Uniform 4-polytope explained

In geometry, a uniform 4-polytope (or uniform polychoron)^[1] is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular 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

Convex Regular polytopes:
- 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 4-polytopes (star polyhedron cells and/or vertex figures)
- 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures and .
- 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 Polyeder Einleitung 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..
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. In four dimensions, this gives the rectified 5-cell, the rectified 600-cell, and the snub 24-cell.^[2]
- 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 4-polytopes, corresponding to the nonprismatic forms listed below. The snub 24-cell and grand antiprism were missing from her list.^[3]
- 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.^[4]
Convex uniform polytopes:
- 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
- Convex uniform 4-polytopes:
  - 1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism.
  - 1966 Norman 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.
  - 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, and the symmetry of the anomalous grand antiprism.
  - 1998^[5] -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 the Greek roots poly ("many") and choros ("room" or "space").^[6] 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.^[7]
  - 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.^[8]
  - 2008: The Symmetries of Things^[9] was published by John H. Conway and contains the first print-published listing of the convex uniform 4-polytopes and higher dimensional polytopes by Coxeter group family, with general vertex figure diagrams for each ringed Coxeter diagram permutation—snub, grand antiprism, and duoprisms—which he called proprisms for product prisms. He used his own ijk-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 the nonconvex uniform polyhedra)
- 1966: Johnson describes three nonconvex uniform antiprisms in 4-space in his dissertation.^[10]
- 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,^[11] with an additional four discovered in 2006 for a total of 1849. The count includes the 74 prisms of the 75 non-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.^[12]
- 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.^[12] ^[13]

Regular 4-polytopes

Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. Regular 4-polytopes can be expressed with Schläfli symbol have cells of type, faces of type, edge figures, and vertex figures .

The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which becomes cells, and which becomes the vertex figure.

Existence as a finite 4-polytope is dependent upon an inequality:^[14]

\sin\left(

	\pi
	p

\right)\sin\left(

	\pi
	r

\right)>\cos\left(

	\pi
	q

\right).

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

6 regular convex 4-polytopes: 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell .
10 regular star 4-polytopes: icosahedral 120-cell, small stellated 120-cell, great 120-cell, grand 120-cell, great stellated 120-cell, grand stellated 120-cell, great grand 120-cell, great icosahedral 120-cell, grand 600-cell, and great grand stellated 120-cell .

Convex uniform 4-polytopes

Symmetry of uniform 4-polytopes in four dimensions

See main article: Point groups in four dimensions.

Orthogonal subgroups
The 24 mirrors of F₄ can be decomposed into 2 orthogonal D₄ groups: = (12 mirrors) = (12 mirrors)
The 10 mirrors of B₃×A₁ can be decomposed into orthogonal groups, 4A₁ and D₃: = (3+1 mirrors) = (6 mirrors)

There are 5 fundamental mirror symmetry point group families in 4-dimensions: A₄ =, B₄ =, D₄ =, F₄ =, H₄ = . There are also 3 prismatic groups A₃A₁ =, B₃A₁ =, H₃A₁ =, and duoprismatic groups: I₂(p)×I₂(q) = . Each group defined by a Goursat 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 a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes 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), an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not 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<sup>1,1,1</sup>]	[3<sup>1,1,1</sup>]⁺	6		12
B₄	±1/6[O×O].2	²S₄ = S₂≀S₄	384	[4,3,3]	[3<sup>1,1,1</sup>]⁺	8	4	12
F₄	±1/2[O×O].2₃	3.²S₄	1152	[3,4,3]	[3<sup>+</sup>,4,3<sup>+</sup>]	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<sub>2''p''</sub>×D<sub>2''q''</sub>]	D_p×D_q	4pq	[''p'',2,''q''] = [''p'']×[''q'']	[''p''<sup>+</sup>,2,''q''<sup>+</sup>]	-		p	q
I₂(2p)I₂(q)	±1/2[D<sub>4''p''</sub>×D<sub>2''q''</sub>]	D_2p×D_q	8pq	[2''p'',2,''q''] = [2''p'']×[''q'']		-	p	p	q
I₂(2p)I₂(2q)	±1/2[D<sub>4''p''</sub>×D<sub>4''q''</sub>]	D_2p×D_2q	16pq	[2''p'',2,2''q''] = [2''p'']×[2''q'']		-	p	p	q	q

