complete list will be. Participating researchers include Jonathan Bowers, George Olshevsky and Norman Johnson.

[edit] Regular polychora

The uniform polychora include two special subsets, which satisfy additional requirements:

The 16 regular polychora, with the property that all cells, faces, edges, and vertices are congruent:
- 6 convex regular 4-polytopes;
- 10 Schläfli-Hess polychora.

[edit] Convex uniform polychora

There are 64 convex uniform polychora, including the 6 regular convex polychora, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms.

5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
13 are polyhedral prisms based on the Archimedean solids
9 are in the self-dual regular A₄ [3,3,3] group (5-cell) family.
9 are in the self-dual regular F₄ [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
15 are in the regular B₄ [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
15 are in the regular H₄ [3,3,5] group (120-cell/600-cell) family.
1 special snub form in the [3,4,3] group (24-cell) family.
1 special non-Wythoffian polychoron, the grand antiprism.
TOTAL: 68 − 4 = 64

These 64 uniform polychora are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.

In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:

Set of uniform antiprismatic prisms - s{p,2}x{} - Polyhedral prisms of two antiprisms.
Set of uniform duoprisms - {p}x{q} - A product of two polygons.

[edit] The A₄ [3,3,3] family - (5-cell)

The 5-cell has diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

The pictures are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 3 (5)	Pos. 2 (10)	Pos. 1 (10)	Pos. 0 (5)	Cells	Faces	Edges	Vertices
1	5-cell			{3,3,3}	(4) (3.3.3)				5	10	10	5
2	rectified 5-cell			t₁{3,3,3}	(3) (3.3.3.3)			(2) (3.3.3)	10	30	30	10
3	truncated 5-cell			t_0,1{3,3,3}	(3) (3.6.6)			(1) (3.3.3)	10	30	40	20
4	cantellated 5-cell			t_0,2{3,3,3}	(1) (3.4.3.4)		(2) (3.4.4)	(1) (3.3.3.3)	20	80	90	30
5	*runcinated 5-cell			t_0,3{3,3,3}	(1) (3.3.3)	(3) (3.4.4)	(3) (3.4.4)	(1) (3.3.3)	30	70	60	20
6	*bitruncated 5-cell			t_1,2{3,3,3}	(2) (3.6.6)			(2) (3.6.6)	10	40	60	30
7	cantitruncated 5-cell			t_0,1,2{3,3,3}	(2) (4.6.6)		(1) (3.4.4)	(1) (3.6.6)	20	80	120	60
8	runcitruncated 5-cell			t_0,1,3{3,3,3}	(1) (3.6.6)	(2) (4.4.6)	(1) (3.4.4)	(1) (3.4.3.4)	30	120	150	60
9	*omnitruncated 5-cell			t_0,1,2,3{3,3,3}	(1) (4.6.6)	(1) (4.4.6)	(1) (4.4.6)	(1) (4.6.6)	30	150	240	120

(*) The three forms marked with an asterisk have the higher extended pentachoric symmetry, of order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.

[edit] The B/C₄ [4,3,3] family - (tesseract/16-cell)

This family has diploid hexadecachoric symmetry, of order 24*16=384: 4!=24 permutations of the four axes, 2⁴=16 for reflection in each axis.

[edit] Tesseract family

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.

