File:Schlegel half-solid truncated 120-cell.png Schlegel diagram for the truncated 120-cell with tetrahedral cells visibleFile:120-cell t01 H3.svg Orthographic projection of the truncated 120-cell, in the H3 Coxeter plane (D10 symmetry). Only vertices and edges are drawn.In geometry , a uniform 4-polytope (or uniform polychoron )[ 1] 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 )
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] : 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] 0,1 , cantellation, t0,2 , runcination t0,3 , with single ringed forms called rectified, and bi, tri-prefixes added when the first ring was on the second or third nodes.[ 7] [ 8] 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.[ 9] 2008 : The Symmetries of Things [ 10] 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.[ 11] 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,[ 12] 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.[ 13] 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.[ 13] [ 14] 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 {p ,q ,r } have cells of type {p ,q }, faces of type {p }, edge figures {r }, and vertex 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 the vertex figure .
Existence as a finite 4-polytope is dependent upon an inequality:[ 15]
sin ( π p ) sin ( π r ) > cos ( π q ) . The 16 regular 4-polytopes , with the property that all cells, faces, edges, and vertices are congruent:
6 regular 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}, and 600-cell {3,3,5}.
10 regular 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 }, and great grand stellated 120-cell {5 /2 ,3,3}. Convex uniform 4-polytopes Symmetry of uniform 4-polytopes in four dimensions There are 5 fundamental mirror symmetry point group families in 4-dimensions: A 4 = File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png B 4 = File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png D 4 = File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png F 4 = File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png H 4 = File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png [ 7] A 3 A 1 = File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png B 3 A 1 = File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png H 3 A 1 = File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png 2 (p)×I2 (q) = File:CDel node.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel q.png File:CDel node.png 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
Conway Abstract
Order
Coxeter
Coxeter
Commutator
Coxeter Mirrors m =2h
Irreducible
A4
+1/60[I×I].21
S5 120
File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png [3,3,3]
[3,3,3]+
5
10File:CDel node c1.png
D4
±1/3[T×T].2
1/2.2 S4
192
File:CDel nodes.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodeab c1.png File:CDel split2.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png [31,1,1 ]
[31,1,1 ]+
6
12File:CDel node c1.png
B4
±1/6[O×O].2
2 S4 = S2 ≀S4 384
File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node c2.png File:CDel 4.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png [4,3,3]
8
4File:CDel node c2.png
12File:CDel node c1.png
F4
±1/2[O×O].23
3.2 S4
1152
File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node c2.png File:CDel 3.png File:CDel node c2.png File:CDel 4.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png [3,4,3]
[3+ ,4,3+ ]
12
12File:CDel node c2.png
12File:CDel node c1.png
H4
±[I×I].2
2.(A5 ×A5 ).2
14400
File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node c1.png File:CDel 5.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png [5,3,3]
[5,3,3]+
30
60File:CDel node c1.png
Prismatic groups
A3 A1
+1/24[O×O].23
S4 ×D1
48
File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 2.png File:CDel node c3.png [3,3,2] = [3,3]×[ ]
[3,3]+
-
6File:CDel node c1.png
1File:CDel node c3.png
B3 A1
±1/24[O×O].2
S4 ×D1
96
File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel node c2.png File:CDel 4.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 2.png File:CDel node c3.png [4,3,2] = [4,3]×[ ]
-
3File:CDel node c2.png
6File:CDel node c1.png
1File:CDel node c3.png
H3 A1
±1/60[I×I].2
A5 ×D1
240
File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel node c1.png File:CDel 5.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 2.png File:CDel node c3.png [5,3,2] = [5,3]×[ ]
[5,3]+
-
15File:CDel node c1.png
1File:CDel node c3.png
Duoprismatic groups (Use 2p,2q for even integers)
