Uniform polychoron

Schlegel diagram for the truncated 120-cell with tetrahedral cells visible. This perspective projection makes edges look smaller towards the center of the projection.

orthographic projection of the truncated 120-cell, in the H₃ Coxeter plane (D₁₀ symmetry). Only vertices and edges are drawn.

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

This article contains the complete list of 47 non-prismatic convex uniform polychora, and describes three sets of convex prismatic forms, two being infinite.

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 [2].
• Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
  • 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.
  • 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. This construction enumerated 45 semiregular polychora.
  • 1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell.
  • 1912: E. L. Elte independently expanded on Gosset's list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.
• Convex uniform polytopes:
  • 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.
    • 1966 N.W. Johnson completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher
    • 1997: A complete enumeration of the names and elements of the convex uniform polychora is given online by George Olshevsky.
    • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope.
• 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, although 1849 convex and nonconvex uniform polychora are currently known. Participating researchers include Jonathan Bowers, George Olshevsky and Norman Johnson.

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: 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell.
  • 10 Schläfli-Hess polychora.

Convex uniform polychora

Enumeration

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 BC₄ [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.

The A₄ family

The 5-cell has diploid pentachoric [3,3,3] 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, [[3,3,3]] 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.

Facets (cells) are given, grouped in their Coxeter-Dynkin locations by removing specified nodes.

#	Johnson Name Bowers name (and acronym)	Vertex figure	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
1	5-cell Pentachoron (pen)		{3,3,3}	(4) (3.3.3)				5	10	10	5
2	rectified 5-cell Rectified pentachoron (rap)		t1{3,3,3}	(3) (3.3.3.3)			(2) (3.3.3)	10	30	30	10
3	truncated 5-cell Truncated pentachoron (tip)		t0,1{3,3,3}	(3) (3.6.6)			(1) (3.3.3)	10	30	40	20
4	cantellated 5-cell Small rhombated pentachoron (srip)		t0,2{3,3,3}	(2) (3.4.3.4)		(2) (3.4.4)	(1) (3.3.3.3)	20	80	90	30
5	*runcinated 5-cell Small prismated decachoron (spid)		t0,3{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 Decachoron (deca)		t1,2{3,3,3}	(2) (3.6.6)			(2) (3.6.6)	10	40	60	30
7	cantitruncated 5-cell Great rhombated pentachoron (grip)		t0,1,2{3,3,3}	(2) (4.6.6)		(1) (3.4.4)	(1) (3.6.6)	20	80	120	60
8	runcitruncated 5-cell Prismatotrhombated pentachoron (prip)		t0,1,3{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 Great prismated decachoron (gippid)		t0,1,2,3{3,3,3}	(1) (4.6.6)	(1) (4.4.6)	(1) (4.4.6)	(1) (4.6.6)	30	150	240	120

Graphs

Three Coxeter plane 2D projections are given, for the A₄, A₃, A₂ Coxeter groups, showing symmetry order 5,4,3, and doubled on even A_k orders to 10,4,6 for symmetric Coxeter diagrams.

The 3D picture 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.

#	Johnson Name Bowers name (and acronym)	Coxeter-Dynkin and Schläfli symbols	Coxeter plane graphs			Schlegel diagram
			A4 [5]	A3 [4]	A2 [3]	Tetrahedron centered	Dual tetrahedron centered
1	5-cell Pentachoron (pen)	{3,3,3}
2	rectified 5-cell Rectified pentachoron (rap)	t1{3,3,3}
3	truncated 5-cell Truncated pentachoron (tip)	t0,1{3,3,3}
4	cantellated 5-cell Small rhombated pentachoron (srip)	t0,2{3,3,3}
5	*runcinated 5-cell Small prismatodecachoron (spid)	t0,3{3,3,3}
6	*bitruncated 5-cell Decachoron (deca)	t1,2{3,3,3}
7	cantitruncated 5-cell Great rhombated pentachoron (grip)	t0,1,2{3,3,3}
8	runcitruncated 5-cell Prismatotrhombated pentachoron (prip)	t0,1,3{3,3,3}
9	*omnitruncated 5-cell Great prismatodecachoron (gippid)	t0,1,2,3{3,3,3}

The coordinates of uniform 4-polytopes with pentachoric symmetry can be generated as permutations of simple integers in 5-space, all in hyperplanes with normal vector (1,1,1,1,1). The A₄ Coxeter group is palindromic, so repeated polytopes exist in pairs of dual configurations. There are 3 symmetric positions, and 6 pairs making the total 15 permutations of one or more rings. All 15 are listed here in order of binary arithmetic for clarity of the coordinate generation from the rings in each corresponding Coxeter-Dynkin diagram.

The number of vertices can be deduced here from the permutations of the number of coordinates, peaking at 5 factorial for the omnitruncated form with 5 unique coordinate values.

Pentachora truncations in 5-space:

#	Base point	Name (symmetric name)	Coxeter-Dynkin	Vertices
1	(0, 0, 0, 0, 1)	5-cell		5
2	(0, 0, 0, 1, 1)	Rectified 5-cell		10
3	(0, 0, 0, 1, 2)	Truncated 5-cell		20
4	(0, 0, 1, 1, 1)	Birectified 5-cell (rectified 5-cell)		10
5	(0, 0, 1, 1, 2)	Cantellated 5-cell		30
6	(0, 0, 1, 2, 2)	Bitruncated 5-cell		30
7	(0, 0, 1, 2, 3)	Cantitruncated 5-cell		60
8	(0, 1, 1, 1, 1)	Trirectified 5-cell (5-cell)		5
9	(0, 1, 1, 1, 2)	Runcinated 5-cell		20
10	(0, 1, 1, 2, 2)	Bicantellated 5-cell (cantellated 5-cell)		30
11	(0, 1, 1, 2, 3)	Runcitruncated 5-cell		60
12	(0, 1, 2, 2, 2)	Tritruncated 5-cell (truncated 5-cell)		20
13	(0, 1, 2, 2, 3)	Runcicantellated 5-cell (runcitruncated 5-cell)		60
14	(0, 1, 2, 3, 3)	Bicantitruncated 5-cell (cantitruncated 5-cell)		60
15	(0, 1, 2, 3, 4)	Omnitruncated 5-cell		120

The BC₄ family

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.