#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Cells	Faces	Edges	Vertices
10	8-cell or tesseract			{4,3,3}	(4) (4.4.4)				8	24	32	16
11	rectified 8-cell			t₁{4,3,3}	(3) (3.4.3.4)			(2) (3.3.3)	24	88	96	32
13	truncated 8-cell			t_0,1{4,3,3}	(3) (3.8.8)			(1) (3.3.3)	24	88	128	64
14	cantellated 8-cell			t_0,2{4,3,3}	(1) (3.4.4.4)		(2) (3.4.4)	(1) (3.3.3.3)	56	248	288	96
15	runcinated 8-cell (also runcinated 16-cell)			t_0,3{4,3,3}	(1) (4.4.4)	(3) (4.4.4)	(3) (3.4.4)	(1) (3.3.3)	80	208	192	64
16	bitruncated 8-cell (also bitruncated 16-cell)			t_1,2{4,3,3}	(2) (4.6.6)			(2) (3.6.6)	24	120	192	96
18	cantitruncated 8-cell			t_0,1,2{4,3,3}	(2) (4.6.8)		(1) (3.4.4)	(1) (3.6.6)	56	248	384	192
19	runcitruncated 8-cell			t_0,1,3{4,3,3}	(1) (3.8.8)	(2) (4.4.8)	(1) (3.4.4)	(1) (3.4.3.4)	80	368	480	192
21	omnitruncated 8-cell (also omnitruncated 16-cell)			t_0,1,2,3{3,3,4}	(1) (4.6.8)	(1) (4.4.8)	(1) (4.4.6)	(1) (4.6.6)	80	464	768	384

[edit] 16-cell family

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at position 0, with a consistent orientation, and the 8 cells at position 3 are shown solid, bicolored in two prismatic sets.

#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Cells	Faces	Edges	Vertices
12	16-cell			{3,3,4}				(8) (3.3.3)	16	32	24	8
[22]	rectified 16-cell (Same as 24-cell*)			t₁{3,3,4}	(2) (3.3.3.3)			(4) (3.3.3.3)	24	96	96	24
17	truncated 16-cell			t_0,1{3,3,4}	(1) (3.3.3.3)			(4) (3.6.6)	24	96	120	48
[23]	cantellated 16-cell (Same as rectified 24-cell*)			t_0,2{3,3,4}	(1) (3.4.3.4)	(2) (4.4.4)		(2) (3.4.3.4)	48	240	288	96
[15]	runcinated 16-cell (also runcinated 8-cell)			t_0,3{3,3,4}	(1) (4.4.4)	(3) (4.4.4)	(3) (3.4.4)	(1) (3.3.3)	80	208	192	64
[16]	bitruncated 16-cell (also bitruncated 8-cell)			t_1,2{3,3,4}	(2) (4.6.6)			(2) (3.6.6)	24	120	192	96
[24]	cantitruncated 16-cell (Same as truncated 24-cell*)			t_0,1,2{3,3,4}	(1) (4.6.6)	(1) (4.4.4)		(2) (4.6.6)	48	240	384	192
20	runcitruncated 16-cell			t_0,1,3{3,3,4}	(1) (3.4.4.4)	(1) (4.4.4)	(2) (4.4.6)	(1) (3.6.6)	80	368	480	192
[21]	omnitruncated 16-cell (also omnitruncated 8-cell)			t_0,1,2,3{3,3,4}	(1) (4.6.8)	(1) (4.4.8)	(1) (4.4.6)	(1) (4.6.6)	80	464	768	384
[31]	alternated cantitruncated 16-cell (Same as the snub 24-cell)			h_0,1,2{3,3,4}	(1) (3.3.3.3.3)	(1) (3.3.3)	(4) (96) (3.3.3)	(2) (3.3.3.3.3)	144	480	432	96

(*) Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family.

The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell. The truncated octahedral cells become icosahedra. The cube becomes a tetrahedron, and 96 new tetrahedra are created in the gaps from the removed vertices.

[edit] The F₄ [3,4,3] family - (24-cell)

This family has diploid icositetrachoric symmetry, of order 24*48=1152: the 48 symmetries of the octahedron for each of the 24 cells.