Enumeration

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

5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
13 are polyhedral prisms based on the Archimedean solids
9 are in the self-dual regular A₄ [3,3,3] group (5-cell) family.
9 are in the self-dual regular F₄ [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
15 are in the regular B₄ [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
15 are in the regular H₄ [3,3,5] group (120-cell/600-cell) family.
1 special snub form in the [3,4,3] group (24-cell) family.
1 special non-Wythoffian 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 of uniform antiprismatic prisms - sr× - Polyhedral prisms of two antiprisms.
Set of uniform duoprisms - × - A Cartesian product of two polygons.

The A₄ family

See also: A4 polytope. 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.

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 and Schläfli symbols	Cell counts by location				Element counts
	Name Bowers name (and acronym)	Vertex figure	Coxeter diagram and Schläfli symbols	Pos. 3 (5)	Pos. 2 (10)	Pos. 1 (10)	Pos. 0 (5)	Cells	Faces	Edges	Vertices
1	5-cell Pentachoron (pen)			(4) (3.3.3)				5	10	10	5
2	rectified 5-cell Rectified pentachoron (rap)		r	(3) (3.3.3.3)			(2) (3.3.3)	10	30	30	10
3	truncated 5-cell Truncated pentachoron (tip)		t	(3) (3.6.6)			(1) (3.3.3)	10	30	40	20
4	cantellated 5-cell Small rhombated pentachoron (srip)		rr	(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	(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	(1) (3.6.6)	(2) (4.4.6)	(1) (3.4.4)	(1) (3.4.3.4)	30	120	150	60

The H₄ family

120-cell truncations

	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram and Schläfli symbols	Cell counts by location					Element counts
	Name (Bowers name and acronym)	Vertex figure	Coxeter diagram and Schläfli symbols	Pos. 3 (120)	Pos. 2 (720)	Pos. 1 (1200)	Pos. 0 (600)	Alt	Cells	Faces	Edges	Vertices
32	120-cell (hecatonicosachoron or dodecacontachoron) Hecatonicosachoron (hi)			(4) (5.5.5)					120	720	1200	600
33	rectified 120-cell Rectified hecatonicosachoron (rahi)		r	(3) (3.5.3.5)			(2) (3.3.3)		720	3120	3600	1200
36	truncated 120-cell Truncated hecatonicosachoron (thi)		t	(3) (3.10.10)			(1) (3.3.3)		720	3120	4800	2400
37	cantellated 120-cell Small rhombated hecatonicosachoron (srahi)		rr	(2) (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) Small disprismatohexacosihecatonicosachoron (sidpixhi)		t_0,3	(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) Hexacosihecatonicosachoron (xhi)		2t	(2) (5.6.6)			(2) (3.6.6)		720	4320	7200	3600
42	cantitruncated 120-cell Great rhombated hecatonicosachoron (grahi)		tr	(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	(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) Great disprismatohexacosihecatonicosachoron (gidpixhi)		t_0,1,2,3	(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 (snahi)^[18] (Same as the omnisnub 600-cell)		ht_0,1,2,3	(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

	Name (Bowers style acronym)	Vertex figure	Coxeter diagram and Schläfli symbols	Symmetry	Cell counts by location				Element counts
	Name (Bowers style acronym)	Vertex figure	Coxeter diagram and Schläfli symbols	Symmetry	Pos. 3 (120)	Pos. 2 (720)	Pos. 1 (1200)	Pos. 0 (600)	Cells	Faces	Edges	Vertices
35	600-cell Hexacosichoron (ex)			[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<sup>+</sup>,4,3] order 576 index 25	(3) (3.3.3.3.3)				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	[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	[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	[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 (also runcinated 120-cell) (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 600-cell (also bitruncated 120-cell) (xhi)		2t	[5,3,3]	(2) (5.6.6)			(2) (3.6.6)	720	4320	7200	3600
45	cantitruncated 600-cell Great rhombated hexacosichoron (grix)		tr	[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	[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 (also omnitruncated 120-cell) (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