Tesseract truncations

#	Johnson Name (Bowers style acronym)	Vertex figure	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
10	8-cell or tesseract (tes)		{4,3,3}	(4) (4.4.4)				8	24	32	16
11	rectified 8-cell (rit)		t1{4,3,3}	(3) (3.4.3.4)			(2) (3.3.3)	24	88	96	32
13	truncated 8-cell (tat)		t0,1{4,3,3}	(3) (3.8.8)			(1) (3.3.3)	24	88	128	64
14	cantellated 8-cell (srit)		t0,2{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) (sidpith)		t0,3{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) (tah)		t1,2{4,3,3}	(2) (4.6.6)			(2) (3.6.6)	24	120	192	96
18	cantitruncated 8-cell (grit)		t0,1,2{4,3,3}	(2) (4.6.8)		(1) (3.4.4)	(1) (3.6.6)	56	248	384	192
19	runcitruncated 8-cell (proh)		t0,1,3{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) (gidpith)		t0,1,2,3{3,3,4}	(1) (4.6.8)	(1) (4.4.8)	(1) (4.4.6)	(1) (4.6.6)	80	464	768	384
12	Demitesseract 16-cell (hex)		h0{4,3,3}	(3.3.3)			(half) (3.3.3)	16	32	24	8

16-cell truncations

#	Johnson Name (Bowers style acronym)	Vertex figure	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
[12]	16-cell (hex)		{3,3,4}				(8) (3.3.3)	16	32	24	8
[22]	*rectified 16-cell (Same as 24-cell) (ico)		t1{3,3,4}	(2) (3.3.3.3)			(4) (3.3.3.3)	24	96	96	24
17	truncated 16-cell (thex)		t0,1{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) (rico)		t0,2{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) (sidpith)		t0,3{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) (tah)		t1,2{3,3,4}	(2) (4.6.6)			(2) (3.6.6)	24	120	192	96
[24]	*cantitruncated 16-cell (Same as truncated 24-cell) (tico)		t0,1,2{3,3,4}	(1) (4.6.6)	(1) (4.4.4)		(2) (4.6.6)	48	240	384	192
20	runcitruncated 16-cell (prit)		t0,1,3{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) (gidpith)		t0,1,2,3{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) (sadi)		h0,1,2{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, with the half symmetry group [(3,3)⁺,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.

Graphs

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.

#	Johnson Name (Bowers style acronym)	Coxeter plane projections					Schlegel diagrams
		F4 [12/3]	B4 [8]	B3 [6]	B2 [4]	A3 [4]	Cube centered	Tetrahedron centered
10	8-cell or tesseract (tes)
11	rectified 8-cell (rit)
12	16-cell (hex)
13	truncated 8-cell (tat)
14	cantellated 8-cell (srit)
15	runcinated 8-cell (also runcinated 16-cell) (sidpith)
16	bitruncated 8-cell (also bitruncated 16-cell) (tah)
17	truncated 16-cell (thex)
18	cantitruncated 8-cell (grit)
19	runcitruncated 8-cell (proh)
20	runcitruncated 16-cell (prit)
21	omnitruncated 8-cell (also omnitruncated 16-cell) (gidpith)
[22]	*rectified 16-cell (Same as 24-cell) (ico)
[23]	*cantellated 16-cell (Same as rectified 24-cell) (rico)
[24]	*cantitruncated 16-cell (Same as truncated 24-cell) (tico)
[31]	alternated cantitruncated 16-cell (Same as the snub 24-cell) (sadi)

Coordinates

The tesseractic family of polychora are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform polychoron. All coordinates correspond with uniform polychora of edge length 2.

Coordinates for uniform polychora in Tesseract/16-cell family

#	Base point	Johnson Name Bowers Name (Bowers style acronym)	Coxeter-Dynkin
1	(0,0,0,1)√2	16-cell Hexadecachoron (hex)
2	(0,0,1,1)√2	Rectified 16-cell Icositetrachoron (ico)
3	(0,0,1,2)√2	Truncated 16-cell Truncated hexadecachoron (thex)
4	(0,1,1,1)√2	Rectified tesseract (birectified 16-cell) Rectified tesseract (rit)
5	(0,1,1,2)√2	Cantellated 16-cell Rectified icositetrachoron (rico)
6	(0,1,2,2)√2	Bitruncated 16-cell Tesseractihexadecachoron (tah)
7	(0,1,2,3)√2	cantitruncated 16-cell Truncated icositetrachoron (tico)
8	(1,1,1,1)	Tesseract Tesseract (tes)
9	(1,1,1,1) + (0,0,0,1)√2	Runcinated tesseract (runcinated 16-cell) Small disprismatotesseractihexadecachoron (sidpith)
10	(1,1,1,1) + (0,0,1,1)√2	Cantellated tesseract Small rhombated tesseract (srit)
11	(1,1,1,1) + (0,0,1,2)√2	Runcitruncated 16-cell Prismatorhombated tesseract (prit)
12	(1,1,1,1) + (0,1,1,1)√2	Truncated tesseract Truncated tesseract (tat)
13	(1,1,1,1) + (0,1,1,2)√2	Runcitruncated tesseract (runcicantellated 16-cell) Prismatorhombated hexadecachoron (proh)
14	(1,1,1,1) + (0,1,2,2)√2	Cantitruncated tesseract Great rhombated tesseract (grit)
15	(1,1,1,1) + (0,1,2,3)√2	Omnitruncated 16-cell (omnitruncated tesseract) Great disprismatotesseractihexadecachoron (gidpith)