#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 3 (24)	Pos. 2 (96)	Pos. 1 (96)	Pos. 0 (24)	Cells	Faces	Edges	Vertices
22	24-cell (Same as rectified 16-cell)			{3,4,3}	(6) (3.3.3.3)				24	96	96	24
23	rectified 24-cell (Same as cantellated 16-cell)			t₁{3,4,3}	(3) (3.4.3.4)			(2) (4.4.4)	48	240	288	96
24	truncated 24-cell (Same as cantitruncated 16-cell)			t_0,1{3,4,3}	(3) (4.6.6)			(1) (4.4.4)	48	240	384	192
25	cantellated 24-cell			t_0,2{3,4,3}	(2) (3.4.4.4)		(2) (3.4.4)	(1) (3.4.3.4)	144	720	864	288
26	*runcinated 24-cell			t_0,3{3,4,3}	(1) (3.3.3.3)	(3) (3.4.4)	(3) (3.4.4)	(1) (3.3.3.3)	240	672	576	144
27	*bitruncated 24-cell			t_1,2{3,4,3}	(2) (3.8.8)			(2) (3.8.8)	48	336	576	288
28	cantitruncated 24-cell			t_0,1,2{3,4,3}	(2) (4.6.8)		(1) (3.4.4)	(1) (3.8.8)	144	720	1152	576
29	runcitruncated 24-cell			t_0,1,3{3,4,3}	(1) (4.6.6)	(2) (4.4.6)	(1) (3.4.4)	(1) (3.4.4.4)	240	1104	1440	576
30	*omnitruncated 24-cell			t_0,1,2,3{3,4,3}	(1) (4.6.8)	(1) (4.4.6)	(1) (4.4.6)	(1) (4.6.8)	240	1392	2304	1152
31	Alternated truncated 24-cell †(Same as snub 24-cell)			h_0,1{3,4,3}	(3) (3.3.3.3.3)		(4) (3.3.3)	(1) (3.3.3)	144	480	432	96

(*) Like the 5-cell, the 24-cell is self-dual, and so the three asterisked forms have twice as many symmetries, bringing their total to 2304 (the extended icositetrachoric group).

(†) The snub 24-cell here, despite its common name, is not analogous to the snub cube; rather, is derived by an alternation of the truncated 24-cell. Its symmetry number is only 576 (the ionic diminished icositetrachoric group).

[edit] The H₄ [5,3,3] family — (120-cell/600-cell)

This family has diploid hexacosichoric symmetry, of order 120*120=24*600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra.

[edit] 120-cell family

#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 3 (120)	Pos. 2 (720)	Pos. 1 (1200)	Pos. 0 (600)	Cells	Faces	Edges	Vertices
32	120-cell			{5,3,3}	(4) (5.5.5)				120	720	1200	600
33	rectified 120-cell			t₁{5,3,3}	(3) (3.5.3.5)			(2) (3.3.3)	720	3120	3600	1200
36	truncated 120-cell			t_0,1{5,3,3}	(3) (3.10.10)			(1) (3.3.3)	720	3120	4800	2400
37	cantellated 120-cell			t_0,2{5,3,3}	(1) (3.4.5.4)		(2) (3.4.4)	(1) (3.3.3.3)	1920	9120	10800	3600
38	runcinated 120-cell (also runcinated 600-cell)			t_0,3{5,3,3}	(1) (5.5.5)	(3) (4.4.5)	(3) (3.4.4)	(1) (3.3.3)	2640	7440	7200	2400
39	bitruncated 120-cell (also bitruncated 600-cell)			t_1,2{5,3,3}	(2) (5.6.6)			(2) (3.6.6)	720	4320	7200	3600
42	cantitruncated 120-cell			t_0,1,2{5,3,3}	(2) (4.6.10)		(1) (3.4.4)	(1) (3.6.6)	1920	9120	14400	7200
43	runcitruncated 120-cell			t_0,1,3{5,3,3}	(1) (3.10.10)	(2) (4.4.10)	(1) (3.4.4)	(1) (3.4.3.4)	2640	13440	18000	7200
46	omnitruncated 120-cell (also omnitruncated 600-cell)			t_0,1,2,3{5,3,3}	(1) (4.6.10)	(1) (4.4.10)	(1) (4.4.6)	(1) (4.6.6)	2640	17040	28800	14400