The D₄ family

[3<sup>1,1,1</sup>] uniform 4-polytopes
	Name (Bowers style acronym)	Vertex figure	Coxeter diagram = =	Cell counts by location	Element counts
Pos. 0 (8)	Pos. 2 (24)	Pos. 1 (8)	Pos. 3 (8)	Pos. Alt (96)	3	2	1	0
[12]	demitesseract half tesseract (Same as 16-cell) (hex)		= h			(4) (3.3.3)	(4) (3.3.3)		16	32	24	8
[17]	cantic tesseract (Same as truncated 16-cell) (thex)		= h₂	(1) (3.3.3.3)		(2) (3.6.6)	(2) (3.6.6)		24	96	120	48
[11]	runcic tesseract (Same as rectified tesseract) (rit)		= h₃	(1) (3.3.3)		(1) (3.3.3)	(3) (3.4.3.4)		24	88	96	32
[16]	runcicantic tesseract (Same as bitruncated tesseract) (tah)		= h_2,3	(1) (3.6.6)		(1) (3.6.6)	(2) (4.6.6)		24	96	96	24

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

The grand antiprism

There is one non-Wythoffian uniform convex 4-polytope, 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 is the ionic diminished Coxeter group, 10,2⁺,10, order 400.

Prismatic uniform 4-polytopes

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 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 and antiprisms.
Duoprisms: products of two polygons.

Convex polyhedral prisms

The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract).

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

Tetrahedral prisms: A₃ × A₁

This prismatic 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.

!rowspan=2
Cells	Picture	Vertex figure	Coxeter diagram and Schläfli symbols	Faces	Edges	Vertices			Net
Name (Bowers style acronym)	Picture	Vertex figure	Coxeter diagram and Schläfli symbols	Cells by type	Element counts				Net
48	Tetrahedral prism (tepe)			× t_0,3	2 3.3.3	4 3.4.4		6	8 6	16	8
49	Truncated tetrahedral prism (tuttip)			t× t_0,1,3	2 3.6.6	4 3.4.4	4 4.4.6	10	8 18 8	48	24

Octahedral prisms: B₃ × A₁

This prismatic 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]⁺.

Icosahedral prisms: H₃ × A₁

This prismatic 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.

Duoprisms: [p] × [q]

The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is . Its vertex figure is a disphenoid 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 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 extended f-vector of × is (p,p,1)*(q,q,1) = (pq,2pq,pq+p+q,p+q).

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:

Alternations are possible. = gives the family of duoantiprisms, but they generally cannot be made uniform. p=q=2 is the only convex 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-called great duoantiprism. gives the p-2q-gonal prismantiprismoid (an edge-alternation of the 2p-4q duoprism), but this cannot be made uniform in any cases.^[19]

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) The second polytope in the series is a lower symmetry of the regular tesseract, ×.

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

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

Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings alternated (shown with empty circle nodes). The first is, s which represented an index 24 subgroup (symmetry [2,2,2]⁺, order 8) form of the demitesseract,, h (symmetry [1+,4,3,3] = [31,1,1], order 192). The second is, s, which is an index 6 subgroup (symmetry [31,1,1]⁺, order 96) form of the snub 24-cell,, s, (symmetry [3+,4,3], order 576).

