Uniform polychoron

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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.

Contents 1 History of discovery 2 Regular polychora 3 Convex uniform polychora 3.1 The A4 [3,3,3] family - (5-cell) 3.2 The C4 [4,3,3] family - (tesseract/16-cell) 3.2.1 Tesseract family 3.2.2 16-cell family 3.3 The F4 [3,4,3] family - (24-cell) 3.4 The G4 [5,3,3] family - (120-cell/600-cell) 3.5 The B4 [31,1,1] group family 3.6 The grand antiprism 3.7 Prismatic uniform polychora 3.7.1 Polyhedral prisms 3.7.2 Tetrahedral prisms: A3xA1 - [3,3] x [ ] 3.7.3 Octahedral prisms: C3xA1 - [4,3] x [ ] 3.7.4 Icosahedral prisms: G3xA1 - [5,3] x [ ] 3.7.5 Duoprisms: D2pxD2q - [p] x [q] 3.7.6 Polygonal prismatic prisms: D2xA1xA1 - [p] x [ ] x [ ] 3.8 Geometric derivations for 46 nonprismatic Wythoffian uniform polychora 4 See also 5 References 6 External links

[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].
• Semiregular polytopes: (convex 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.
• 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 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. [2]
    • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope (in German).
• 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 [3,3,3] group (5-cell) family.
• 9 are in the self-dual regular [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
• 15 are in the regular [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
• 15 are in the regular [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

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

Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
			Pos. 3 (5)	Pos. 2 (10)	Pos. 1 (10)	Pos. 0 (5)	Cells	Faces	Edges	Vertices
5-cell		{3,3,3}	(3.3.3)				5	10	10	5
truncated 5-cell		t0,1{3,3,3}	(3.6.6)			(3.3.3)	10	30	40	20
rectified 5-cell		t1{3,3,3}	(3.3.3.3)			(3.3.3)	10	30	30	10
cantellated 5-cell		t0,2{3,3,3}	(3.4.3.4)		(3.4.4)	(3.3.3.3)	20	80	90	30
cantitruncated 5-cell		t0,1,2{3,3,3}	(4.6.6)		(3.4.4)	(3.6.6)	20	80	120	60
runcitruncated 5-cell		t0,1,3{3,3,3}	(3.6.6)	(4.4.6)	(3.4.4)	(3.4.3.4)	30	120	150	60
*bitruncated 5-cell		t1,2{3,3,3}	(3.6.6)			(3.6.6)	10	40	60	30
*runcinated 5-cell		t0,3{3,3,3}	(3.3.3)	(3.4.4)	(3.4.4)	(3.3.3)	30	70	60	20
*omnitruncated 5-cell		t0,1,2,3{3,3,3}	(4.6.6)	(4.4.6)	(4.4.6)	(4.6.6)	30	150	240	120

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

[edit] Tesseract family

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

Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
			Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Cells	Faces	Edges	Vertices
8-cell or tesseract		{4,3,3}	(4.4.4)				8	24	32	16
truncated 8-cell		t0,1{4,3,3}	(3.8.8)			(3.3.3)	24	88	128	64
rectified 8-cell		t1{4,3,3}	(3.4.3.4)			(3.3.3)	24	88	96	32
cantellated 8-cell		t0,2{4,3,3}	(3.4.4.4)		(3.4.4)	(3.3.3.3)	56	248	288	96
cantitruncated 8-cell		t0,1,2{4,3,3}	(4.6.8)		(3.4.4)	(3.6.6)	56	248	384	192
runcitruncated 8-cell		t0,1,3{4,3,3}	(3.8.8)	(4.4.8)	(3.4.4)	(3.4.3.4)	80	368	480	192
bitruncated 8-cell (also bitruncated 16-cell)		t1,2{4,3,3}	(4.6.6)			(3.6.6)	24	120	192	96
runcinated 8-cell (also runcinated 16-cell)		t0,3{4,3,3}	(4.4.4)	(4.4.4)	(3.4.4)	(3.3.3)	80	208	192	64
omnitruncated 8-cell (also omnitruncated 16-cell)		t0,1,2,3{3,3,4}	(4.6.8)	(4.4.8)	(4.4.6)	(4.6.6)	80	464	768	384

