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Uniform 4-polytope

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Class of 4-dimensional polytopes

Schlegel diagram for thetruncated 120-cell withtetrahedral cells visible

Orthographic projection of the truncated 120-cell, in theH₃Coxeter plane (D₁₀ symmetry). Only vertices and edges are drawn.

Ingeometry, auniform 4-polytope (oruniform polychoron)^[1] is a 4-dimensionalpolytope which isvertex-transitive and whose cells areuniform polyhedra, and faces areregular polygons.

There are 47 non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms.

History of discovery

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ConvexRegular polytopes:
- 1852:Ludwig Schläfli proved in his manuscriptTheorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4dimensions and only 3 in 5 or more dimensions.
Regular star 4-polytopes (star polyhedron cells and/orvertex figures)
- 1852:Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, 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 4-polytopes, 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 PolyederEinleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder, von dr. Edmund Hess. Mit sechzehn lithographierten tafeln..
Convexsemiregular polytopes: (Various definitions before Coxeter'suniform category)
- 1900:Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publicationOn the Regular and Semi-Regular Figures in Space of n Dimensions. In four dimensions, this gives therectified 5-cell, therectified 600-cell, and thesnub 24-cell.^[2]
- 1910:Alicia Boole Stott, in her publicationGeometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowingArchimedean solid andprism cells. This construction enumerated 45 semiregular 4-polytopes, corresponding to the nonprismatic forms listed below.^[3] The snub 24-cell andgrand antiprism were missing from her list.^[4]
- 1911:Pieter Hendrik Schoute publishedAnalytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on5-cell,8-cell/16-cell, and24-cell.^[5]^[6]
- 1912:E. L. Elte independently expanded on Gosset's list with the publicationThe Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.^[7]
Convex uniform polytopes:
- 1940: The search was expanded systematically byH.S.M. Coxeter in his publicationRegular and Semi-Regular Polytopes.
- Convex uniform 4-polytopes:
  - 1965: The complete list of convex forms was finally enumerated byJohn Horton Conway andMichael Guy, in their publicationFour-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism.
  - 1966Norman Johnson completes his Ph.D. dissertationThe Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher.
  - 1986 Coxeter published a paperRegular and Semi-Regular Polytopes II which included analysis of the uniquesnub 24-cell structure, and the symmetry of the anomalous grand antiprism.
  - 1998^[8]-2000: The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration (used as a basis for this listing). Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from theGreek rootspoly ("many") andchoros ("room" or "space").^[9] The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t_0,1, cantellation, t_0,2, runcination t_0,3, with single ringed forms called rectified, and bi, tri-prefixes added when the first ring was on the second or third nodes.^[10]^[11]
  - 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation,Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's naming system in his listing.^[12]
  - 2008:The Symmetries of Things^[13] was published byJohn H. Conway and contains the first print-published listing of the convex uniform 4-polytopes and higher dimensional polytopes by Coxeter group family, with generalvertex figure diagrams for each ringedCoxeter diagram permutation—snub, grand antiprism, and duoprisms—which he called proprisms for product prisms. He used his ownijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, and all of Johnson's names were included in the book index.
Nonregular uniform star 4-polytopes: (similar to thenonconvex uniform polyhedra)
- 1966: Johnson describes three nonconvex uniform antiprisms in 4-space in his dissertation.^[14]
- 1990-2006: In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky,^[15] with an additional four discovered in 2006 for a total of 1849. The count includes the 74 prisms of the 75non-prismatic uniform polyhedra (since that is a finite set – the cubic prism is excluded as it duplicates the tesseract), but not the infinite categories of duoprisms or prisms of antiprisms.^[16]
- 2020-2023: 342 new polychora were found, bringing up the total number of known uniform 4-polytopes to 2191. The list has not been proven complete.^[16]^[17]

Regular 4-polytopes

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Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements.Regular 4-polytopes can be expressed withSchläfli symbol {p,q,r} have cells of type {p,q}, faces of type {p}, edge figures {r}, andvertex figures {q,r}.

The existence of a regular 4-polytope {p,q,r} is constrained by the existence of the regular polyhedra {p,q} which becomes cells, and {q,r} which becomes thevertex figure.

Existence as a finite 4-polytope is dependent upon an inequality:^[18]

\sin \left({\frac {\pi }{p}}\right)\sin \left({\frac {\pi }{r}}\right)>\cos \left({\frac {\pi }{q}}\right).

The 16regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent:

6regular convex 4-polytopes:5-cell {3,3,3},8-cell {4,3,3},16-cell {3,3,4},24-cell {3,4,3},120-cell {5,3,3}, and600-cell {3,3,5}.
10regular star 4-polytopes:icosahedral 120-cell {3,5,⁵/₂},small stellated 120-cell {⁵/₂,5,3},great 120-cell {5,⁵/₂,5},grand 120-cell {5,3,⁵/₂},great stellated 120-cell {⁵/₂,3,5},grand stellated 120-cell {⁵/₂,5,⁵/₂},great grand 120-cell {5,⁵/₂,3},great icosahedral 120-cell {3,⁵/₂,5},grand 600-cell {3,3,⁵/₂}, andgreat grand stellated 120-cell {⁵/₂,3,3}.

Convex uniform 4-polytopes

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Symmetry of uniform 4-polytopes in four dimensions

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Main article:Point groups in four dimensions

Orthogonal subgroups
The 24 mirrors ofF₄ can be decomposed into 2 orthogonalD₄ groups: = (12 mirrors) = (12 mirrors)
The 10 mirrors ofB₃×A₁ can be decomposed into orthogonal groups, 4A₁ andD₃: = (3+1 mirrors) = (6 mirrors)

There are 5 fundamental mirror symmetrypoint group families in 4-dimensions:A₄ =,B₄ =,D₄ =,F₄ =,H₄ =.^[10] There are also 3 prismatic groupsA₃A₁ =,B₃A₁ =,H₃A₁ =, and duoprismatic groups: I₂(p)×I₂(q) =. Each group defined by aGoursat tetrahedron fundamental domain bounded by mirror planes.

Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by aWythoff construction, represented by rings around permutations of nodes in aCoxeter diagram. Mirrorhyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, [[a,b,a]], doubling the symmetry order. This includes [3,3,3], [3,4,3], and [p,2,p]. Uniform polytopes in these group with symmetric rings contain this extended symmetry.

If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), analternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry isnot generally adjustable to create uniform solutions.

Weyl group	Conway Quaternion	Abstract structure	Order	Coxeter notation	Commutator subgroup	Coxeter number (h)	Mirrors m=2h
Irreducible
A₄	+1/60[I×I].21	S₅	120	[3,3,3]	[3,3,3]⁺	5		10
D₄	±1/3[T×T].2	1/2.²S₄	192	[3^1,1,1]	[3^1,1,1]⁺	6		12
B₄	±1/6[O×O].2	²S₄ = S₂≀S₄	384	[4,3,3]	[3^1,1,1]⁺	8	4	12
F₄	±1/2[O×O].2₃	3.²S₄	1152	[3,4,3]	[3⁺,4,3⁺]	12	12	12
H₄	±[I×I].2	2.(A₅×A₅).2	14400	[5,3,3]	[5,3,3]⁺	30		60
Prismatic groups
A₃A₁	+1/24[O×O].2₃	S₄×D₁	48	[3,3,2] = [3,3]×[ ]	[3,3]⁺	-		6	1
B₃A₁	±1/24[O×O].2	S₄×D₁	96	[4,3,2] = [4,3]×[ ]	[3,3]⁺	-	3	6	1
H₃A₁	±1/60[I×I].2	A₅×D₁	240	[5,3,2] = [5,3]×[ ]	[5,3]⁺	-		15	1
Duoprismatic groups (Use 2p,2q for even integers)
I₂(p)I₂(q)	±1/2[D_2p×D_2q]	D_p×D_q	4pq	[p,2,q] = [p]×[q]	[p⁺,2,q⁺]	-		p	q
I₂(2p)I₂(q)	±1/2[D_4p×D_2q]	D_2p×D_q	8pq	[2p,2,q] = [2p]×[q]		-	p	p	q
I₂(2p)I₂(2q)	±1/2[D_4p×D_4q]	D_2p×D_2q	16pq	[2p,2,2q] = [2p]×[2q]		-	p	p	q	q

Enumeration

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There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of theduoprisms and theantiprismatic prisms.