Other alternations, such as, as an alternation from the omnitruncated tesseract, can not be made uniform as solving for equal edge lengths are in general overdetermined (there are six equations but only four variables). Such nonuniform alternated figures can be constructed as vertex-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 the alternated omnitruncated tesseract.^[20]

Notes and References

[Norman Johnson (mathematician)|N.W. Johnson]
[Thorold Gosset|T. Gosset]
Web site: Archived copy . 2010-08-13 . dead . https://web.archive.org/web/20091229071352/http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf . 2009-12-29 .
Elte (1912)
https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html Uniform Polytopes in Four Dimensions
https://books.google.com/books?id=HrOxRdtYYaMC&q=polychoron&pg=PP1 The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes
Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups
Vierdimensionale Archimedische Polytope . Möller . Marco . 2004 . University of Hamburg . Doctoral thesis . de .
Conway (2008)
https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html Multidimensional Glossary
https://www.mit.edu/~hlb/Associahedron/program.pdf Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract
Web site: Uniform Polychora. www.polytope.net. February 20, 2020.
Web site: Uniform polytope. 6 November 2023. Polytope Wiki. 11 November 2023.
Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135
Web site: S3s3s3s.
Web site: S3s3s4s.
Web site: S3s4s3s.
Web site: S3s3s5s.
https://bendwavy.org/klitzing/incmats/sns2s2mx.htm sns2s2mx
Web site: Polytope-tree.
Web site: tuta.
http://www.polytope.net/hedrondude/simplesc.htm Category S1: Simple Scaliforms
Web site: Prissi.
http://www.polytope.net/hedrondude/special.htm Category S3: Special Scaliforms
Web site: bidex. bendwavy.org. 11 November 2023.
http://www.polytope.net/hedrondude/special.htm Category S3: Special Scaliforms
http://eusebeia.dyndns.org/4d/bi24dim600cell The Bi-icositetradiminished 600-cell
Web site: spidrox. bendwavy.org. 11 November 2023.
http://www.polytope.net/hedrondude/scaleswirl.htm Category S4: Scaliform Swirlprisms

H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) p. 582-588 2.7 ''The four-dimensional analogues of the snub cube''</ref> Wythoff constructions with alternations produce [[vertex-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 is scaliform polytopes.^[20] This category allows a subset of Johnson solids as cells, for example triangular cupola.

Each vertex 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.

Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

The 46 Wythoffian 4-polytopes include the six convex regular 4-polytopes. 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 4-polytopes from the regular 4-polytopes are truncating 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.

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	Symmetry	Description
Parent	t₀	[p,q,r]	Original regular form
Rectification	t₁		Truncation operation applied until the original edges are degenerated into points.
Birectification (Rectified dual)	t₂		Face are fully truncated to points. Same as rectified dual.
Trirectification (dual)	t₃		Cells are truncated to points. Regular dual
Truncation	t_0,1		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	t_1,2		A truncation between a rectified form and the dual rectified form.
Tritruncation	t_2,3		Truncated dual .
Cantellation	t_0,2		A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
Bicantellation	t_1,3		Cantellated dual .
Runcination (or expansion)	t_0,3		A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual.
Cantitruncation	t_0,1,2		Both the cantellation and truncation operations applied together.
Bicantitruncation	t_1,2,3		Cantitruncated dual .
Runcitruncation	t_0,1,3		Both the runcination and truncation operations applied together.
Runcicantellation	t_0,2,3		Runcitruncated dual .
Omnitruncation (runcicantitruncation)	t_0,1,2,3		Application of all three operators.
Half	h	[1<sup>+</sup>,2p,3,q] =[(3,p,3),q]	Alternation of, same as
Cantic	h₂		Same as
Runcic	h₃		Same as
Runcicantic	h_2,3		Same as
Quarter	q	[1<sup>+</sup>,2p,3,2q,1<sup>+</sup>]	Same as
Snub	s	[p<sup>+</sup>,2q,r]	Alternated truncation
Cantic snub	s₂		Cantellated alternated truncation
Runcic snub	s₃		Runcinated alternated truncation
Runcicantic snub	s_2,3		Runcicantellated alternated truncation
Snub rectified	sr	[(p,q)<sup>+</sup>,2r]	Alternated truncated rectification
	ht_0,3	[(2p,q,2r,2<sup>+</sup>)]	Alternated runcination
Bisnub	2s	[2p,q<sup>+</sup>,2r]	Alternated bitruncation
Omnisnub	ht_0,1,2,3	[p,q,r]⁺	Alternated omnitruncation

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.