[edit] 600-cell family

#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Pos. 3 (120)	Pos. 2 (720)	Pos. 1 (1200)	Pos. 0 (600)	Cells	Faces	Edges	Vertices
35	600-cell			{3,3,5}				(20) (3.3.3)	600	1200	720	120
34	rectified 600-cell			t₁{3,3,5}	(2) (3.3.3.3.3)			(5) (3.3.3.3)	720	3600	3600	720
41	truncated 600-cell			t_0,1{3,3,5}	(1) (3.3.3.3.3)			(5) (3.6.6)	720	3600	4320	1440
40	cantellated 600-cell			t_0,2{3,3,5}	(1) (3.5.3.5)	(2) (4.4.5)		(1) (3.4.3.4)	1440	8640	10800	3600
[38]	runcinated 600-cell (also runcinated 120-cell)			t_0,3{3,3,5}	(1) (5.5.5)	(3) (4.4.5)	(3) (3.4.4)	(1) (3.3.3)	2640	7440	7200	2400
[39]	bitruncated 600-cell (also bitruncated 120-cell)			t_1,2{3,3,5}	(2) (5.6.6)			(2) (3.6.6)	720	4320	7200	3600
45	cantitruncated 600-cell			t_0,1,2{3,3,5}	(1) (5.6.6)	(1) (4.4.5)		(2) (4.6.6)	1440	8640	14400	7200
44	runcitruncated 600-cell			t_0,1,3{3,3,5}	(1) (3.4.5.4)	(1) (4.4.5)	(2) (4.4.6)	(1) (3.6.6)	2640	13440	18000	7200
[46]	omnitruncated 600-cell (also omnitruncated 120-cell)			t_0,1,2,3{3,3,5}	(1) (4.6.10)	(1) (4.4.10)	(1) (4.4.6)	(1) (4.6.6)	2640	17040	28800	14400

[edit] The D₄ [3^1,1,1] group family (Demitesseract)

This demitesseract family introduces no new uniform polychora, but it is worthy to repeat these alternative constructions.

#	Name	Picture	Vertex figure	Coxeter-Dynkin	Cell counts by location					Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin	Pos. 0 (8)	Pos. 1 (24)	Pos. 2 (8)	Pos. 3 (8)	Pos. Alt (96)	Cells	Faces	Edges	Vertices
[12]	demitesseract (Same as 16-cell)			t₀{3^1,1,1}			(4) (3.3.3)	(4) (3.3.3)		16	32	24	8
[17]	truncated demitesseract (Same as truncated 16-cell)			t_0,1{3^1,1,1}	(1) (3.3.3.3)		(2) (3.6.6)	(2) (3.6.6)		24	96	120	48
[11]	cantellated demitesseract (Same as rectified tesseract)			t_0,2{3^1,1,1}	(1) (3.3.3)		(1) (3.3.3)	(3) (3.4.3.4)		24	88	96	32
[16]	cantitruncated demitesseract (Same as bitruncated tesseract)			t_0,1,2{3^1,1,1}	(1) (3.6.6)		(1) (3.6.6)	(2) (4.6.6)		24	120	192	96
[22]	rectified demitesseract (Same as rectified 16-cell) (Same as 24-cell)			t₁{3^1,1,1}	(2) (3.3.3.3)		(2) (3.3.3.3)	(2) (3.3.3.3)		24	96	96	24
[23]	runcicantellated demitesseract (Same as cantellated 16-cell) (Same as rectified 24-cell)			t_0,2,3{3^1,1,1}	(1) (3.4.3.4)	(2) (4.4.4)	(1) (3.4.3.4)	(1) (3.4.3.4)		48	240	288	96
[24]	omnitruncated demitesseract (Same as cantitruncated 16-cell) (Same as truncated 24-cell)			t_0,1,2,3{3^1,1,1}	(1) (4.6.6)	(1) (4.4.4)	(1) (4.6.6)	(1) (4.6.6)		48	240	384	192
[31]	snub demitesseract (Same as snub 24-cell)			s{3^1,1,1}	(1) (3.3.3.3.3)	(1) (3.3.3)	(1) (3.3.3.3.3)	(1) (3.3.3.3.3)	(4) (3.3.3)	144	480	432	96

Here again the snub 24-cell represents an alternated truncation of the truncated 24-cell, creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed polychoron, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron.