5 are polyhedral prisms based on thePlatonic solids (1 overlap with regular since a cubic hyperprism is atesseract)
13 are polyhedral prisms based on theArchimedean 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 4-polytope, the grand antiprism.
TOTAL: 68 − 4 = 64

These 64 uniform 4-polytopes 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 ofuniform antiprismatic prisms - sr{p,2}×{ } - Polyhedral prisms of twoantiprisms.
Set of uniformduoprisms - {p}×{q} - ACartesian product of two polygons.

The A₄ family

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Further information:A4 polytope

The 5-cell hasdiploid pentachoric [3,3,3] symmetry,^[10] oforder 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

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

[3,3,3] uniform polytopes
#	Name Bowers name (and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location				Element counts
#	Name Bowers name (and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (5)	Pos. 2 (10)	Pos. 1 (10)	Pos. 0 (5)	Cells	Faces	Edges	Vertices
1	5-cell Pentachoron^[10] (pen)		{3,3,3}	(4) (3.3.3)				5	10	10	5
2	rectified 5-cell Rectified pentachoron (rap)		r{3,3,3}	(3) (3.3.3.3)			(2) (3.3.3)	10	30	30	10
3	truncated 5-cell Truncated pentachoron (tip)		t{3,3,3}	(3) (3.6.6)			(1) (3.3.3)	10	30	40	20
4	cantellated 5-cell Small rhombated pentachoron (srip)		rr{3,3,3}	(2) (3.4.3.4)		(2) (3.4.4)	(1) (3.3.3.3)	20	80	90	30
7	cantitruncated 5-cell Great rhombated pentachoron (grip)		tr{3,3,3}	(2) (4.6.6)		(1) (3.4.4)	(1) (3.6.6)	20	80	120	60
8	runcitruncated 5-cell Prismatorhombated pentachoron (prip)		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

[[3,3,3]] uniform polytopes
#	Name Bowers name (and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location			Element counts
#	Name Bowers name (and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3-0 (10)	Pos. 1-2 (20)	Alt	Cells	Faces	Edges	Vertices
5	*runcinated 5-cell Small prismatodecachoron (spid)		t_0,3{3,3,3}	(2) (3.3.3)	(6) (3.4.4)		30	70	60	20
6	*bitruncated 5-cell Decachoron (deca)		2t{3,3,3}	(4) (3.6.6)			10	40	60	30
9	*omnitruncated 5-cell Great prismatodecachoron (gippid)		t_0,1,2,3{3,3,3}	(2) (4.6.6)	(2) (4.4.6)		30	150	240	120
Nonuniform	omnisnub 5-cell Snub decachoron (snad) Snub pentachoron (snip)^[19]		ht_0,1,2,3{3,3,3}	(2) (3.3.3.3.3)	(2) (3.3.3.3)	(4) (3.3.3)	90	300	270	60

The three uniform 4-polytopes forms marked with anasterisk,*, have the higherextended 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. There is one small index subgroup [3,3,3]⁺, order 60, or its doubling [[3,3,3]]⁺, order 120, defining anomnisnub 5-cell which is listed for completeness, but is not uniform.

The B₄ family

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Further information:B4 polytope

This family hasdiploid hexadecachoric symmetry,^[10] [4,3,3], oforder 24×16=384: 4!=24 permutations of the four axes, 2⁴=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1⁺,4,3,3], [4,(3,3)⁺], and [4,3,3]⁺, all order 192.

Tesseract truncations

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#	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location					Element counts
#	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Cells	Faces	Edges	Vertices
10	tesseract or 8-cell Tesseract (tes)		{4,3,3}	(4) (4.4.4)				8	24	32	16
11	Rectified tesseract (rit)		r{4,3,3}	(3) (3.4.3.4)			(2) (3.3.3)	24	88	96	32
13	Truncated tesseract (tat)		t{4,3,3}	(3) (3.8.8)			(1) (3.3.3)	24	88	128	64
14	Cantellated tesseract Small rhombated tesseract (srit)		rr{4,3,3}	(2) (3.4.4.4)		(2) (3.4.4)	(1) (3.3.3.3)	56	248	288	96
15	Runcinated tesseract (alsoruncinated 16-cell) Small disprismatotesseractihexadecachoron (sidpith)		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 tesseract (alsobitruncated 16-cell) Tesseractihexadecachoron (tah)		2t{4,3,3}	(2) (4.6.6)			(2) (3.6.6)	24	120	192	96
18	Cantitruncated tesseract Great rhombated tesseract (grit)		tr{4,3,3}	(2) (4.6.8)		(1) (3.4.4)	(1) (3.6.6)	56	248	384	192
19	Runcitruncated tesseract Prismatorhombated hexadecachoron (proh)		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 tesseract (alsoomnitruncated 16-cell) Great disprismatotesseractihexadecachoron (gidpith)		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

Related half tesseract, [1⁺,4,3,3] uniform 4-polytopes
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location					Element counts
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Alt	Cells	Faces	Edges	Vertices
[12]	Half tesseract Demitesseract =16-cell (hex)		= h{4,3,3}={3,3,4}	(4) (3.3.3)				(4) (3.3.3)	16	32	24	8
[17]	Cantic tesseract =Truncated 16-cell (thex)		= h₂{4,3,3}=t{4,3,3}	(4) (6.6.3)			(1) (3.3.3.3)		24	96	120	48
[11]	Runcic tesseract =Rectified tesseract (rit)		= h₃{4,3,3}=r{4,3,3}	(3) (3.4.3.4)			(2) (3.3.3)		24	88	96	32
[16]	Runcicantic tesseract =Bitruncated tesseract (tah)		= h_2,3{4,3,3}=2t{4,3,3}	(2) (3.4.3.4)			(2) (3.6.6)		24	120	192	96
[11]	=Rectified tesseract (rat)		= h₁{4,3,3}=r{4,3,3}						24	88	96	32
[16]	=Bitruncated tesseract (tah)		= h_1,2{4,3,3}=2t{4,3,3}						24	120	192	96
[23]	=Rectified 24-cell (rico)		= h_1,3{4,3,3}=rr{3,3,4}						48	240	288	96
[24]	=Truncated 24-cell (tico)		= h_1,2,3{4,3,3}=tr{3,3,4}						48	240	384	192

#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location					Element counts
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Alt	Cells	Faces	Edges	Vertices
Nonuniform	omnisnub tesseract Snub tesseract (snet)^[20] (Oromnisnub 16-cell)		ht_0,1,2,3{4,3,3}	(1) (3.3.3.3.4)	(1) (3.3.3.4)	(1) (3.3.3.3)	(1) (3.3.3.3.3)	(4) (3.3.3)	272	944	864	192