[edit] 16-cell family

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

Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
			Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Cells	Faces	Edges	Vertices
16-cell		{3,3,4}				(3.3.3)	16	32	24	8
truncated 16-cell		t0,1{3,3,4}	(3.3.3.3)			(3.6.6)	24	96	120	48
*rectified 16-cell (Same as 24-cell)		t1{3,3,4}	(3.3.3.3)			(3.3.3.3)	24	96	96	24
*cantellated 16-cell (Same as rectified 24-cell)		t0,2{3,3,4}	(3.4.3.4)	(4.4.4)		(3.4.3.4)	48	240	288	96
*cantitruncated 16-cell (Same as truncated 24-cell)		t0,1,2{3,3,4}	(4.6.6)	(4.4.4)		(4.6.6)	48	240	384	192
runcitruncated 16-cell		t0,1,3{3,3,4}	(3.4.4.4)	(4.4.4)	(4.4.6)	(3.6.6)	80	368	480	192
bitruncated 16-cell (also bitruncated 8-cell)		t1,2{3,3,4}	(4.6.6)			(3.6.6)	24	120	192	96
runcinated 16-cell (also runcinated 8-cell)		t0,3{3,3,4}	(4.4.4)	(4.4.4)	(3.4.4)	(3.3.3)	80	208	192	64
omnitruncated 16-cell (also omnitruncated 8-cell)		t0,1,2,3{3,3,4}	(4.6.8)	(4.4.8)	(4.4.6)	(4.6.6)	80	464	768	384
Alternated cantitruncated 16-cell (Same as the snub 24-cell)		h0,1,2{3,3,4}	(3.3.3.3.3)	(3.3.3)	(96) (3.3.3)	(3.3.3.3.3)	144	480	432	96

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.

(*) 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 cantellated 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)

Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
			Pos. 3 (24)	Pos. 2 (96)	Pos. 1 (96)	Pos. 0 (24)	Cells	Faces	Edges	Vertices
24-cell		{3,4,3}	(3.3.3.3)				24	96	96	24
truncated 24-cell		t0,1{3,4,3}	(4.6.6)			(4.4.4)	48	240	384	192
rectified 24-cell		t1{3,4,3}	(3.4.3.4)			(4.4.4)	48	240	288	96
cantellated 24-cell		t0,2{3,4,3}	(3.4.4.4)		(3.4.4)	(3.4.3.4)	144	720	864	288
cantitruncated 24-cell		t0,1,2{3,4,3}	(4.6.8)		(3.4.4)	(3.8.8)	144	720	1152	576
runcitruncated 24-cell		t0,1,3{3,4,3}	(4.6.6)	(4.4.6)	(3.4.4)	(3.4.4.4)	240	1104	1440	576
*bitruncated 24-cell		t1,2{3,4,3}	(3.8.8)			(3.8.8)	48	336	576	288
*runcinated 24-cell		t0,3{3,4,3}	(3.3.3.3)	(3.4.4)	(3.4.4)	(3.3.3.3)	240	672	576	144
*omnitruncated 24-cell		t0,1,2,3{3,4,3}	(4.6.8)	(4.4.6)	(4.4.6)	(4.6.8)	240	1392	2304	1152
Alternated truncated 24-cell †(Same as snub 24-cell)		h0,1{3,4,3}	(3.3.3.3.3)		(3.3.3) (oblique)	(3.3.3)	144	480	432	96

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

*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 G₄ [5,3,3] family - (120-cell/600-cell)

Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cell counts by location				Element counts
			Pos. 3 (120)	Pos. 2 (720)	Pos. 1 (1200)	Pos. 0 (600)	Cells	Faces	Edges	Vertices
120-cell		{5,3,3}	(5.5.5)				120	720	1200	600
600-cell		{3,3,5}				(3.3.3)	600	1200	720	120
truncated 120-cell		t0,1{5,3,3}	(3.10.10)			(3.3.3)	720	3120	4800	2400
truncated 600-cell		t0,1{3,3,5}	(3.3.3.3.3)			(3.6.6)	720	3600	4320	1440
rectified 120-cell		t1{5,3,3}	(3.5.3.5)			(3.3.3)	720	3120	3600	1200
rectified 600-cell		t1{3,3,5}	(3.3.3.3.3)			(3.3.3.3)	720	3600	3600	720
cantellated 120-cell		t0,2{5,3,3}	(3.4.5.4)		(3.4.4)	(3.3.3.3)	1920	9120	10800	3600
cantellated 600-cell		t0,2{3,3,5}	(3.5.3.5)	(4.4.5)		(3.4.3.4)	1440	8640	10800	3600
cantitruncated 120-cell		t0,1,2{5,3,3}	(4.6.10)		(3.4.4)	(3.6.6)	1920	9120	14400	7200
cantitruncated 600-cell		t0,1,2{3,3,5}	(5.6.6)	(4.4.5)		(4.6.6)	1440	8640	14400	7200
runcitruncated 120-cell		t0,1,3{5,3,3}	(3.10.10)	(4.4.10)	(3.4.4)	(3.4.3.4)	2640	13440	18000	7200
runcitruncated 600-cell		t0,1,3{3,3,5}	(3.4.5.4)	(4.4.5)	(4.4.6)	(3.6.6)	2640	13440	18000	7200
bitruncated 120-cell (also bitruncated 600-cell)		t1,2{5,3,3}	(5.6.6)			(3.6.6)	720	4320	7200	3600
runcinated 120-cell (also runcinated 600-cell)		t0,3{5,3,3}	(5.5.5)	(4.4.5)	(3.4.4)	(3.3.3)	2640	7440	7200	2400
omnitruncated 120-cell (also omnitruncated 600-cell)		t0,1,2,3{5,3,3}	(4.6.10)	(4.4.10)	(4.4.6)	(4.6.6)	2640	17040	28800	14400

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] The B₄ [3^1,1,1] group family

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

Name	Picture	Coxeter-Dynkin	Cell counts by location					Element counts
			Pos. 0 (8)	Pos. 1 (24)	Pos. 0' (8)	Pos. 3 (8)	Pos. Alt (96)	Cells	Faces	Edges	Vertices
16-cell		t0{31,1,1}			(3.3.3)	(3.3.3)		16	32	24	8
truncated 16-cell		t0,1{31,1,1}	(3.3.3.3)		(3.6.6)	(3.6.6)		24	96	120	48
rectified 8-cell		t0,2{31,1,1}	(3.3.3)		(3.3.3)	(3.4.3.4)		24	88	96	32
bitruncated 8-cell		t0,1,2{31,1,1}	(3.6.6)		(3.6.6)	(4.6.6)		24	120	192	96
24-cell		t1{31,1,1}	(3.3.3.3)		(3.3.3.3)	(3.3.3.3)		24	96	96	24
rectified 24-cell		t0,2,3{31,1,1}	(3.4.3.4)	(4.4.4)	(3.4.3.4)	(3.4.3.4)		48	240	288	96
truncated 24-cell		t0,1,2,3{31,1,1}	(4.6.6)	(4.4.4)	(4.6.6)	(4.6.6)		48	240	384	192
snub 24-cell		s{31,1,1}	(3.3.3.3.3)	(3.3.3)	(3.3.3.3.3)	(3.3.3.3.3)	(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 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, 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	Coxeter-Dynkin and Schläfli symbols	Cells by type		Element counts
					Cells	Faces	Edges	Vertices
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 duprism set, except for the snubbed form - prisms of antiprisms.