I2 (p )I2 (q )
±1/2[D2p ×D2q ]
Dp q
4pq
File:CDel node.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel node c1.png File:CDel p.png File:CDel node c1.png File:CDel 2.png File:CDel node c3.png File:CDel q.png File:CDel node c3.png [p ,2,q ] = [p ]×[q ]
[p + ,2,q + ]
-
p File:CDel node c1.png q File:CDel node c3.png
I2 (2p )I2 (q )
±1/2[D4p ×D2q ]
D2p ×Dq
8pq
File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel node c2.png File:CDel 2x.png File:CDel p.png File:CDel node c1.png File:CDel 2.png File:CDel node c3.png File:CDel q.png File:CDel node c3.png [2p ,2,q ] = [2p ]×[q ]
-
p File:CDel node c2.png p File:CDel node c1.png q File:CDel node c3.png
I2 (2p )I2 (2q )
±1/2[D4p ×D4q ]
D2p 2q
16pq
File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2x.png File:CDel q.png File:CDel node.png File:CDel node c2.png File:CDel 2x.png File:CDel p.png File:CDel node c1.png File:CDel 2.png File:CDel node c3.png File:CDel 2x.png File:CDel q.png File:CDel node c4.png [2p ,2,2q ] = [2p ]×[2q ]
-
p File:CDel node c2.png p File:CDel node c1.png q File:CDel node c3.png q File:CDel node c4.png
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:
The A4 family The 5-cell has diploid pentachoric [3,3,3] symmetry[ 7] 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
Vertex
Coxeter diagram Schläfli Cell counts by location
Element counts
Pos. 3File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel 2.png
Pos. 2File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel 2.png File:CDel 2.png File:CDel node.png
Pos. 1File:CDel node.png File:CDel 2.png File:CDel 2.png File:CDel 2.png File:CDel node.png File:CDel 3.png File:CDel node.png
Pos. 0File:CDel 2.png File:CDel 2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Cells
Faces
Edges
Vertices
1
5-cell [ 7] File:5-cell verf.svg File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (4)File:Uniform polyhedron-33-t0.png (3.3.3)
5
10
10
5
2
rectified 5-cell File:Rectified 5-cell verf.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-43-t2.png (3.3.3.3)
(2)File:Uniform polyhedron-33-t0.png (3.3.3)
10
30
30
10
3
truncated 5-cell File:Truncated 5-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-33-t01.png (3.6.6)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
10
30
40
20
4
cantellated 5-cell File:Cantellated 5-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-33-t02.png (3.4.3.4)
(2)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t1.svg (3.3.3.3)
20
80
90
30
7
cantitruncated 5-cell File:Cantitruncated 5-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-33-t012.png (4.6.6)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
20
80
120
60
8
runcitruncated 5-cell File:Runcitruncated 5-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,1,3 {3,3,3}
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
(2)File:Hexagonal prism.png (4.4.6)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t02.png (3.4.3.4)
30
120
150
60
[[3,3,3]] uniform polytopes
#
Name
Vertex
Coxeter diagram File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png Schläfli Cell counts by location
Element counts
Pos. 3-0File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel 2.png
Pos. 1-2File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel 2.png File:CDel 2.png File:CDel node.png
Alt
Cells
Faces
Edges
Vertices
5
*runcinated 5-cell
File:Runcinated 5-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,3 {3,3,3}
(2)File:Uniform polyhedron-33-t0.png (3.3.3)
(6)File:Triangular prism.png (3.4.4)
30
70
60
20
6
*bitruncated 5-cell
File:Bitruncated 5-cell verf.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (4)File:Uniform polyhedron-33-t01.png (3.6.6)
10
40
60
30
9
*omnitruncated 5-cell
File:Omnitruncated 5-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,2,3 {3,3,3}
(2)File:Uniform polyhedron-33-t012.png (4.6.6)
(2)File:Hexagonal prism.png (4.4.6)
30
150
240
120
Nonuniform
omnisnub 5-cell [ 16] File:Snub 5-cell verf.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png 0,1,2,3 {3,3,3}
File:Uniform polyhedron-33-s012.png (3.3.3.3.3)
File:Trigonal antiprism.png (3.3.3.3)
File:Uniform polyhedron-33-t0.png (3.3.3)
90
300
270
60
The three uniform 4-polytopes forms marked with an asterisk , * , have the higher extended pentachoric symmetry , of order 240, [[3,3,3]] because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup [3,3,3]+ , order 60, or its doubling [[3,3,3]]+ , order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform.