16-cell truncations

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#	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location					Element counts
#	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (8)	Pos. 2 (24)	Pos. 1 (32)	Pos. 0 (16)	Alt	Cells	Faces	Edges	Vertices
12	16-cell Hexadecachoron^[10] (hex)		{3,3,4}				(8) (3.3.3)		16	32	24	8
[22]	*Rectified 16-cell (Same as24-cell) (ico)		= r{3,3,4}	(2) (3.3.3.3)			(4) (3.3.3.3)		24	96	96	24
17	Truncated 16-cell Truncated hexadecachoron (thex)		t{3,3,4}	(1) (3.3.3.3)			(4) (3.6.6)		24	96	120	48
[23]	*Cantellated 16-cell (Same asrectified 24-cell) (rico)		= rr{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 (alsoruncinated tesseract) (sidpith)		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 (alsobitruncated tesseract) (tah)		2t{3,3,4}	(2) (4.6.6)			(2) (3.6.6)		24	120	192	96
[24]	*Cantitruncated 16-cell (Same astruncated 24-cell) (tico)		= tr{3,3,4}	(1) (4.6.6)	(1) (4.4.4)		(2) (4.6.6)		48	240	384	192
20	Runcitruncated 16-cell Prismatorhombated tesseract (prit)		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 (alsoomnitruncated tesseract) (gidpith)		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 thesnub 24-cell) (sadi)		sr{3,3,4}	(1) (3.3.3.3.3)		(1) (3.3.3)	(2) (3.3.3.3.3)	(4) (3.3.3)	144	480	432	96
Nonuniform	Runcic snub rectified 16-cell Pyritosnub tesseract (pysnet)		sr₃{3,3,4}	(1) (3.4.4.4)	(2) (3.4.4)	(1) (4.4.4)	(1) (3.3.3.3.3)	(2) (3.4.4)	176	656	672	192

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

Thesnub 24-cell is repeat to this family for completeness. It is an alternation of thecantitruncated 16-cell ortruncated 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.

The F₄ family

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Further information:F4 polytope

This family hasdiploid icositetrachoric symmetry,^[10] [3,4,3], oforder 24×48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform 4-polytopes which are also repeated in other families, [3⁺,4,3], [3,4,3⁺], and [3,4,3]⁺, all order 576.

[3,4,3] uniform 4-polytopes
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location				Element counts
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (24)	Pos. 2 (96)	Pos. 1 (96)	Pos. 0 (24)	Cells	Faces	Edges	Vertices
22	24-cell (Same asrectified 16-cell) Icositetrachoron^[10] (ico)		{3,4,3}	(6) (3.3.3.3)				24	96	96	24
23	rectified 24-cell (Same ascantellated 16-cell) Rectified icositetrachoron (rico)		r{3,4,3}	(3) (3.4.3.4)			(2) (4.4.4)	48	240	288	96
24	truncated 24-cell (Same ascantitruncated 16-cell) Truncated icositetrachoron (tico)		t{3,4,3}	(3) (4.6.6)			(1) (4.4.4)	48	240	384	192
25	cantellated 24-cell Small rhombated icositetrachoron (srico)		rr{3,4,3}	(2) (3.4.4.4)		(2) (3.4.4)	(1) (3.4.3.4)	144	720	864	288
28	cantitruncated 24-cell Great rhombated icositetrachoron (grico)		tr{3,4,3}	(2) (4.6.8)		(1) (3.4.4)	(1) (3.8.8)	144	720	1152	576
29	runcitruncated 24-cell Prismatorhombated icositetrachoron (prico)		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

[3⁺,4,3] uniform 4-polytopes
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location					Element counts
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (24)	Pos. 2 (96)	Pos. 1 (96)	Pos. 0 (24)	Alt	Cells	Faces	Edges	Vertices
31	†snub 24-cell Snub disicositetrachoron (sadi)		s{3,4,3}	(3) (3.3.3.3.3)			(1) (3.3.3)	(4) (3.3.3)	144	480	432	96
Nonuniform	runcic snub 24-cell Prismatorhombisnub icositetrachoron (prissi)		s₃{3,4,3}	(1) (3.3.3.3.3)	(2) (3.4.4)		(1) (3.6.6)	(3) Tricup	240	960	1008	288
[25]	cantic snub 24-cell (Same ascantellated 24-cell) (srico)		s₂{3,4,3}	(2) (3.4.4.4)			(1) (3.4.3.4)	(2) (3.4.4)	144	720	864	288
[29]	runcicantic snub 24-cell (Same asruncitruncated 24-cell) (prico)		s_2,3{3,4,3}	(1) (4.6.6)		(1) (3.4.4)	(1) (3.4.4.4)	(2) (4.4.6)	240	1104	1440	576

(†) The snub 24-cell here, despite its common name, is not analogous to thesnub cube; rather, it is derived by analternation of the truncated 24-cell. Itssymmetry number is only 576, (theionic diminished icositetrachoric group, [3⁺,4,3]).

Like the 5-cell, the 24-cell is self-dual, and so the following three forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry [[3,4,3]]).

[[3,4,3]] uniform 4-polytopes
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location		Element counts
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3-0 (48)	Pos. 2-1 (192)	Cells	Faces	Edges	Vertices
26	runcinated 24-cell Small prismatotetracontoctachoron (spic)		t_0,3{3,4,3}	(2) (3.3.3.3)	(6) (3.4.4)	240	672	576	144
27	bitruncated 24-cell Tetracontoctachoron (cont)		2t{3,4,3}	(4) (3.8.8)		48	336	576	288
30	omnitruncated 24-cell Great prismatotetracontoctachoron (gippic)		t_0,1,2,3{3,4,3}	(2) (4.6.8)	(2) (4.4.6)	240	1392	2304	1152

[[3,4,3]]⁺ isogonal 4-polytope
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location			Element counts
#	Name	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3-0 (48)	Pos. 2-1 (192)	Alt	Cells	Faces	Edges	Vertices
Nonuniform	omnisnub 24-cell Snub tetracontoctachoron (snoc) Snub icositetrachoron (sni)^[21]		ht_0,1,2,3{3,4,3}	(2) (3.3.3.3.4)	(2) (3.3.3.3)	(4) (3.3.3)	816	2832	2592	576

The H₄ family

[edit]

Further information:H4 polytope

This family hasdiploid hexacosichoric symmetry,^[10] [5,3,3], oforder 120×120=24×600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]⁺, all order 7200.

120-cell truncations

[edit]

#	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Cell counts by location					Element counts
#	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Pos. 3 (120)	Pos. 2 (720)	Pos. 1 (1200)	Pos. 0 (600)	Alt	Cells	Faces	Edges	Vertices
32	120-cell (hecatonicosachoron or dodecacontachoron)^[10] Hecatonicosachoron (hi)		{5,3,3}	(4) (5.5.5)					120	720	1200	600
33	rectified 120-cell Rectified hecatonicosachoron (rahi)		r{5,3,3}	(3) (3.5.3.5)			(2) (3.3.3)		720	3120	3600	1200
36	truncated 120-cell Truncated hecatonicosachoron (thi)		t{5,3,3}	(3) (3.10.10)			(1) (3.3.3)		720	3120	4800	2400
37	cantellated 120-cell Small rhombated hecatonicosachoron (srahi)		rr{5,3,3}	(2) (3.4.5.4)		(2) (3.4.4)	(1) (3.3.3.3)		1920	9120	10800	3600
38	runcinated 120-cell (alsoruncinated 600-cell) Small disprismatohexacosihecatonicosachoron (sidpixhi)		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 (alsobitruncated 600-cell) Hexacosihecatonicosachoron (xhi)		2t{5,3,3}	(2) (5.6.6)			(2) (3.6.6)		720	4320	7200	3600
42	cantitruncated 120-cell Great rhombated hecatonicosachoron (grahi)		tr{5,3,3}	(2) (4.6.10)		(1) (3.4.4)	(1) (3.6.6)		1920	9120	14400	7200
43	runcitruncated 120-cell Prismatorhombated hexacosichoron (prix)		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 (alsoomnitruncated 600-cell) Great disprismatohexacosihecatonicosachoron (gidpixhi)		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
Nonuniform	omnisnub 120-cell Snub hecatonicosachoron (snixhi)^[22] (Same as theomnisnub 600-cell)		ht_0,1,2,3{5,3,3}	(1) (3.3.3.3.5)	(1) (3.3.3.5)	(1) (3.3.3.3)	(1) (3.3.3.3.3)	(4) (3.3.3)	9840	35040	32400	7200