[edit] The grand antiprism

There is one non-Wythoffian uniform convex polychoron, known as the grand antiprism, consisting of 20 pentagonal antiprisms forming two perpendicular rings joined by 300 tetrahedra. 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 symmetry number is 400 (the ionic diminished Coxeter group).

#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cells by type		Element counts
#	Name	Picture	Vertex figure	Coxeter-Dynkin and Schläfli symbols	Cells by type		Cells	Faces	Edges	Vertices
47	grand antiprism			No symbol	300 (3.3.3)	20 (3.3.3.5)	320	20 {5} 700 {3}	500	100

[edit] Prismatic uniform polychora

There are three infinite families of uniform polychora that are considered prismatic, in that they generalize the properties of the 3-dimensional prisms. A prismatic polytope is a Cartesian product of two polytopes of lower dimension. The third form is contained within the duoprism set, except for the snubbed form - prisms of antiprisms.

1. {p,q} × { } - - {p,q}-hedral prism
2. {p} × {q} - - p-gonal q-gonal duoprism
3. {p} × { } × { } - - p-gonal prismatic prisms - (same as {p} × {4})
4. s{p,2}x{ } - p-gonal antiprism prisms.

[edit] Polyhedral prisms

The more obvious family of prismatic polychora is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a polychoron are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract),

There are 18 convex polyhedral prisms created 5 Platonic solid and 13 Archimedean solid as well as for the infinite families of three-dimensional prisms and antiprisms. The symmetry number of a polyhedral prism is twice that of the base polyhedron.

[edit] Tetrahedral prisms: A₃×A₁ — [3,3] × [ ]

#	Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type			Element counts
#	Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type			Cells	Faces	Edges	Vertices
48	Tetrahedral prism		t₀{3,3}×{}	2 3.3.3	4 3.4.4		6	8 {3} 6 {4}	16	8
49	Truncated tetrahedral prism		t_0,1{3,3}×{}	2 3.6.6	4 3.4.4	4 4.4.6	10	8 (3) 18 {4} 8 {6}	48	24
[51]	Rectified tetrahedral prism (Same as octahedral prism)		t₁{3,3}×{}	2 3.3.3.3	4 3.4.4		6	16 {3} 12 {4}	30	12
[50]	Cantellated tetrahedral prism (Same as cuboctahedral prism)		t_0,2{3,3}×{}	2 3.4.3.4	8 3.4.4	6 4.4.4	16	16 {3} 36 {4}	60	24
[54]	Cantitruncated tetrahedral prism (Same as truncated octahedral prism)		t_0,1,2{3,3}×{}	2 4.6.6	8 3.4.4	6 4.4.4	16	48 {4} 16 {6}	96	48
[59]	Snub tetrahedral prism (Same as icosahedral prism)		s{3,3}×{}	2 3.3.3.3.3	20 3.4.4		22	40 {3} 30 {4}	72	24

[edit] Octahedral prisms: B₃×A₁ - [4,3] × [ ]