The B4 family This family has diploid hexadecachoric symmetry[ 7] order 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
#
Name
Vertex
Coxeter diagram Schläfli Cell counts by location
Element counts
Pos. 3File:CDel node n0.png File:CDel 4.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 2.png File:CDel 2.png
Pos. 2File:CDel node n0.png File:CDel 4.png File:CDel node n1.png File:CDel 2.png File:CDel 2.png File:CDel node n3.png
Pos. 1File:CDel node n0.png File:CDel 2.png File:CDel 2.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Pos. 0File:CDel 2.png File:CDel 2.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Cells
Faces
Edges
Vertices
10
tesseract or 8-cellFile:8-cell verf.svg File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (4)File:Uniform polyhedron-43-t0.png (4.4.4)
8
24
32
16
11
Rectified tesseract (rit)
File:Rectified 8-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-43-t1.png (3.4.3.4)
(2)File:Uniform polyhedron-33-t0.png (3.3.3)
24
88
96
32
13
Truncated tesseract (tat)
File:Truncated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-43-t01.png (3.8.8)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
24
88
128
64
14
Cantellated tesseract File:Cantellated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-43-t02.png (3.4.4.4)
(2)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-43-t2.png (3.3.3.3)
56
248
288
96
15
Runcinated tesseract runcinated 16-cell )File:Runcinated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,3 {4,3,3}
(1)File:Uniform polyhedron-43-t0.png (4.4.4)
(3)File:Uniform polyhedron-43-t0.png (4.4.4)
(3)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
80
208
192
64
16
Bitruncated tesseract bitruncated 16-cell )File:Bitruncated 8-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-43-t12.png (4.6.6)
(2)File:Uniform polyhedron-33-t01.png (3.6.6)
24
120
192
96
18
Cantitruncated tesseract File:Cantitruncated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-43-t012.png (4.6.8)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
56
248
384
192
19
Runcitruncated tesseract File:Runcitruncated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,1,3 {4,3,3}
(1)File:Uniform polyhedron-43-t01.png (3.8.8)
(2)File:Octagonal prism.png (4.4.8)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-43-t1.png (3.4.3.4)
80
368
480
192
21
Omnitruncated tesseract omnitruncated 16-cell )File:Omnitruncated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,2,3 {3,3,4}
(1)File:Uniform polyhedron-43-t012.png (4.6.8)
(1)File:Octagonal prism.png (4.4.8)
(1)File:Hexagonal prism.png (4.4.6)
(1)File:Uniform polyhedron-43-t12.png (4.6.6)
80
464
768
384
Related half tesseract, [1+ ,4,3,3] uniform 4-polytopes
#
Name
Vertex
Coxeter diagram Schläfli Cell counts by location
Element counts
Pos. 3File:CDel node n0.png File:CDel 4.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 2.png File:CDel 2.png
Pos. 2File:CDel node n0.png File:CDel 4.png File:CDel node n1.png File:CDel 2.png File:CDel 2.png File:CDel node n3.png
Pos. 1File:CDel node n0.png File:CDel 2.png File:CDel 2.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Pos. 0File:CDel 2.png File:CDel 2.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Alt
Cells
Faces
Edges
Vertices
12
Half tesseract16-cell (hex)
File:16-cell verf.svg File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png (4)File:Uniform polyhedron-33-t0.png (3.3.3)
(4)File:Uniform polyhedron-33-t0.png (3.3.3)
16
32
24
8
[17]
Cantic tesseract Truncated 16-cell (thex)
File:Truncated demitesseract verf.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png 2 {4,3,3}=t{4,3,3}
(4)File:Uniform polyhedron-33-t01.png (6.6.3)
(1)File:Uniform polyhedron-43-t2.png (3.3.3.3)
24
96
120
48
[11]
Runcic tesseract Rectified tesseract (rit)