600-cell truncations

[edit]

#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Symmetry	Cell counts by location				Element counts
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram andSchläfli symbols	Symmetry	Pos. 3 (120)	Pos. 2 (720)	Pos. 1 (1200)	Pos. 0 (600)	Cells	Faces	Edges	Vertices
35	600-cell Hexacosichoron^[10] (ex)		{3,3,5}	[5,3,3] order 14400				(20) (3.3.3)	600	1200	720	120
[47]	20-diminished 600-cell =Grand antiprism (gap)		Nonwythoffian construction	[[10,2⁺,10]] order 400 Index 36	(2) (3.3.3.5)			(12) (3.3.3)	320	720	500	100
[31]	24-diminished 600-cell =Snub 24-cell (sadi)		Nonwythoffian construction	[3⁺,4,3] order 576 index 25	(3) (3.3.3.3.3)			(5) (3.3.3)	144	480	432	96
Nonuniform	bi-24-diminished 600-cell Bi-icositetradiminished hexacosichoron (bidex)		Nonwythoffian construction	order 144 index 100	(6) tdi				48	192	216	72
34	rectified 600-cell Rectified hexacosichoron (rox)		r{3,3,5}	[5,3,3]	(2) (3.3.3.3.3)			(5) (3.3.3.3)	720	3600	3600	720
Nonuniform	120-diminished rectified 600-cell Swirlprismatodiminished rectified hexacosichoron (spidrox)		Nonwythoffian construction	order 1200 index 12	(2) 3.3.3.5	(2) 4.4.5		(5) P4	840	2640	2400	600
41	truncated 600-cell Truncated hexacosichoron (tex)		t{3,3,5}	[5,3,3]	(1) (3.3.3.3.3)			(5) (3.6.6)	720	3600	4320	1440
40	cantellated 600-cell Small rhombated hexacosichoron (srix)		rr{3,3,5}	[5,3,3]	(1) (3.5.3.5)	(2) (4.4.5)		(1) (3.4.3.4)	1440	8640	10800	3600
[38]	runcinated 600-cell (alsoruncinated 120-cell) (sidpixhi)		t_0,3{3,3,5}	[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 600-cell (alsobitruncated 120-cell) (xhi)		2t{3,3,5}	[5,3,3]	(2) (5.6.6)			(2) (3.6.6)	720	4320	7200	3600
45	cantitruncated 600-cell Great rhombated hexacosichoron (grix)		tr{3,3,5}	[5,3,3]	(1) (5.6.6)	(1) (4.4.5)		(2) (4.6.6)	1440	8640	14400	7200
44	runcitruncated 600-cell Prismatorhombated hecatonicosachoron (prahi)		t_0,1,3{3,3,5}	[5,3,3]	(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 (alsoomnitruncated 120-cell) (gidpixhi)		t_0,1,2,3{3,3,5}	[5,3,3]	(1) (4.6.10)	(1) (4.4.10)	(1) (4.4.6)	(1) (4.6.6)	2640	17040	28800	14400

The D₄ family

[edit]

Further information:D4 polytope

Thisdemitesseract family, [3^1,1,1], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family hasorder 12×16=192: 4!/2=12 permutations of the four axes, half as alternated, 2⁴=16 for reflection in each axis. There is one small index subgroups that generating uniform 4-polytopes, [3^1,1,1]⁺, order 96.

[3^1,1,1] uniform 4-polytopes
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram = =	Cell counts by location					Element counts
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram = =	Pos. 0 (8)	Pos. 2 (24)	Pos. 1 (8)	Pos. 3 (8)	Pos. Alt (96)	3	2	1	0
[12]	demitesseract half tesseract (Same as16-cell) (hex)		= h{4,3,3}			(4) (3.3.3)	(4) (3.3.3)		16	32	24	8
[17]	cantic tesseract (Same astruncated 16-cell) (thex)		= h₂{4,3,3}	(1) (3.3.3.3)		(2) (3.6.6)	(2) (3.6.6)		24	96	120	48
[11]	runcic tesseract (Same asrectified tesseract) (rit)		= h₃{4,3,3}	(1) (3.3.3)		(1) (3.3.3)	(3) (3.4.3.4)		24	88	96	32
[16]	runcicantic tesseract (Same asbitruncated tesseract) (tah)		= h_2,3{4,3,3}	(1) (3.6.6)		(1) (3.6.6)	(2) (4.6.6)		24	96	96	24

When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as [3[3^1,1,1]] = [3,4,3], and thus these polytopes are repeated from the24-cell family.

[3[3^1,1,1]] uniform 4-polytopes
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram = =	Cell counts by location			Element counts
#	Name (Bowers style acronym)	Vertex figure	Coxeter diagram = =	Pos. 0,1,3 (24)	Pos. 2 (24)	Pos. Alt (96)	3	2	1	0
[22]	rectified 16-cell (Same as24-cell) (ico)		= = = {3^1,1,1} = r{3,3,4} = {3,4,3}	(6) (3.3.3.3)			48	240	288	96
[23]	cantellated 16-cell (Same asrectified 24-cell) (rico)		= = = r{3^1,1,1} = rr{3,3,4} = r{3,4,3}	(3) (3.4.3.4)	(2) (4.4.4)		24	120	192	96
[24]	cantitruncated 16-cell (Same astruncated 24-cell) (tico)		= = = t{3^1,1,1} = tr{3,3,4} = t{3,4,3}	(3) (4.6.6)	(1) (4.4.4)		48	240	384	192
[31]	snub 24-cell (sadi)		= = = s{3^1,1,1} = sr{3,3,4} = s{3,4,3}	(3) (3.3.3.3.3)	(1) (3.3.3)	(4) (3.3.3)	144	480	432	96

Here again thesnub 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 4-polytope, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. thesnub cube and thesnub dodecahedron.

The grand antiprism

[edit]

There is one non-Wythoffian uniform convex 4-polytope, known as thegrand antiprism, consisting of 20pentagonal antiprisms forming two perpendicular rings joined by 300tetrahedra. It is loosely analogous to the three-dimensionalantiprisms, which consist of two parallelpolygons joined by a band oftriangles. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes.

Its symmetry is theionic diminished Coxeter group, [[10,2⁺,10]], order 400.

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

Prismatic uniform 4-polytopes

[edit]

A prismatic polytope is aCartesian product of two polytopes of lower dimension; familiar examples are the 3-dimensionalprisms, which are products of apolygon and aline segment. The prismatic uniform 4-polytopes 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 andantiprisms.
Duoprisms: products of two polygons.

Convex polyhedral prisms

[edit]

The most obvious family of prismatic 4-polytopes is thepolyhedral prisms, i.e. products of a polyhedron with aline segment. The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallelhyperplanes (thebase cells) and a layer of prisms joining them (thelateral cells). This family includes prisms for the 75 nonprismaticuniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as thetesseract).^{[citation needed]}

There are18 convex polyhedral prisms created from 5Platonic solids and 13Archimedean solids as well as for the infinite families of three-dimensionalprisms andantiprisms.^{[citation needed]} The symmetry number of a polyhedral prism is twice that of the base polyhedron.