The F₄ family

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

#	Name	Vertex figure	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
22	24-cell (Same as rectified 16-cell) (ico)		{3,4,3}	(6) (3.3.3.3)				24	96	96	24
23	rectified 24-cell (Same as cantellated 16-cell) (rico)		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) (tico)		t0,1{3,4,3}	(3) (4.6.6)			(1) (4.4.4)	48	240	384	192
25	cantellated 24-cell (srico)		t0,2{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 (spic)		t0,3{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 (cont)		t1,2{3,4,3}	(2) (3.8.8)			(2) (3.8.8)	48	336	576	288
28	cantitruncated 24-cell (grico)		t0,1,2{3,4,3}	(2) (4.6.8)		(1) (3.4.4)	(1) (3.8.8)	144	720	1152	576
29	runcitruncated 24-cell (prico)		t0,1,3{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 (gippic)		t0,1,2,3{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) (sadi)		h0,1{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 [[3,4,3]]).

(†) 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, [3⁺,4,3]).

Graphs

#	Name Coxeter-Dynkin Schläfli symbol	Graph				Schlegel diagram		Orthogonal Projection
		F4 [12]	B4 [8]	B3 [6]	B2 [4]	Octahedron centered	Dual octahedron centered	Octahedron centered
22	24-cell (ico) (rectified 16-cell) {3,4,3}
23	rectified 24-cell (rico) (cantellated 16-cell) t1{3,4,3}
24	truncated 24-cell (tico) (cantitruncated 16-cell) t0,1{3,4,3}
25	cantellated 24-cell (srico) t0,2{3,4,3}
26	*runcinated 24-cell (spic) t0,3{3,4,3}
27	*bitruncated 24-cell (cont) t1,2{3,4,3}
28	cantitruncated 24-cell (grico) t0,1,2{3,4,3}
29	runcitruncated 24-cell (prico) t0,1,3{3,4,3}
30	*omnitruncated 24-cell (gippic) t0,1,2,3{3,4,3}
31	Alternated truncated 24-cell †(Same as snub 24-cell) (sadi) h0,1{3,4,3}

Coordinates

Vertex coordinates for all 15 forms are given below, including dual configurations from the two regular 24-cells. (The dual configurations are named in bold.) Active rings in the first and second nodes generate points in the first column. Active rings in the third and fourth nodes generate the points in the second column. The sum of each of these points are then permutated by coordinate positions, and sign combinations. This generates all vertex coordinates. Edge lengths are 2.

The only exception is the snub 24-cell, which is generated by half of the coordinate permutations, only an even number of coordinate swaps. φ=(√5+1)/2.

24-cell family coordinates

Base point(s) t(0,1)	Base point(s) t(2,3)	Schläfli symbol	Name	Coxeter-Dynkin
	(0,0,1,1)√2	t0{3,4,3}	24-cell
	(0,1,1,2)√2	t1{3,4,3}	Rectified 24-cell
	(0,1,2,3)√2	t0,1{3,4,3}	Truncated 24-cell
	(0,1,φ,φ+1)√2	h0,1{3,4,3}	Snub 24-cell
(0,2,2,2) (1,1,1,3)		t2{3,4,3}	Birectified 24-cell (Rectified 24-cell)
(0,2,2,2) + (1,1,1,3) +	(0,0,1,1)√2 "	t0,2{3,4,3}	Cantellated 24-cell
(0,2,2,2) + (1,1,1,3) +	(0,1,1,2)√2 "	t1,2{3,4,3}	Bitruncated 24-cell
(0,2,2,2) + (1,1,1,3) +	(0,1,2,3)√2 "	t0,1,2{3,4,3}	Cantitruncated 24-cell
(0,0,0,2) (1,1,1,1)		t3{3,4,3}	Trirectified 24-cell (24-cell)
(0,0,0,2) + (1,1,1,1) +	(0,0,1,1)√2 "	t0,3{3,4,3}	Runcinated 24-cell
(0,0,0,2) + (1,1,1,1) +	(0,1,1,2)√2 "	t1,3{3,4,3}	bicantellated 24-cell (Cantellated 24-cell)
(0,0,0,2) + (1,1,1,1) +	(0,1,2,3)√2 "	t0,1,3{3,4,3}	Runcitruncated 24-cell
(1,1,1,5) (1,3,3,3) (2,2,2,4)		t2,3{3,4,3}	Tritruncated 24-cell (Truncated 24-cell)
(1,1,1,5) + (1,3,3,3) + (2,2,2,4) +	(0,0,1,1)√2 " "	t0,2,3{3,4,3}	Runcicantellated 24-cell (Runcitruncated 24-cell)
(1,1,1,5) + (1,3,3,3) + (2,2,2,4) +	(0,1,1,2)√2 " "	t1,2,3{3,4,3}	Bicantitruncated 24-cell (Cantitruncated 24-cell)
(1,1,1,5) + (1,3,3,3) + (2,2,2,4) +	(0,1,2,3)√2 " "	t0,1,2,3{3,4,3}	Omnitruncated 24-cell

The H₄ family

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.