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

[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: A₃xA₁ - [3,3] x [ ]

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

[edit] Octahedral prisms: C₃xA₁ - [4,3] x [ ]

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

[edit] Icosahedral prisms: G₃xA₁ - [5,3] x [ ]

Name	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type				Element counts
							Cells	Faces	Edges	Vertices
Dodecahedral prism		t0{5,3}x{}	2 5.5.5	12 4.4.5			14	30 {4} 24 {5}	80	40
Icosahedral prism (same as snub tetrahedral prism)		t2{5,3}x{}	2 3.3.3.3.3	20 3.4.4			22	40 {3} 30 {4}	72	24
Icosidodecahedral prism		t1{5,3}x{}	2 3.5.3.5	20 3.4.4	12 4.4.5		34	40 {3} 60 {4} 24 {5}	150	60
Truncated dodecahedral prism		t0,1{5,3}x{}	2 3.10.10	20 3.4.4	12 4.4.5		34	40 {3} 90 {4} 24 {10}	240	120
Truncated icosahedral prism		t1,2{5,3}x{}	2 5.6.6	12 4.4.5	20 4.4.6		34	90 {4} 24 {5} 40 {6}	240	120
Rhombicosidodecahedral prism		t0,2{5,3}x{}	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
Truncated icosidodecahedral prism		t0,1,2{5,3}x{}	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
Snub dodecahedral prism		s{5,3}x{}	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: D₂^pxD₂^q - [p] x [q]

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

They have a Coxeter-Dynkin diagram as:

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

[edit] Polygonal prismatic prisms: D₂xA₁xA₁ - [p] x [ ] x [ ]

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)

1. Square antiprismatic prism - - 2 square antiprisms connected by 2 cubes and 8 triangular prisms
2. Pentagonal antiprismatic prism - - 2 pentagonal antiprisms connected by 2 pentagonal prisms and 10 triangular prisms
3. Hexagonal antiprismatic prism - - 2 hexagonal antiprisms connected by 2 hexagonal prisms and 12 triangular prisms
4. Heptagonal antiprismatic prism - - 2 heptagonal antiprisms connected by 2 heptagonal prisms and 14 triangular prisms
5. Octagonal antiprismatic prism - - 2 octagonal antiprisms connected by 2 octagonal prisms and 16 triangular prisms
6. ...

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

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. That is mirrors for which the generating point is located off the mirror.

Operation	Schläfli symbol	Coxeter- Dynkin diagram	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	t0,1{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	t1,2{p,q,r}		A truncation between a rectified form and the dual rectified form
Tritruncation	t2,3{p,q,r}		Truncated dual {r,q,p}
Cantellation	t0,2{p,q,r}		A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
Bicantellation	t1,3{p,q,r}		Cantellated dual {r,q,p}
Runcination (or expansion)	t0,3{p,q,r}		A truncation applied to the cells, faces, and edges and defines a progression between a regular form and the dual.
Cantitruncation	t0,1,2{p,q,r}		Both the cantellation and truncation operations applied together.
Bicantitruncation	t1,2,3{p,q,r}		Cantitruncated dual {r,q,p}
Runcitruncation	t0,1,3{p,q,r}		Both the runcination and truncation operations applied together.
Runcicantellation	t0,1,3{p,q,r}		Runcitruncated dual {r,q,p}
Omnitruncation (or more specifically runcicantitruncated)	t0,1,2,3{p,q,r}		Has all three operators applied.

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

• Polychoron
• Semiregular 4-polytopes
• Duoprism

[edit] References

• 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
• 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 [3]
  • (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
• M. Möller: Definitions and computations to the Platonic and Archimedean polyhedrons, thesis (diploma), University of Hamburg, 2001
• 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.

[edit] External links

• Eric W. Weisstein, Uniform polychoron at MathWorld.
• Convex uniform polychora
  • Polytope in R⁴, Marco Möller
    • 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope
  • Uniform Polytopes in Four Dimensions, George Olshevsky
    1. Convex uniform polychora based on the pentachoron (5-cell)
    2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell)
    3. Convex uniform polychora based on the icositetrachoron (24-cell)
    4. Convex uniform polychora based on the hecatonicosachoron (120-cell) and hexacosichoron (600-cell)
    5. Anomalous convex uniform polychoron: (grand antiprism)
    6. Convex uniform prismatic polychora
    7. Uniform polychora derived from glomeric tetrahedron B4
  • Regular and semi-regular convex polytopes a short historical overview
• 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.