#	Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type				Element counts
#	Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type				Cells	Faces	Edges	Vertices
[10]	Cubic prism (Same as tesseract) (Same as 4-4 duoprism)		t₀{4,3}×{}	2 4.4.4	6 4.4.4			8	24 {4}	32	16
50	Cuboctahedral prism (Same as cantellated tetrahedral prism)		t₁{4,3}×{}	2 3.4.3.4	8 3.4.4	6 4.4.4		16	16 {3} 36 {4}	60	24
51	Octahedral prism (Same as rectified tetrahedral prism) (Same as triangular antiprismatic prism)		t₂{4,3}×{}	2 3.3.3.3	8 3.4.4			10	16 {3} 12 {4}	30	12
52	Rhombicuboctahedral prism		t_0,2{4,3}×{}	2 3.4.4.4	8 3.4.4	18 4.4.4		28	16 {3} 84 {4}	120	96
53	Truncated cubic prism		t_0,1{4,3}×{}	2 3.8.8	8 3.4.4	6 4.4.8		16	16 {3} 36 {4} 12 {8}	96	48
54	Truncated octahedral prism (Same as cantitruncated tetrahedral prism)		t_1,2{4,3}×{}	2 4.6.6	6 4.4.4	8 4.4.6		16	48 {4} 16 {6}	96	48
55	Truncated cuboctahedral prism		t_0,1,2{4,3}×{}	2 4.6.8	12 4.4.4	8 4.4.6	6 4.4.8	28	96 {4} 16 {6} 12 {8}	192	96
56	Snub cubic prism		s{4,3}×{}	2 3.3.3.3.4	32 3.4.4	6 4.4.4		40	64 {3} 72 {4}	144	48

[edit] Icosahedral prisms: H₃×A₁ - [5,3] × [ ]

#	Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type				Element counts
#	Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type				Cells	Faces	Edges	Vertices
57	Dodecahedral prism		t₀{5,3}×{}	2 5.5.5	12 4.4.5			14	30 {4} 24 {5}	80	40
58	Icosidodecahedral prism		t₁{5,3}×{}	2 3.5.3.5	20 3.4.4	12 4.4.5		34	40 {3} 60 {4} 24 {5}	150	60
59	Icosahedral prism (same as snub tetrahedral prism)		t₂{5,3}×{}	2 3.3.3.3.3	20 3.4.4			22	40 {3} 30 {4}	72	24
60	Truncated dodecahedral prism		t_0,1{5,3}×{}	2 3.10.10	20 3.4.4	12 4.4.5		34	40 {3} 90 {4} 24 {10}	240	120
61	Rhombicosidodecahedral prism		t_0,2{5,3}×{}	2 3.4.5.4	20 3.4.4	30 4.4.4	12 4.4.5	64	40 {3} 180 {4} 24 {5}	300	120
62	Truncated icosahedral prism		t_1,2{5,3}×{}	2 5.6.6	12 4.4.5	20 4.4.6		34	90 {4} 24 {5} 40 {6}	240	120
63	Truncated icosidodecahedral prism		t_0,1,2{5,3}×{}	2 4.6.4.10	30 4.4.4	20 4.4.6	12 4.4.10	64	240 {4} 40 {6} 24 {5}	480	240
64	Snub dodecahedral prism		s{5,3}×{}	2 3.3.3.3.5	80 3.4.4	12 4.4.5		94	240 {4} 40 {6} 24 {10}	360	120

[edit] Duoprisms: I₂(p)×I₂(q) - [p] × [q]

The simplest of the duoprisms, the 3,3-duoprism, in Schlegel diagram, one of 6 triangular prism cells shown.

The second is the infinite family of uniform duoprisms, products of two regular polygons.

Their Coxeter-Dynkin diagram is of the form

This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p². The tesseract can also be considered a 4,4-duoprism.

The elements of a p,q-duoprism (p ≥ 3, q ≥ 3) are:

Cells: p q-gonal prisms, q p-gonal prisms
Faces: pq squares, p q-gons, q p-gons
Edges: 2pq
Vertices: pq

There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms.

Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms:

3-3 duoprism - - 6 triangular prisms
3-4 duoprism - - 3 cubes, 4 triangular prisms
4-4 duoprism - - 8 cubes (same as tesseract)
3-5 duoprism - - 3 pentagonal prisms, 5 triangular prisms
4-5 duoprism - - 4 pentagonal prisms, 5 cubes
5-5 duoprism - - 10 pentagonal prisms
3-6 duoprism - - 3 hexagonal prisms, 6 triangular prisms
4-6 duoprism - - 4 hexagonal prisms, 6 cubes
5-6 duoprism - - 5 hexagonal prisms, 6 pentagonal prisms
6-6 duoprism - - 12 hexagonal prisms
...