120-cell truncations

#	Johnson Name (Bowers style Acronym)	Vertex figure	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
32	120-cell (hi)		{5,3,3}	(4) (5.5.5)				120	720	1200	600
33	rectified 120-cell (rahi)		t1{5,3,3}	(3) (3.5.3.5)			(2) (3.3.3)	720	3120	3600	1200
36	truncated 120-cell (thi)		t0,1{5,3,3}	(3) (3.10.10)			(1) (3.3.3)	720	3120	4800	2400
37	cantellated 120-cell (srahi)		t0,2{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) (sidpixhi)		t0,3{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) (xhi)		t1,2{5,3,3}	(2) (5.6.6)			(2) (3.6.6)	720	4320	7200	3600
42	cantitruncated 120-cell (grahi)		t0,1,2{5,3,3}	(2) (4.6.10)		(1) (3.4.4)	(1) (3.6.6)	1920	9120	14400	7200
43	runcitruncated 120-cell (prix)		t0,1,3{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) (gidpixhi)		t0,1,2,3{5,3,3}	(1) (4.6.10)	(1) (4.4.10)	(1) (4.4.6)	(1) (4.6.6)	2640	17040	28800	14400

600-cell truncations

#	Johnson Name (Bowers style acronym)	Vertex figure	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
35	600-cell (ex)		{3,3,5}				(20) (3.3.3)	600	1200	720	120
34	rectified 600-cell (rox)		t1{3,3,5}	(2) (3.3.3.3.3)			(5) (3.3.3.3)	720	3600	3600	720
41	truncated 600-cell (tex)		t0,1{3,3,5}	(1) (3.3.3.3.3)			(5) (3.6.6)	720	3600	4320	1440
40	cantellated 600-cell (srix)		t0,2{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) (sidpixhi)		t0,3{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) (xhi)		t1,2{3,3,5}	(2) (5.6.6)			(2) (3.6.6)	720	4320	7200	3600
45	cantitruncated 600-cell (grix)		t0,1,2{3,3,5}	(1) (5.6.6)	(1) (4.4.5)		(2) (4.6.6)	1440	8640	14400	7200
44	runcitruncated 600-cell (prahi)		t0,1,3{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) (gidpixhi)		t0,1,2,3{3,3,5}	(1) (4.6.10)	(1) (4.4.10)	(1) (4.4.6)	(1) (4.6.6)	2640	17040	28800	14400

Graphs

#	Johnson Name (Bowers style Acronym)	Coxeter plane projections						Schlegel diagrams
		F4 [12]	[20]	H4 [30]	H3 [10]	A3 [4]	A2 [3]	Dodecahedron centered	Tetrahedron centered
32	120-cell (hi)
33	rectified 120-cell (rahi)
34	rectified 600-cell (rox)
35	600-cell (ex)
36	truncated 120-cell (thi)
37	cantellated 120-cell (srahi)
38	runcinated 120-cell (also runcinated 600-cell) (sidpixhi)
39	bitruncated 120-cell (also bitruncated 600-cell) (xhi)
40	cantellated 600-cell (srix)
41	truncated 600-cell (tex)
42	cantitruncated 120-cell (grahi)
43	runcitruncated 120-cell (prix)
44	runcitruncated 600-cell (prahi)
45	cantitruncated 600-cell (grix)
46	omnitruncated 120-cell (also omnitruncated 600-cell) (gidpixhi)

The D₄ family

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

This family has order 12*16=192: 4!/2=12 permutations of the four axes, half as alternated, 2⁴=16 for reflection in each axis.

#	Johnson Name (Bowers style acronym)	Vertex figure	Coxeter-Dynkin	Cell counts by location					Element counts
				Pos. 0 (8)	Pos. 1 (24)	Pos. 2 (8)	Pos. 3 (8)	Pos. Alt (96)	3	2	1	0
[12]	demitesseract (Same as 16-cell) (hex)		t0{31,1,1}			(4) (3.3.3)	(4) (3.3.3)		16	32	24	8
[17]	truncated demitesseract (Same as truncated 16-cell) (thex)		t0,1{31,1,1}	(1) (3.3.3.3)		(2) (3.6.6)	(2) (3.6.6)		24	96	120	48
[11]	cantellated demitesseract (Same as rectified tesseract) (rit)		t0,2{31,1,1}	(1) (3.3.3)		(1) (3.3.3)	(3) (3.4.3.4)		24	88	96	32
[16]	cantitruncated demitesseract (Same as bitruncated tesseract) (tah)		t0,1,2{31,1,1}	(1) (3.6.6)		(1) (3.6.6)	(2) (4.6.6)		24	96	96	24
[22]	rectified demitesseract (Same as rectified 16-cell) (Same as 24-cell) (ico)		t1{31,1,1}	(2) (3.3.3.3)		(2) (3.3.3.3)	(2) (3.3.3.3)		48	240	288	96
[23]	runcicantellated demitesseract (Same as cantellated 16-cell) (Same as rectified 24-cell) (rico)		t0,2,3{31,1,1}	(1) (3.4.3.4)	(2) (4.4.4)	(1) (3.4.3.4)	(1) (3.4.3.4)		24	120	192	96
[24]	omnitruncated demitesseract (Same as cantitruncated 16-cell) (Same as truncated 24-cell) (tico)		t0,1,2,3{31,1,1}	(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) (sadi)		s{31,1,1}	(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, with the symmetry group [3^1,1,1]⁺ this time, 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.