Tetrahedral prisms: A₃ × A₁

[edit]

Thisprismatic tetrahedral symmetry is [3,3,2], order 48. There are two index 2 subgroups, [(3,3)⁺,2] and [3,3,2]⁺, but the second doesn't generate a uniform 4-polytope.

[3,3,2] uniform 4-polytopes
#	Name (Bowers style acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type			Element counts				Net
#	Name (Bowers style acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type			Cells	Faces	Edges	Vertices	Net
48	Tetrahedral prism (tepe)			{3,3}×{ } t_0,3{3,3,2}	2 3.3.3	4 3.4.4		6	8 {3} 6 {4}	16	8
49	Truncated tetrahedral prism (tuttip)			t{3,3}×{ } t_0,1,3{3,3,2}	2 3.6.6	4 3.4.4	4 4.4.6	10	8 {3} 18 {4} 8 {6}	48	24

[[3,3],2] uniform 4-polytopes
#	Name (Bowers style acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type			Element counts				Net
#	Name (Bowers style acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type			Cells	Faces	Edges	Vertices	Net
[51]	Rectified tetrahedral prism (Same asoctahedral prism) (ope)			r{3,3}×{ } t_1,3{3,3,2}	2 3.3.3.3	4 3.4.4		6	16 {3} 12 {4}	30	12
[50]	Cantellated tetrahedral prism (Same ascuboctahedral prism) (cope)			rr{3,3}×{ } t_0,2,3{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 astruncated octahedral prism) (tope)			tr{3,3}×{ } t_0,1,2,3{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 asicosahedral prism) (ipe)			sr{3,3}×{ }	2 3.3.3.3.3	20 3.4.4		22	40 {3} 30 {4}	72	24
Nonuniform	omnisnub tetrahedral antiprism Pyritohedral icosahedral antiprism (pikap)			$s\left\{{\begin{array}{l}3\\3\\2\end{array}}\right\}$	2 3.3.3.3.3	8 3.3.3.3	6+24 3.3.3	40	16+96 {3}	96	24

Octahedral prisms: B₃ × A₁

[edit]

Thisprismatic octahedral family symmetry is [4,3,2], order 96. There are 6 subgroups of index 2, order 48 that are expressed in alternated 4-polytopes below.Symmetries are [(4,3)⁺,2], [1⁺,4,3,2], [4,3,2⁺], [4,3⁺,2], [4,(3,2)⁺], and [4,3,2]⁺.

#	Name (Bowers style acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type				Element counts				Net
#	Name (Bowers style acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type				Cells	Faces	Edges	Vertices	Net
[10]	Cubic prism (Same astesseract) (Same as4-4 duoprism) (tes)			{4,3}×{ } t_0,3{4,3,2}	2 4.4.4	6 4.4.4			8	24 {4}	32	16
50	Cuboctahedral prism (Same ascantellated tetrahedral prism) (cope)			r{4,3}×{ } t_1,3{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 asrectified tetrahedral prism) (Same astriangular antiprismatic prism) (ope)			{3,4}×{ } t_2,3{4,3,2}	2 3.3.3.3	8 3.4.4			10	16 {3} 12 {4}	30	12
52	Rhombicuboctahedral prism (sircope)			rr{4,3}×{ } t_0,2,3{4,3,2}	2 3.4.4.4	8 3.4.4	18 4.4.4		28	16 {3} 84 {4}	120	48
53	Truncated cubic prism (ticcup)			t{4,3}×{ } t_0,1,3{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 ascantitruncated tetrahedral prism) (tope)			t{3,4}×{ } t_1,2,3{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)			tr{4,3}×{ } t_0,1,2,3{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)			sr{4,3}×{ }	2 3.3.3.3.4	32 3.4.4	6 4.4.4		40	64 {3} 72 {4}	144	48
[48]	Tetrahedral prism (tepe)			h{4,3}×{ }	2 3.3.3	4 3.4.4			6	8 {3} 6 {4}	16	8
[49]	Truncated tetrahedral prism (tuttip)			h₂{4,3}×{ }	2 3.3.6	4 3.4.4	4 4.4.6		6	8 {3} 6 {4}	16	8
[50]	Cuboctahedral prism (cope)			rr{3,3}×{ }	2 3.4.3.4	8 3.4.4	6 4.4.4		16	16 {3} 36 {4}	60	24
[52]	Rhombicuboctahedral prism (sircope)			s₂{3,4}×{ }	2 3.4.4.4	8 3.4.4	18 4.4.4		28	16 {3} 84 {4}	120	48
[54]	Truncated octahedral prism (tope)			tr{3,3}×{ }	2 4.6.6	6 4.4.4	8 4.4.6		16	48 {4} 16 {6}	96	48
[59]	Icosahedral prism (ipe)			s{3,4}×{ }	2 3.3.3.3.3	20 3.4.4			22	40 {3} 30 {4}	72	24
[12]	16-cell (hex)			s{2,4,3}	2+6+8 3.3.3.3				16	32 {3}	24	8
Nonuniform	Omnisnub tetrahedral antiprism = Pyritohedral icosahedral antiprism (pikap)			sr{2,3,4}	2 3.3.3.3.3	8 3.3.3.3	6+24 3.3.3		40	16+96 {3}	96	24
Nonuniform	Edge-snub octahedral hosochoron Pyritosnub alterprism (pysna)			sr₃{2,3,4}	2 3.4.4.4	6 4.4.4	8 3.3.3.3	24 3.4.4	40	16+48 {3} 12+12+24+24 {4}	144	48
Nonuniform	Omnisnub cubic antiprism Snub cubic antiprism (sniccap)			$s\left\{{\begin{array}{l}4\\3\\2\end{array}}\right\}$	2 3.3.3.3.4	12+48 3.3.3	8 3.3.3.3	6 3.3.3.4	76	16+192 {3} 12 {4}	192	48
Nonuniform	Runcic snub cubic hosochoron Truncated tetrahedral alterprism (tuta)			s₃{2,4,3}	2 3.6.6	6 3.3.3	8 triangular cupola		16	52	60	24

Icosahedral prisms: H₃ × A₁

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Thisprismatic icosahedral symmetry is [5,3,2], order 240. There are two index 2 subgroups, [(5,3)⁺,2] and [5,3,2]⁺, but the second doesn't generate a uniform polychoron.

#	Name (Bowers name and acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type				Element counts				Net
#	Name (Bowers name and acronym)	Picture	Vertex figure	Coxeter diagram andSchläfli symbols	Cells by type				Cells	Faces	Edges	Vertices	Net
57	Dodecahedral prism (dope)			{5,3}×{ } t_0,3{5,3,2}	2 5.5.5	12 4.4.5			14	30 {4} 24 {5}	80	40
58	Icosidodecahedral prism (iddip)			r{5,3}×{ } t_1,3{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 assnub tetrahedral prism) (ipe)			{3,5}×{ } t_2,3{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)			t{5,3}×{ } t_0,1,3{5,3,2}	2 3.10.10	20 3.4.4	12 4.4.10		34	40 {3} 90 {4} 24 {10}	240	120
61	Rhombicosidodecahedral prism (sriddip)			rr{5,3}×{ } t_0,2,3{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)			t{3,5}×{ } t_1,2,3{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)			tr{5,3}×{ } t_0,1,2,3{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 {10}	480	240
64	Snub dodecahedral prism (sniddip)			sr{5,3}×{ }	2 3.3.3.3.5	80 3.4.4	12 4.4.5		94	160 {3} 150 {4} 24 {5}	360	120
Nonuniform	Omnisnub dodecahedral antiprism Snub dodecahedral antiprism (sniddap)			$s\left\{{\begin{array}{l}5\\3\\2\end{array}}\right\}$	2 3.3.3.3.5	30+120 3.3.3	20 3.3.3.3	12 3.3.3.5	184	20+240 {3} 24 {5}	220	120

Duoprisms: [p] × [q]

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

The second is the infinite family ofuniform duoprisms, products of tworegular polygons. A duoprism'sCoxeter-Dynkin diagram is. Itsvertex figure is adisphenoid tetrahedron,.