[edit] Polygonal prismatic prisms: I₂(p)×A₁×A₁ - [p] × [ ] × [ ]

The infinte set of uniform prismatic prism overlap with the 4-p duoprisms: (p≥3) - - p cubes and 4 p-gonal prisms - (All are the same as 4-p duoprism)

Triangular prismatic prism - - 3 cubes and 4 triangular prisms - (same as 3-4 duoprism)
Square prismatic prism - - 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract)
Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms - (same as 4-5 duoprism)
Hexagonal prismatic prism - - 6 cubes and 4 hexagonal prisms - (same as 4-6 duoprism)
Heptagonal prismatic prism - - 7 cubes and 4 heptagonal prisms - (same as 4-7 duoprism)
Octagonal prismatic prism - - 8 cubes and 4 octagonal prisms - (same as 4-8 duoprism)
...

The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥3) - - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms.

Triangular antiprismatic prism - - 2 octahedras connected by 8 triangular prisms (same as the octahedral prism)
Square antiprismatic prism - - 2 square antiprisms connected by 2 cubes and 8 triangular prisms
Pentagonal antiprismatic prism - - 2 pentagonal antiprisms connected by 2 pentagonal prisms and 10 triangular prisms
Hexagonal antiprismatic prism - - 2 hexagonal antiprisms connected by 2 hexagonal prisms and 12 triangular prisms
Heptagonal antiprismatic prism - - 2 heptagonal antiprisms connected by 2 heptagonal prisms and 14 triangular prisms
Octagonal antiprismatic prism - - 2 octagonal antiprisms connected by 2 octagonal prisms and 16 triangular prisms
...

A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices.

[edit] Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

Summary chart of truncation operations

Example locations of kaleidoscopic generator point on fundamental domain.

The 46 Wythoffian polychora include the six convex regular polychora. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their symmetries, and therefore may be classified by the symmetry groups that they have in common.

The geometric operations that derive the 40 uniform polychora from the regular polychora are truncating operations. A polychoron may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.

The Coxeter-Dynkin diagram shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors. (180/n degrees) Circled nodes show which mirrors are active for each form; a mirror is active with respect to a vertex that does not lie on it.

Operation	Schläfli symbol	Description
Parent	t₀{p,q,r}	Original regular form {p,q,r}
Rectification	t₁{p,q,r}	Truncation operation applied until the original edges are degenerated into points.
Birectification	t₂{p,q,r}	Face are fully truncated to points. Same as rectified dual.
Trirectification (dual)	t₃{p,q,r}	Cells are truncated to points. Regular dual {r,q,p}
Truncation	t_0,1{p,q,r}	Each vertex cut off so that the middle of each original edge remains. Where the vertex was, there appears a new cell, the parent's vertex figure. Each original cell is likewise truncated.
Bitruncation	t_1,2{p,q,r}	A truncation between a rectified form and the dual rectified form
Tritruncation	t_2,3{p,q,r}	Truncated dual {r,q,p}
Cantellation	t_0,2{p,q,r}	A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
Bicantellation	t_1,3{p,q,r}	Cantellated dual {r,q,p}
Runcination (or expansion)	t_0,3{p,q,r}	A truncation applied to the cells, faces, and edges and defines a progression between a regular form and the dual.
Cantitruncation	t_0,1,2{p,q,r}	Both the cantellation and truncation operations applied together.
Bicantitruncation	t_1,2,3{p,q,r}	Cantitruncated dual {r,q,p}
Runcitruncation	t_0,1,3{p,q,r}	Both the runcination and truncation operations applied together.
Runcicantellation	t_0,1,3{p,q,r}	Runcitruncated dual {r,q,p}
Omnitruncation (or more specifically runcicantitruncation)	t_0,1,2,3{p,q,r}	Has all three operators applied.
Snub	s{p,q,r}	An alternation of an omnitruncated form, (rings are replaced by holes)

See also convex uniform honeycombs, some of which illustrate these operations as applied to the regular cubic honeycomb.