Graphs

#	Johnson Name (Bowers style acronym) Coxeter-Dynkin	Coxeter plane projections			Schlegel diagrams		Parallel 3D
		B4 [8/2]	D4 [6]	D3 [2]	Cube centered	Tetrahedron centered	D4 [6]
[12]	demitesseract (Same as 16-cell) (hex) t0{31,1,1}
[17]	truncated demitesseract (Same as truncated 16-cell) (thex) t0,1{31,1,1}
[11]	cantellated demitesseract (Same as rectified tesseract) (rit) t0,2{31,1,1}
[16]	cantitruncated demitesseract (Same as bitruncated tesseract) (tah) t0,1,2{31,1,1}
[22]	rectified demitesseract (Same as rectified 16-cell) (Same as 24-cell) (ico) t1{31,1,1}
[23]	runcicantellated demitesseract (Same as cantellated 16-cell) (Same as rectified 24-cell) (rico) t0,2,3{31,1,1}
[24]	omnitruncated demitesseract (Same as cantitruncated 16-cell) (Same as truncated 24-cell) (tico) t0,1,2,3{31,1,1}
[31]	Snub demitesseract (snub 24-cell) (sadi) s{31,1,1}

Coordinates

The base point can generate the coordinates of the polytope by taking all coordinate permutations and sign combinations. The edges' length will be √2. Some polytopes have two possible generator points. Points are prefixed by Even to imply only an even count of sign permutations should be included.

#	Base point	Johnson and Bowers Names	Coxeter diagram	Related B4 diagram	Related F4 diagram
[12]	(0,0,0,2)	16-cell
[22]	(0,0,2,2)	Rectified 16-cell
[17]	(0,0,2,4)	Truncated 16-cell
[11]	(0,2,2,2)	Cantellated 16-cell
[23]	(0,2,2,4)	Cantellated 16-cell
[16]	(0,2,4,4)	Bitruncated 16-cell
[24]	(0,2,4,6)	Cantitruncated 16-cell
[31]	(0,1,φ,φ+1)/√2	snub 24-cell
[12]	Even (1,1,1,1)	demitesseract (16-cell)
[11]	Even (1,1,1,3)	Cantellated demitesseract (cantellated 16-cell)
[17]	Even (1,1,3,3)	Truncated demitesseract (truncated 16-cell)
[16]	Even (1,3,3,3)	Cantitruncated demitesseract (cantitruncated 16-cell)

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).

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

Prismatic uniform polychora

A prismatic polytope is a Cartesian product of two polytopes of lower dimension; familiar examples are the 3-dimensional prisms, which are products of a polygon and a line segment. The prismatic uniform polychora consist of two infinite families:

• Polyhedral prisms: products of a line segment and a uniform polyhedron. This family is infinite because it includes prisms built on 3-dimensional prisms and antiprisms.
• Duoprisms: products of two polygons.

Convex polyhedral prisms

The most 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).^{[citation needed]}

There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms.^{[citation needed]} The symmetry number of a polyhedral prism is twice that of the base polyhedron.

Tetrahedral prisms: A₃ × A₁

#	Johnson Name (Bowers style acronym)	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type			Element counts
							Cells	Faces	Edges	Vertices
48	Tetrahedral prism (tepe)		t0{3,3}×{} t0,3{3,3,2}	2 3.3.3	4 3.4.4		6	8 {3} 6 {4}	16	8
49	Truncated tetrahedral prism (tuttip)		t0,1{3,3}×{} t0,1,3{3,3,2}	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) (ope)		t1{3,3}×{} t1,3{3,3,2}	2 3.3.3.3	4 3.4.4		6	16 {3} 12 {4}	30	12
[50]	Cantellated tetrahedral prism (Same as cuboctahedral prism) (cope)		t0,2{3,3}×{} t0,2,3{3,3,2}	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) (tope)		t0,1,2{3,3}×{} t0,1,2,3{3,3,2}	2 4.6.6	8 6.4.4	6 4.4.4	16	48 {4} 16 {6}	96	48
[59]	Snub tetrahedral prism (Same as icosahedral prism) (ipe)		s{3,3}×{}	2 3.3.3.3.3	20 3.4.4		22	40 {3} 30 {4}	72	24

Octahedral prisms: BC₃ × A₁

#	Johnson Name (Bowers style acronym)	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type				Element counts
								Cells	Faces	Edges	Vertices
[10]	Cubic prism (Same as tesseract) (Same as 4-4 duoprism) (tes)		t0{4,3}×{} t0,3{4,3,2}	2 4.4.4	6 4.4.4			8	24 {4}	32	16
50	Cuboctahedral prism (Same as cantellated tetrahedral prism) (cope)		t1{4,3}×{} t1,3{4,3,2}	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) (ope)		t2{4,3}×{} t2,3{4,3,2}	2 3.3.3.3	8 3.4.4			10	16 {3} 12 {4}	30	12
52	Rhombicuboctahedral prism (sircope)		t0,2{4,3}×{} t0,2,3{4,3,2}	2 3.4.4.4	8 3.4.4	18 4.4.4		28	16 {3} 84 {4}	120	96
53	Truncated cubic prism (ticcup)		t0,1{4,3}×{} t0,1,3{4,3,2}	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) (tope)		t1,2{4,3}×{} t1,2,3{4,3,2}	2 4.6.6	6 4.4.4	8 4.4.6		16	48 {4} 16 {6}	96	48
55	Truncated cuboctahedral prism (gircope)		t0,1,2{4,3}×{} t0,1,2,3{4,3,2}	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 (sniccup)		s{4,3}×{}	2 3.3.3.3.4	32 3.4.4	6 4.4.4		40	64 {3} 72 {4}	144	48