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 ap-gon and aq-gon (a "p,q-duoprism") is 4pq ifp≠q; if the factors are bothp-gons, the symmetry number is 8p². The tesseract can also be considered a 4,4-duoprism.

The extendedf-vector of {p}×{q} is (p,p,1)*(q,q,1) = (pq,2pq,pq+p+q,p+q).

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

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

Infinite set ofp-q duoprism - -pq-gonal prisms,qp-gonal prisms:

Name	Coxeter graph	Cells	Images	Net
3-3 duoprism (triddip)		3+3 triangular prisms
3-4 duoprism (tisdip)		3 cubes 4 triangular prisms
4-4 duoprism (tes) (same as tesseract)		4+4 cubes
3-5 duoprism (trapedip)		3 pentagonal prisms 5 triangular prisms
4-5 duoprism (squipdip)		4 pentagonal prisms 5 cubes
5-5 duoprism (pedip)		5+5 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)		6+6 hexagonal prisms

3-3	3-4	3-5	3-6	3-7	3-8
4-3	4-4	4-5	4-6	4-7	4-8
5-3	5-4	5-5	5-6	5-7	5-8
6-3	6-4	6-5	6-6	6-7	6-8
7-3	7-4	7-5	7-6	7-7	7-8
8-3	8-4	8-5	8-6	8-7	8-8

Alternations are possible. = gives the family ofduoantiprisms, but they generally cannot be made uniform. p=q=2 is the onlyconvex case that can be made uniform, giving the regular 16-cell. p=5, q=5/3 is the only nonconvex case that can be made uniform, giving the so-calledgreat duoantiprism. gives the p-2q-gonalprismantiprismoid (an edge-alternation of the 2p-4q duoprism), but this cannot be made uniform in any cases.^[23]

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

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The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - -p cubes and 4p-gonal prisms - (All are the same as4-p duoprism) The second polytope in the series is a lower symmetry of the regulartesseract, {4}×{4}.

Convexp-gonal prismatic prisms
Name	{3}×{4}	{4}×{4}	{5}×{4}	{6}×{4}	{7}×{4}	{8}×{4}	{p}×{4}
Coxeter diagrams
Image
Cells	3{4}×{} 4{3}×{}	4{4}×{} 4{4}×{}	5{4}×{} 4{5}×{}	6{4}×{} 4{6}×{}	7{4}×{} 4{7}×{}	8{4}×{} 4{8}×{}	p{4}×{} 4{p}×{}
Net

Polygonal antiprismatic prisms: [p] × [ ] × [ ]

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The infinite sets ofuniform antiprismatic prisms are constructed from two parallel uniformantiprisms): (p≥2) - - 2p-gonal antiprisms, connected by 2p-gonal prisms and2p triangular prisms.

Convexp-gonal antiprismatic prisms
Name	s{2,2}×{}	s{2,3}×{}	s{2,4}×{}	s{2,5}×{}	s{2,6}×{}	s{2,7}×{}	s{2,8}×{}	s{2,p}×{}
Coxeter diagram
Image
Vertex figure
Cells	2s{2,2} (2) {2}×{}={4} 4{3}×{}	2s{2,3} 2{3}×{} 6{3}×{}	2s{2,4} 2{4}×{} 8{3}×{}	2s{2,5} 2{5}×{} 10{3}×{}	2s{2,6} 2{6}×{} 12{3}×{}	2s{2,7} 2{7}×{} 14{3}×{}	2s{2,8} 2{8}×{} 16{3}×{}	2 s{2,p} 2 {p}×{} 2p {3}×{}
Net

Ap-gonal antiprismatic prism has4p triangle,4p square and4 p-gon faces. It has10p edges, and4p vertices.

Nonuniform alternations

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Like the 3-dimensionalsnub cube,, analternation removes half the vertices, in two chiral sets of vertices from the ringed form, however theuniform solution requires the vertex positions be adjusted for equal lengths. In four dimensions, this adjustment is only possible for 2 alternated figures, while the rest only exist as nonequilateral alternated figures.

Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all ringsalternated (shown with empty circle nodes). The first is, s{2^1,1,1} which represented an index 24 subgroup (symmetry [2,2,2]⁺, order 8) form of thedemitesseract,, h{4,3,3} (symmetry [1⁺,4,3,3] = [3^1,1,1], order 192). The second is, s{3^1,1,1}, which is an index 6 subgroup (symmetry [3^1,1,1]⁺, order 96) form of thesnub 24-cell,, s{3,4,3}, (symmetry [3⁺,4,3], order 576).

Other alternations, such as, as an alternation from theomnitruncated tesseract, can not be made uniform as solving for equal edge lengths are in generaloverdetermined (there are six equations but only four variables). Such nonuniform alternated figures can be constructed asvertex-transitive 4-polytopes by the removal of one of two half sets of the vertices of the full ringed figure, but will have unequal edge lengths. Just like uniform alternations, they will have half of the symmetry of uniform figure, like [4,3,3]⁺, order 192, is the symmetry of thealternated omnitruncated tesseract.^[24]

Wythoff constructions with alternations producevertex-transitive figures that can be made equilateral, but not uniform because the alternated gaps (around the removed vertices) create cells that are not regular or semiregular. A proposed name for such figures isscaliform polytopes.^[25] This category allows a subset ofJohnson solids as cells, for exampletriangular cupola.

Eachvertex configuration within a Johnson solid must exist within the vertex figure. For example, a square pyramid has two vertex configurations: 3.3.4 around the base, and 3.3.3.3 at the apex.

The nets and vertex figures of the four convex equilateral cases are given below, along with a list of cells around each vertex.

Four convex vertex-transitive equilateral 4-polytopes with nonuniform cells
Coxeter diagram	s₃{2,4,3},	s₃{3,4,3},	Others
Relation	24 of 48 vertices of rhombicuboctahedral prism	288 of 576 vertices of runcitruncated 24-cell	72 of 120 vertices of600-cell	600 of 720 vertices ofrectified 600-cell
Projection				Two rings of pyramids
Net	runcic snub cubic hosochoron^[26]^[27]	runcic snub 24-cell^[28]^[29]	^[30]^[31]^[32]	^[33]^[34]
Cells
Vertex figure	(1) 3.4.3.4:triangular cupola (2) 3.4.6: triangular cupola (1) 3.3.3:tetrahedron (1) 3.6.6:truncated tetrahedron	(1) 3.4.3.4: triangular cupola (2) 3.4.6: triangular cupola (2) 3.4.4:triangular prism (1) 3.6.6:truncated tetrahedron (1) 3.3.3.3.3:icosahedron	(2) 3.3.3.5:tridiminished icosahedron (4) 3.5.5: tridiminished icosahedron	(1) 3.3.3.3:square pyramid (4) 3.3.4: square pyramid (2) 4.4.5:pentagonal prism (2) 3.3.3.5pentagonal antiprism

Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

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The 46 Wythoffian 4-polytopes include the sixconvex regular 4-polytopes. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of theirsymmetries, and therefore may be classified by thesymmetry groups that they have in common.

Summary chart of truncation operations

Example locations of kaleidoscopic generator point on fundamental domain.

The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes aretruncating operations. A 4-polytope 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.