If two polytopes are duals of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent.

[edit] See also

[edit] References

^ 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope (in German)

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 141817968X
H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
B. Grünbaum Convex polytopes, New York ; London : Springer, c2003. ISBN 0-387-00424-6.
Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

[edit] External links

Weisstein, Eric W., "Uniform polychoron" from MathWorld.
Convex uniform polychora
Nonconvex uniform polychora
- Uniform polychora by Jonathan Bowers
- Stella4D Stella (software) produces interactive views of all 1849 known uniform polychora including the 64 convex forms and the infinite prismatic families. Was used to create most images on this page.

[0] 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope (in German)

[1]

Uniform polychoron

From Wikipedia, the free encyclopedia

Contents

[edit] History of discovery

[edit] Regular polychora

[edit] Convex uniform polychora

[edit] The A₄ [3,3,3] family - (5-cell)

[edit] The B/C₄ [4,3,3] family - (tesseract/16-cell)

[edit] Tesseract family

[edit] 16-cell family

[edit] The F₄ [3,4,3] family - (24-cell)

[edit] The H₄ [5,3,3] family — (120-cell/600-cell)

[edit] 120-cell family

[edit] 600-cell family

[edit] The D₄ [3^1,1,1] group family (Demitesseract)

[edit] The grand antiprism

[edit] Prismatic uniform polychora

[edit] Polyhedral prisms

[edit] Tetrahedral prisms: A₃×A₁ — [3,3] × [ ]

[edit] Octahedral prisms: B₃×A₁ - [4,3] × [ ]

[edit] Icosahedral prisms: H₃×A₁ - [5,3] × [ ]

[edit] Duoprisms: I₂(p)×I₂(q) - [p] × [q]

[edit] Polygonal prismatic prisms: I₂(p)×A₁×A₁ - [p] × [ ] × [ ]

[edit] Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

[edit] See also

[edit] References

[edit] External links

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Uniform polychoron

From Wikipedia, the free encyclopedia

Contents

[edit] History of discovery

[edit] Regular polychora

[edit] Convex uniform polychora

[edit] The A4 [3,3,3] family - (5-cell)

[edit] The B/C4 [4,3,3] family - (tesseract/16-cell)

[edit] Tesseract family

[edit] 16-cell family

[edit] The F4 [3,4,3] family - (24-cell)

[edit] The H4 [5,3,3] family — (120-cell/600-cell)

[edit] 120-cell family

[edit] 600-cell family

[edit] The D4 [31,1,1] group family (Demitesseract)

[edit] The grand antiprism

[edit] Prismatic uniform polychora

[edit] Polyhedral prisms

[edit] Tetrahedral prisms: A3×A1 — [3,3] × [ ]

[edit] Octahedral prisms: B3×A1 - [4,3] × [ ]

[edit] Icosahedral prisms: H3×A1 - [5,3] × [ ]

[edit] Duoprisms: I2(p)×I2(q) - [p] × [q]

[edit] Polygonal prismatic prisms: I2(p)×A1×A1 - [p] × [ ] × [ ]

[edit] Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

[edit] See also

[edit] References

[edit] External links

Views

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[edit] The A₄ [3,3,3] family - (5-cell)

[edit] The B/C₄ [4,3,3] family - (tesseract/16-cell)

[edit] The F₄ [3,4,3] family - (24-cell)

[edit] The H₄ [5,3,3] family — (120-cell/600-cell)

[edit] The D₄ [3^1,1,1] group family (Demitesseract)

[edit] Tetrahedral prisms: A₃×A₁ — [3,3] × [ ]

[edit] Octahedral prisms: B₃×A₁ - [4,3] × [ ]

[edit] Icosahedral prisms: H₃×A₁ - [5,3] × [ ]

[edit] Duoprisms: I₂(p)×I₂(q) - [p] × [q]

[edit] Polygonal prismatic prisms: I₂(p)×A₁×A₁ - [p] × [ ] × [ ]