Summary of constructions by extended symmetry

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 the snub 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<sup>+</sup>[3,3,3]] (order 240)	3	₍₅₎	₍₆₎	₍₉₎	[2<sup>+</sup>[3,3,3]]⁺ (order 120)	(1)	₍₋₎
[3,3<sup>1,1</sup>]	[3,3<sup>1,1</sup>] (order 192)	0	(none)
	[1[3,3<sup>1,1</sup>]]=[4,3,3] = (order 384)	(4)	₍₁₂₎	₍₁₇₎	₍₁₁₎	₍₁₆₎
	[3[3<sup>1,1,1</sup>]]=[3,4,3] = (order 1152)	(3)	₍₂₂₎	₍₂₃₎	₍₂₄₎	[3[3,3<sup>1,1</sup>]]⁺ =[3,4,3]⁺ (order 576)	(1)	₍₃₁₎ (=) ₍₋₎
[4,3,3]	[3[1<sup>+</sup>,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<sup>+</sup>,4,3,3]⁺ (order 96)	(2)	₍₁₂₎ (=) ₍₃₁₎ ₍₋₎
	[4,3,3] (order 384)	12	₍₁₀₎	[4,3,3]⁺ (order 192)	(1)	₍₋₎
[3,4,3]	[3,4,3] (order 1152)	6	₍₂₂₎	₍₂₃₎	₍₂₄₎ ₍₂₅₎	₍₂₈₎	₍₂₉₎	[2<sup>+</sup>[3<sup>+</sup>,4,3<sup>+</sup>]] (order 576)	1	₍₃₁₎
[3,4,3]	[2<sup>+</sup>[3,4,3]] (order 2304)	3	₍₂₆₎	₍₂₇₎	₍₃₀₎	[2<sup>+</sup>[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<sup>+</sup>[3,2,3]] (order 72)	0		[2<sup>+</sup>[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<sup>+</sup>,4)[3,2,3]]=[2<sup>+</sup>[6,2,6]] = (order 288)	1		[(2<sup>+</sup>,4)[3,2,3]]⁺=[2<sup>+</sup>[6,2,6]]⁺ (order 144)	(1)
[4,2,4]	[4,2,4] (order 64)	0	(none)	[4,2,4]⁺ (order 32)	0	(none)
	[2<sup>+</sup>[4,2,4]] (order 128)	0	(none)	[2<sup>+</sup>[(4,2<sup>+</sup>,4,2<sup>+</sup>)]] (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<sup>+</sup>,4)[4,2,4]]=[2<sup>+</sup>[8,2,8]] = (order 512)	(1)		[(2<sup>+</sup>,4)[4,2,4]]⁺=[2<sup>+</sup>[8,2,8]]⁺ (order 256)	(1)

Uniform star polychora

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 four regular star polychora:,,, and . In 1883, Edmund Hess found the other six:,,,,, and . Norman Johnson described three uniform antiprism-like star polychora in his doctoral dissertation of 1966: they are based on the three ditrigonal 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

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

B. Grünbaum Convex Polytopes, New York; London : Springer, c2003. .
Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.
The semiregular polytopes of the hyperspaces. The semiregular polytopes of the hyperspaces.
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, Londen, 1954
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
- (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 p. 92, p. 122.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 26)
John 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

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, 329-344, (2010) http://bendwavy.org/klitzing/pdf/Stott_v8.pdf
Googlebook, 370-381

External links

Convex uniform 4-polytopes
- Uniform, convex polytopes in four dimensions, Marco Möller . 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.
- 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) https://arxiv.org/ftp/arxiv/papers/1102/1102.1132.pdf