TheCoxeter-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/nradians 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	Symmetry	Description
Parent	t₀{p,q,r}	[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 (Rectified dual)	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 is cut off so that the middle of each original edge remains. Where the vertex was, there appears a new cell, the parent'svertex 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 (orexpansion)	t_0,3{p,q,r}		A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual.
Cantitruncation	t_0,1,2{p,q,r}		Both thecantellation andtruncation 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 theruncination andtruncation operations applied together.
Runcicantellation	t_0,2,3{p,q,r}		Runcitruncated dual {r,q,p}.
Omnitruncation (runcicantitruncation)	t_0,1,2,3{p,q,r}		Application of all three operators.
Half	h{2p,3,q}	[1⁺,2p,3,q] =[(3,p,3),q]	Alternation of, same as
Cantic	h₂{2p,3,q}		Same as
Runcic	h₃{2p,3,q}		Same as
Runcicantic	h_2,3{2p,3,q}		Same as
Quarter	q{2p,3,2q}	[1⁺,2p,3,2q,1⁺]	Same as
Snub	s{p,2q,r}	[p⁺,2q,r]	Alternated truncation
Cantic snub	s₂{p,2q,r}		Cantellated alternated truncation
Runcic snub	s₃{p,2q,r}		Runcinated alternated truncation
Runcicantic snub	s_2,3{p,2q,r}		Runcicantellated alternated truncation
Snub rectified	sr{p,q,2r}	[(p,q)⁺,2r]	Alternated truncated rectification
	ht_0,3{2p,q,2r}	[(2p,q,2r,2⁺)]	Alternated runcination
Bisnub	2s{2p,q,2r}	[2p,q⁺,2r]	Alternated bitruncation
Omnisnub	ht_0,1,2,3{p,q,r}	[p,q,r]⁺	Alternated omnitruncation

See alsoconvex uniform honeycombs, some of which illustrate these operations as applied to the regularcubic honeycomb.

If two polytopes areduals of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), thenbitruncating,runcinating oromnitruncating 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.

Summary of constructions by extended symmetry

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The 46 uniform polychora constructed from the A₄, B₄, F₄, H₄ symmetry are given in this table by their full extended symmetry and Coxeter diagrams. The D₄ symmetry is also included, though it only creates duplicates. Alternations are grouped by their chiral symmetry. All alternations are given, although thesnub 24-cell, with its 3 constructions from different families is the only one that is uniform. Counts in parentheses are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46. The 3-3 and 4-4 duoprismatic family is included, the second for its relation to the B₄ family.

Coxeter group	Extended symmetry	Polychora		Chiral extended symmetry	Alternation honeycombs
[3,3,3]	[3,3,3] (order 120)	6	₍₁₎ \|₍₂₎ \|₍₃₎ ₍₄₎ \|₍₇₎ \|₍₈₎
[3,3,3]	[2⁺[3,3,3]] (order 240)	3	₍₅₎\|₍₆₎ \|₍₉₎	[2⁺[3,3,3]]⁺ (order 120)	(1)	₍₋₎
[3,3^1,1]	[3,3^1,1] (order 192)	0	(none)
	[1[3,3^1,1]]=[4,3,3] = (order 384)	(4)	₍₁₂₎ \|₍₁₇₎ \|₍₁₁₎ \|₍₁₆₎
	[3[3^1,1,1]]=[3,4,3] = (order 1152)	(3)	₍₂₂₎ \|₍₂₃₎ \|₍₂₄₎	[3[3,3^1,1]]⁺ =[3,4,3]⁺ (order 576)	(1)	₍₃₁₎ (=) ₍₋₎
[4,3,3]	[3[1⁺,4,3,3]]=[3,4,3] = (order 1152)	(3)	₍₂₂₎ \|₍₂₃₎ \|₍₂₄₎
	[4,3,3] (order 384)	12	₍₁₀₎ \|₍₁₁₎ \|₍₁₂₎ \|₍₁₃₎ \|₍₁₄₎ ₍₁₅₎ \|₍₁₆₎ \|₍₁₇₎ \|₍₁₈₎ \|₍₁₉₎ ₍₂₀₎ \|₍₂₁₎	[1⁺,4,3,3]⁺ (order 96)	(2)	₍₁₂₎ (=) ₍₃₁₎ ₍₋₎
	[4,3,3] (order 384)	12		[4,3,3]⁺ (order 192)	(1)	₍₋₎
[3,4,3]	[3,4,3] (order 1152)	6	₍₂₂₎ \|₍₂₃₎ \|₍₂₄₎ ₍₂₅₎ \|₍₂₈₎ \|₍₂₉₎	[2⁺[3⁺,4,3⁺]] (order 576)	1	₍₃₁₎
[3,4,3]	[2⁺[3,4,3]] (order 2304)	3	₍₂₆₎ \|₍₂₇₎ \|₍₃₀₎	[2⁺[3,4,3]]⁺ (order 1152)	(1)	₍₋₎
[5,3,3]	[5,3,3] (order 14400)	15	₍₃₂₎ \|₍₃₃₎ \|₍₃₄₎ \|₍₃₅₎ \|₍₃₆₎ ₍₃₇₎ \|₍₃₈₎ \|₍₃₉₎ \|₍₄₀₎ \|₍₄₁₎ ₍₄₂₎ \|₍₄₃₎ \|₍₄₄₎ \|₍₄₅₎ \|₍₄₆₎	[5,3,3]⁺ (order 7200)	(1)	₍₋₎
[3,2,3]	[3,2,3] (order 36)	0	(none)	[3,2,3]⁺ (order 18)	0	(none)
	[2⁺[3,2,3]] (order 72)	0		[2⁺[3,2,3]]⁺ (order 36)	0	(none)
	[[3],2,3]=[6,2,3] = (order 72)	1		[1[3,2,3]]=[[3],2,3]⁺=[6,2,3]⁺ (order 36)	(1)
	[(2⁺,4)[3,2,3]]=[2⁺[6,2,6]] = (order 288)	1		[(2⁺,4)[3,2,3]]⁺=[2⁺[6,2,6]]⁺ (order 144)	(1)
[4,2,4]	[4,2,4] (order 64)	0	(none)	[4,2,4]⁺ (order 32)	0	(none)
	[2⁺[4,2,4]] (order 128)	0	(none)	[2⁺[(4,2⁺,4,2⁺)]] (order 64)	0	(none)
	[(3,3)[4,2*,4]]=[4,3,3] = (order 384)	(1)	₍₁₀₎	[(3,3)[4,2*,4]]⁺=[4,3,3]⁺ (order 192)	(1)	₍₁₂₎
	[[4],2,4]=[8,2,4] = (order 128)	(1)		[1[4,2,4]]=[[4],2,4]⁺=[8,2,4]⁺ (order 64)	(1)
	[(2⁺,4)[4,2,4]]=[2⁺[8,2,8]] = (order 512)	(1)		[(2⁺,4)[4,2,4]]⁺=[2⁺[8,2,8]]⁺ (order 256)	(1)

Uniform star polychora

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Other than the aforementioned infinite duoprism and antiprism prism families, which have infinitely many nonconvex members, many uniform star polychora have been discovered. In 1852, Ludwig Schläfli discovered fourregular star polychora: {5,3,5/2}, {5/2,3,5}, {3,3,5/2}, and {5/2,3,3}. In 1883, Edmund Hess found the other six: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5/2,5,5/2}, {5,5/2,3}, and {3,5/2,5}. Norman Johnson described three uniform antiprism-like star polychora in his doctoral dissertation of 1966: they are based on the threeditrigonal polyhedra sharing the edges and vertices of the regular dodecahedron. Many more have been found since then by other researchers, including Jonathan Bowers and George Olshevsky, creating a total count of 2127 known uniform star polychora at present (not counting the infinite set of duoprisms based on star polygons). There is currently no proof of the set's completeness.