Icosahedral prisms: H₃ × A₁

#	Johnson Name (Bowers style acronym)	Picture	Coxeter-Dynkin and Schläfli symbols	Cells by type				Element counts
								Cells	Faces	Edges	Vertices
57	Dodecahedral prism (dope)		t0{5,3}×{} t0,3{5,3,2}	2 5.5.5	12 4.4.5			14	30 {4} 24 {5}	80	40
58	Icosidodecahedral prism (iddip)		t1{5,3}×{} t1,3{5,3,2}	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) (ipe)		t2{5,3}×{} t2,3{5,3,2}	2 3.3.3.3.3	20 3.4.4			22	40 {3} 30 {4}	72	24
60	Truncated dodecahedral prism (tiddip)		t0,1{5,3}×{} t0,1,3{5,3,2}	2 3.10.10	20 3.4.4	12 4.4.5		34	40 {3} 90 {4} 24 {10}	240	120
61	Rhombicosidodecahedral prism (sriddip)		t0,2{5,3}×{} t0,2,3{5,3,2}	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 (tipe)		t1,2{5,3}×{} t1,2,3{5,3,2}	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 (griddip)		t0,1,2{5,3}×{} t0,1,2,3{5,3,2}	2 4.6.10	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 (sniddip)		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

Duoprisms: [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:

Name	Coxeter graph	Cells
3-3 duoprism (triddip)		6 triangular prisms
3-4 duoprism (tisdip)		3 cubes, 4 triangular prisms
4-4 duoprism (tes)		8 cubes (same as tesseract)
3-5 duoprism (trapedip)		3 pentagonal prisms, 5 triangular prisms
4-5 duoprism (squipdip)		4 pentagonal prisms, 5 cubes
5-5 duoprism (pedip)		10 pentagonal prisms
3-6 duoprism (thiddip)		3 hexagonal prisms, 6 triangular prisms
4-6 duoprism (shiddip)		4 hexagonal prisms, 6 cubes
5-6 duoprism (phiddip)		5 hexagonal prisms, 6 pentagonal prisms
6-6 duoprism (hiddip)		12 hexagonal prisms

Polygonal prismatic prisms: [p] × [ ] × [ ]

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

Name	Coxeter graph	Cells
Triangular prismatic prism (tisdip)		3 cubes and 4 triangular prisms (same as 3-4 duoprism)
Square prismatic prism (tes)		4 cubes and 4 cubes (same as 4-4 duoprism and same as a tesseract)
Pentagonal prismatic prism (squipdip)		5 cubes and 4 pentagonal prisms (same as 4-5 duoprism)
Hexagonal prismatic prism (shiddip)		6 cubes and 4 hexagonal prisms (same as 4-6 duoprism)
Heptagonal prismatic prism (shedip)		7 cubes and 4 heptagonal prisms (same as 4-7 duoprism)
Octagonal prismatic prism (sodip)		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.

Name	Coxeter graph	Cells	Image
Triangular antiprismatic prism (ope)		2 octahedra connected by 8 triangular prisms (same as the octahedral prism)
Square antiprismatic prism (squapip)		2 square antiprisms connected by 2 cubes and 8 triangular prisms
Pentagonal antiprismatic prism (pappip)		2 pentagonal antiprisms connected by 2 pentagonal prisms and 10 triangular prisms
Hexagonal antiprismatic prism (happip)		2 hexagonal antiprisms connected by 2 hexagonal prisms and 12 triangular prisms
Heptagonal antiprismatic prism (heappip)		2 heptagonal antiprisms connected by 2 heptagonal prisms and 14 triangular prisms
Octagonal antiprismatic prism (oappip)		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.

Nonuniform alternations

There are a number of alternations of the uniform polychora that can not be made uniform as they have too many parameters to satisfy.

Four snubs are not uniform unlike their 3-dimensional analogies. Only the snub 24-cell is uniform, although it is more accurately called a semisnub 24-cell or snub demitesseract for being a full snub of the bifurcating family D₄ with the demitesseract as the alternated tesseract.

• Snub 5-cell (snip), s{3,3,3}, , 10 icosahedrons, 20 octahedrons, and 60 tetrahedrons
• Snub tesseract (snit), s{4,3,3}, , with 16 icosahedrons, 32 octahedra, 24 square antiprisms, 8 snub cubes and 192 tetrahedrons
• Full snub 24-cell (snico), s{3,4,3}, , from 48 snub cubes, 192 octahedrons, and 576 tetrahedrons
• Snub 120-cell (snahi), s{5,3,3}, , 1200 octahedrons, 600 icosahedrons, 720 pentagonal antiprisms, 120 snub dodecahedrons, and 7200 tetrahedrons

The polyhedral prisms , can be alternated into , but do not generate uniform solutions.

1. Snub tetrahedral antiprism, s{3,3,2} , 2 icosahedrons connected by 6 tetrahedrons, and 8 octahedrons, with 24 tetrahedra in the alternated gaps.
2. Snub cubic antiprism, s{4,3,2} , 2 snub cubes connected by 12 tetrahedrons, 6 square antiprisms, and 8 octahedrons, with 48 tetrahedra in the alternated gaps.
3. Snub dodecahedral antiprism, s{5,3,2} , 2 snub dodecahedrons connected by 30 tetrahedrons, 12 pentagonal antiprisms, and 20 octahedrons, with 120 tetrahedra in the alternated gaps.

The duoprisms , t_0,1,2,3{p,2,q}, can be alternated into , s{p,2,q}, called duoantiprisms, which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the tesseract , t_0,1,2,3{2,2,2}, with its alternation as the 16-cell, , s{2,2,2}.

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 (π/n radians or 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	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 is 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; 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 runcicantitruncation)	t0,1,2,3{p,q,r}		Application of all three operators.
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.