References

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^N.W. Johnson:Geometries and Transformations, (2018)ISBN 978-1-107-10340-5 Chapter 11:Finite Symmetry Groups, 11.1Polytopes and Honeycombs, p.224
^T. Gosset:On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
^A. Boole Stott (1910)."Geometrical deduction of semiregular from regular polytopes and space fillings"(PDF).Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam.XI (1). Amsterdam: Johannes Müller. Archived fromthe original(PDF) on 29 April 2025.
^Polo-Blanco, Irene (2007). "Alicia Boole Stott and four-dimensional polytopes".Theory and history of geometric models(PDF) (PhD thesis). University of Groningen. pp. 155–6. Archived fromthe original(PDF) on 22 January 2025.
^P. H. Schoute (1911)."Analytical treatment of the polytopes regularly derived from the regular polytopes"(PDF).Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Section I.XI (3). Amsterdam: Johannes Müller. Archived fromthe original(PDF) on 22 January 2025.
^P. H. Schoute (1913)."Analytical treatment of the polytopes regularly derived from the regular polytopes"(PDF).Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Sections II, III, IV.XI (5). Amsterdam: Johannes Müller. Archived fromthe original(PDF) on 22 February 2025.
^Elte (1912)
^Uniform Polytopes in Four Dimensions December 6, 1998 oldest archive
^The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes By David Darling, (2004) ASIN: B00SB4TU58
^^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^kJohnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5full polychoric groups
^Uniform Polytopes in Four Dimensions, George Olshevsky.
^Möller, Marco (2004).Vierdimensionale Archimedische Polytope(PDF) (Doctoral thesis) (in German). University of Hamburg.
^Conway (2008)
^Multidimensional Glossary, George Olshevsky
^https://www.mit.edu/~hlb/Associahedron/program.pdfConvex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract
^^a ^b"Uniform Polychora".www.polytope.net. RetrievedFebruary 20, 2020.
^"Uniform polytope".Polytope Wiki. 6 November 2023. Retrieved11 November 2023.
^Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135
^"S3s3s3s".
^"S3s3s4s".
^"S3s4s3s".
^"S3s3s5s".
^sns2s2mx, Richard Klitzing
^H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) p. 582-588 2.7The four-dimensional analogues of the snub cube
^"Polytope-tree".
^"tuta".
^Category S1: Simple Scaliforms tutcup
^"Prissi".
^Category S3: Special Scaliforms prissi
^"bidex".bendwavy.org. Retrieved11 November 2023.
^Category S3: Special Scaliforms bidex
^The Bi-icositetradiminished 600-cell
^"spidrox".bendwavy.org. Retrieved11 November 2023.
^Category S4: Scaliform Swirlprisms spidrox

B. GrünbaumConvex Polytopes, New York, London: Springer, 2003.ISBN 0-387-00424-6.
Second edition prepared by Volker Kaibel,Victor Klee, and Günter M. Ziegler.
Elte, Emanuel Lodewijk (2006).The Semiregular Polytopes of the Hyperspaces. Scholarly Publishing Office, University of Michigan Library.ISBN 1-4181-7968-X.
H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins and J.C.P. Miller:Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, 1954
- H.S.M. Coxeter,Regular Polytopes, 3rd Edition, Dover, New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited 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. pp. 92, 122.
John H. Conway, Heidi Burgiel,Chaim Goodman-Strauss,The Symmetries of Things 2008,ISBN 978-1-56881-220-5 (Chapter 26)
John H. Conway andM.J.T. Guy:Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, pp. 38–39, 1965
N.W. Johnson:The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
N.W. Johnson:Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups
Richard Klitzing,Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, pp. 329–344, (2010)

External links

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Convex uniform 4-polytopes
- Uniform, convex polytopes in four dimensions, Marco Möller(in German). Includes alternative names for these figures, including those from Jonathan Bowers, George Olshevsky, and Norman Johnson.
- Regular and semi-regular convex polytopes a short historical overview
- Java3D Applets with sources
Nonconvex uniform 4-polytopes
- Uniform polychora by Jonathan Bowers
- Stella4D Stella (software) produces interactive views of known uniform polychora including the 64 convex forms and the infinite prismatic families.
Klitzing, Richard."4D uniform polytopes".
4D-Polytopes and Their Dual Polytopes of the Coxeter Group W(A4) Represented by Quaternions International Journal of Geometric Methods in Modern Physics, Vol. 9, No. 4 (2012) Mehmet Koca, Nazife Ozdes Koca, Mudhahir Al-Ajmi (2012)[1]

v t e Fundamental convexregular anduniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) /D_n	E₆ /E₇ /E₈ /F₄ /G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron •Cube	Demicube		Dodecahedron •Icosahedron
Uniform polychoron	Pentachoron	16-cell •Tesseract	Demitesseract	24-cell	120-cell •600-cell
Uniform 5-polytope	5-simplex	5-orthoplex •5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex •6-cube	6-demicube	1₂₂ •2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex •7-cube	7-demicube	1₃₂ •2₃₁ •3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex •8-cube	8-demicube	1₄₂ •2₄₁ •4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex •9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex •10-cube	10-demicube
Uniformn-polytope	n-simplex	n-orthoplex •n-cube	n-demicube	1_k2 •2_k1 •k₂₁	n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds •Polytope operations

Retrieved from "https://en.wikipedia.org/w/index.php?title=Uniform_4-polytope&oldid=1320305807"

Category:

Uniform 4-polytopes

Hidden categories:

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3-3	3-4	3-5	3-6	3-7	3-8
4-3	4-4	4-5	4-6	4-7	4-8
5-3	5-4	5-5	5-6	5-7	5-8
6-3	6-4	6-5	6-6	6-7	6-8
7-3	7-4	7-5	7-6	7-7	7-8
8-3	8-4	8-5	8-6	8-7	8-8

3-3	3-4	3-5	3-6	3-7	3-8
4-3	4-4	4-5	4-6	4-7	4-8
5-3	5-4	5-5	5-6	5-7	5-8
6-3	6-4	6-5	6-6	6-7	6-8
7-3	7-4	7-5	7-6	7-7	7-8
8-3	8-4	8-5	8-6	8-7	8-8

Movatterモバイル変換

History of discovery

Regular 4-polytopes

Convex uniform 4-polytopes

Symmetry of uniform 4-polytopes in four dimensions

Enumeration

The A4 family

The B4 family

Tesseract truncations

16-cell truncations

The F4 family

The H4 family

120-cell truncations

600-cell truncations

The D4 family

The grand antiprism

Prismatic uniform 4-polytopes

Convex polyhedral prisms

Tetrahedral prisms: A3 × A1

Octahedral prisms: B3 × A1

Icosahedral prisms: H3 × A1

Duoprisms: [p] × [q]

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

Polygonal antiprismatic prisms: [p] × [ ] × [ ]

Nonuniform alternations

Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

Summary of constructions by extended symmetry

Uniform star polychora

See also

References

External links

The A₄ family

The B₄ family

The F₄ family

The H₄ family

The D₄ family

Tetrahedral prisms: A₃ × A₁

Octahedral prisms: B₃ × A₁

Icosahedral prisms: H₃ × A₁

3-3	3-4	3-5	3-6	3-7	3-8
4-3	4-4	4-5	4-6	4-7	4-8
5-3	5-4	5-5	5-6	5-7	5-8
6-3	6-4	6-5	6-6	6-7	6-8
7-3	7-4	7-5	7-6	7-7	7-8
8-3	8-4	8-5	8-6	8-7	8-8