Polychoric symmetry groups

Further information: Coxeter group#Finite Coxeter groups and Goursat_tetrahedron#3-sphere (finite) solutions

BC₄, D₄, and F₄ have a correspondence of mirror nodes in the Coxeter diagrams, allowing for multiple Wythoff constructions of the same uniform polychora. A₄ and F₄ also have a symmetry doubling when the nodes have front to back symmetry.
Other extended symmetries divide the fundamental domain by mirrors and polytopes with symmetric rings can be related between families. For example there are three constructions of the rectified 24-cell: , ,

A hierarchy of 4D polychoric point groups and some subgroups. Vertical positioning is grouped by order. Blue, green, and pink colors show reflectional, hybrid, and rotational groups.

Some 4D point groups in Conway's notation

There are 5 fundamental mirror symmetry point group families in 4-dimensions, each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes. The dihedral angles between the mirrors determine order of dihedral symmetry. The Coxeter-Dynkin diagram is a graph where nodes represent mirror planes, and edges are called branches, and labeled by their dihedral angle order between the mirrors. The 6 families are: A₄: , BC₄: , D₄: , F₄: , and H₄: .

Like the 3D polyhedral groups, the names of these groups given are constructed by the greek prefixes of the cell counts of the corresponding triangle-faced regular polytopes. Extended symmetries exist in uniform polychora with symmetric ring-patterns within the Coxeter diagram construct. Chiral symmetries exist in alternated uniform polychora. The groups are named in this article in Coxeter's Bracket notation (1985), but also given here are quaternion based notations by Patrick du Val (1964) and John Conway (2003). Conway's notation allows the order of the group to be computed as a product of elements with chiral polyhedral group orders: (T=12, O=24, I=60). In Conway's notation, a (±) prefix implies central inversion, and a suffix (.2) implies mirror symmetry. Simiarly Du Val's notation has an asterisk (*) superscript for mirror symmetry.

• Pentachoric group‎ - A₄, [3,3,3], , order 120, (Du Val (I^†/C₁;I/C₁)^†*, Conway +¹/₆₀[I×I].2₁), named for the 5-cell (pentachoron), given by ringed Coxeter diagram . It is also sometimes called the hypertetrahedral group for extending the tetrahedral group [3,3]. It is isomorphic to the abstract symmetric group, S₅.
  • The extended pentachoric group is [[3,3,3]] order 240, (Du Val (I^†*/C₂;I/C₂)^†*, Conway ±¹/₆₀[I×I].2).
    • The chiral extended pentachoric group is [[3,3,3]]⁺ order 120, (Du Val (I^†/C₂;I/C₂)^†, Conway ±¹/₆₀[IxI]). This group represents the construction of the snub 5-cell, , although it can not be made uniform.
  • The chiral pentachoric group is [3,3,3]⁺, order 60, (Du Val (I^†/C₁;I/C₁)^†, Conway +¹/₆₀[I×I]). It is isomorphic to the abstract alternating group, A₅.
    • The extended chiral pentachoric group is [[3,3,3]⁺] order 120, (Du Val (I^†/C₁;I/C₁)_-^†*, Conway +¹/₆₀[IxI].2₃).
• Hexadecachoric group - BC₄, [4,3,3], , order 384, (Du Val (O/V;O/V)^*, Conway ±¹/₆[O×O].2), named for the 16-cell (hexadecachoron), . It is also called a hyperoctahedral group for extending the 3D octahedral group [4,3], and the tesseractic group for the tesseract, .
  • The chiral hexadecachoric group is [4,3,3]⁺, order 192, (Du Val (O/V;O/V), Conway ±¹/₆[O×O]). This group represents the construction of a snub tesseract, , although it can not be made uniform.
  • The ionic diminished hexadecachoric group is [4,(3,3)⁺], order 192, (Du Val (T/V;T/V)^*, Conway ±¹/₃[TxT].2). This group leads to the snub 24-cell with construction .
  • The half hexadecachoric group is [1⁺,4,3,3], order 192, and same as the demitesseractic group [3^1,1,1] (). This group is expressed in the tesseract alternated construction of the 16-cell, .
    • The group [1⁺,4,(3,3)⁺], order 96, and same as the chiral demitesseractic group [3^1,1,1]⁺ and also is the commutator subgroup of [4,3,3].
  • A higher-index reflective subgroup is [4,3,2] (), index 4, order 96, (Du Val (O/C₂;O/C₂)^*, Conway ±¹/₂₄[OxO].2).
    • Its chiral subgroup is [4,3,2]⁺, order 48, (Du Val (O/C₂;O/C₂), Conway ±¹/₂₄[OxO]).
    • Hybrid subgroups include:
      • [(3,4)⁺,2], order 48, (Du Val (O/C₁;O/C₁)_-^*, Conway +¹/₂₄[OxO].2₁).
        • [(3,4)⁺,2⁺], order 24, (Conway ±¹/₁₂[TxT].2₃).
      • [4,3⁺,2], order 48, (Du Val (T/C₂;T/C₂)_c^*, Conway ±¹/₁₂[TxT].2).
      • [3,4,2⁺], order 48, (Conway ±¹/₁₂[TxT].2).
      • [4,(3,2)⁺], order 48, (Conway +¹/₂₄[OxO].2₁).
    • A half subgroup [4,3,2,1⁺], order 48 (Conway +¹/₂₄[OxO].2₃).
      • A chiral half subgroup [(4,3)⁺,2,1⁺], order 24 (Conway +¹/₂₄[OxO]).
  • Another high-index reflective subgroup is [3,3,2], index 8, order 48, (Du Val (O/C₁;O/C₁)^*, Conway +¹/₂₄[OxO].2₃).
    • Its chiral subgroup is [3,3,2]⁺, order 24, (Du Val (T/C₂;T/C₂), Conway +¹/₂₄[OxO]). An example is the snub tetrahedral antiprism, , although it can not be made uniform.
    • A hybrid subgroup is [(3,3)⁺,2], order 24, (Du Val (T/C₁;T/C₁)_c^*, Conway ±¹/₁₂[TxT].2₃). An example is the snub tetrahedral prism, .
    • A half subgroup is [3,3,2,1⁺], order 24, (Conway ±¹/₁₂[TxT].2₁)
      • A chiral half subgroup [(3,3)⁺,2,1⁺], order 12, (Conway +¹/₁₂[TxT])
  • Other higher-index reflective subgroups of [4,3,3] are: [4,2,4] (), [4,2,2] (), and [2,2,2] (), with subgroup indices 6, 12, and 24, and order 64, 32, and 16.
• Icositetrachoric group‎ - F₄, [3,4,3], , order 1152, (Du Val (O/T;O/T)^*, Conway [O×O].2₃), named for the 24-cell (icositetrachoron), .
  • The extended icositetrachoric group is [[3,4,3]] has order 2304, (Du Val (O/O;O/O)^*, Conway ±[O×O].2).
    • The chiral extended pentachoric group, [[3,4,3]]⁺ has order 1152, (Du Val (O/O;O/O), Conway ±[OxO]). This group represents the construction of the full snub 24-cell, , although it can not be made uniform.
  • The ionic diminished hexadecachoric groups, [3⁺,4,3] and [3,4,3⁺], have order 576, (Du Val (T/T;T/T)^*, Conway ±[T×T].2). This group leads to the snub 24-cell with construction .
    • The double diminished icositetrachoric group, [3⁺,4,3⁺], order 288, (Du Val (T/T;T/T), Conway ±[T×T]) is the commutator subgroup of [3,4,3].
      • It can be extended as [[3⁺,4,3⁺]], order 576, (Du Val (T/T;O/O), Conway ±[OxT]).
  • The chiral icositetrachoric group is [3,4,3]⁺, order 576, (Du Val (O/T;O/T), Conway ±¹/₂[O×O]).
    • The extended chiral pentachoric group, [[3,4,3]⁺] has order 1152, (Du Val (O/T;O/T)_-^*, Conway ±¹/₂[OxO].2).
• Demitesseractic group - D₄, [3^1,1,1], [3,3^1,1] or [1⁺,4,3,3], , order 192, (Du Val (T/V;T/V)_-^*, Conway ±¹/₃[T×T].2), named for the (demitesseract) 4-demicube construction of the 16-cell, or .
  • There are two types of extended symmetries by adding mirrors: <[3,3^1,1]> which becomes [4,3,3] by bisecting the fundamental domain by a mirror, with 3 orientations possible; and the full extended group [3[3^1,1,1]] becomes [3,4,3].
  • The chiral demitesseractic group is [3^1,1,1]⁺ or [1⁺,4,(3,3)⁺], order 96, (Du Val (T/V;T/V), Conway ±¹/₃[T×T]). This group leads to the snub 24-cell with construction .
• Hexacosichoric group‎ - H₄, [5,3,3], , order 14400, (Du Val (I/I;I/I)^*, Conway ±[I×I].2), named for the 600-cell (hexacosichoron), . It is also sometimes called the hypericosahedral group for extending the 3D icosahedral group [5,3], and hecatonicosachoric group from the 120-cell, .
  • The chiral hexacosichoric group is [5,3,3]⁺, order 7200, (Du Val (I/I;I/I), Conway ±[I×I]). This group represents the construction of the snub 120-cell, , although it can not be made uniform.
  • A higher index reflective subgroup is [5,3,2], index 60, order 240, (Du Val (I/C₂;I/C₂)^*, Conway ±¹/₆₀[IxI].2).
    • Its chiral subgroup is [5,3,2]⁺, order 120, (Du Val (I/C₂;I/C₂), Conway ±¹/₆₀[IxI]).
    • A hybrid subgroup is [(5,3)⁺,2], order 120, (Du Val (I/C₁;I/C₁)^*, Conway +¹/₆₀[IxI].2₁).
    • A half subgroup is [5,3,2,1⁺], order 120, (Conway +¹/₆₀[IxI].2₃).
      • A chiral half subgroup is [(5,3)⁺,2,1⁺], order 60, (Conway +¹/₆₀[IxI]).

See also

• Regular skew polyhedron#Finite regular skew polyhedra of 4-space
• Convex uniform honeycomb - related infinite 4-polytopes in Euclidean 3-space.
• Convex uniform honeycombs in hyperbolic space - related infinite 4-polytopes in Hyperbolic 3-space.

References

• 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
• Schoute, Pieter Hendrik (1911), "Analytic treatment of the polytopes regularly derived from the regular polytopes", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam 11 (3): 87 pp.  Googlebook, 370-381
  • 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
  • (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]
• H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980 p92, p122.
• 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)

External links

• Weisstein, Eric W., "Uniform polychoron", MathWorld.
• Convex uniform polychora
  • uniform, convex polytopes in four dimensions:, Marco Möller (German)
  • Uniform Polytopes in Four Dimensions, George Olshevsky.
    1. Convex uniform polychora based on the pentachoron, George Olshevsky.
    2. Convex uniform polychora based on the tesserract/16-cell, George Olshevsky.
    3. Convex uniform polychora based on the 24-cell, George Olshevsky.
    4. Convex uniform polychora based on the 120-cell/600-cell, George Olshevsky.
    5. Anomalous convex uniform polychoron: (grand antiprism), George Olshevsky.
    6. Convex uniform prismatic polychora, George Olshevsky.
    7. Uniform polychora derived from glomeric tetrahedron B4, George Olshevsky.
  • Regular and semi-regular convex polytopes a short historical overview
  • Java3D Applets with sources
• 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.
• Richard Klitzing, 4D, uniform polytopes