File:Cantellated demitesseract verf.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 3 {4,3,3}=r{4,3,3}
(3)File:Uniform polyhedron-43-t1.png (3.4.3.4)
(2)File:Uniform polyhedron-33-t0.png (3.3.3)
24
88
96
32
[16]
Runcicantic tesseract Bitruncated tesseract (tah)
File:Cantitruncated demitesseract verf.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 2,3 {4,3,3}=2t{4,3,3}
(2)File:Uniform polyhedron-43-t12.png (3.4.3.4)
(2)File:Uniform polyhedron-33-t01.png (3.6.6)
24
120
192
96
[11]
= Rectified tesseract (rat)
File:Cantellated demitesseract verf.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png 1 {4,3,3}=r{4,3,3}
24
88
96
32
[16]
= Bitruncated tesseract (tah)
File:Cantitruncated demitesseract verf.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png 1,2 {4,3,3}=2t{4,3,3}
24
120
192
96
[23]
= Rectified 24-cell (rico)
File:Runcicantellated demitesseract verf.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 1,3 {4,3,3}=rr{3,3,4}
48
240
288
96
[24]
= Truncated 24-cell (tico)
File:Omnitruncated demitesseract verf.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 1,2,3 {4,3,3}=tr{3,3,4}
48
240
384
192
16-cell truncations
#
Name (Bowers name and acronym)
Vertex
Coxeter diagram Schläfli Cell counts by location
Element counts
Pos. 3File:CDel node n0.png File:CDel 4.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 2.png File:CDel 2.png
Pos. 2File:CDel node n0.png File:CDel 4.png File:CDel node n1.png File:CDel 2.png File:CDel 2.png File:CDel node n3.png
Pos. 1File:CDel node n0.png File:CDel 2.png File:CDel 2.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Pos. 0File:CDel 2.png File:CDel 2.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Alt
Cells
Faces
Edges
Vertices
[12]
16-cell [ 7] File:16-cell verf.svg File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
(8)File:Uniform polyhedron-33-t0.png (3.3.3)
16
32
24
8
[22]
*Rectified 16-cell24-cell
File:Rectified 16-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-43-t2.png (3.3.3.3)
(4)File:Uniform polyhedron-43-t2.png (3.3.3.3)
24
96
96
24
17
Truncated 16-cell File:Truncated 16-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (1)File:Uniform polyhedron-43-t2.png (3.3.3.3)
(4)File:Uniform polyhedron-33-t01.png (3.6.6)
24
96
120
48
[23]
*Cantellated 16-cellrectified 24-cell
File:Cantellated 16-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (1)File:Uniform polyhedron-43-t1.png (3.4.3.4)
(2)File:Tetragonal prism.png (4.4.4)
(2)File:Uniform polyhedron-43-t1.png (3.4.3.4)
48
240
288
96
[15]
Runcinated 16-cell runcinated tesseract ) (sidpith)
File:Runcinated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,3 {3,3,4}
(1)File:Uniform polyhedron-43-t0.png (4.4.4)
(3)File:Tetragonal prism.png (4.4.4)
(3)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
80
208
192
64
[16]
Bitruncated 16-cell bitruncated tesseract ) (tah)
File:Bitruncated 8-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-43-t12.png (4.6.6)
(2)File:Uniform polyhedron-33-t01.png (3.6.6)
24
120
192
96
[24]
*Cantitruncated 16-celltruncated 24-cell
File:Cantitruncated 16-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (1)File:Uniform polyhedron-43-t12.png (4.6.6)
(1)File:Tetragonal prism.png (4.4.4)
(2)File:Uniform polyhedron-43-t12.png (4.6.6)
48
240
384
192
20
Runcitruncated 16-cell File:Runcitruncated 16-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,3 {3,3,4}
(1)File:Uniform polyhedron-43-t02.png (3.4.4.4)
(1)File:Tetragonal prism.png (4.4.4)
(2)File:Hexagonal prism.png (4.4.6)
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
80
368
480
192
[21]
Omnitruncated 16-cell omnitruncated tesseract ) (gidpith)
File:Omnitruncated 8-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,2,3 {3,3,4}
(1)File:Uniform polyhedron-43-t012.png (4.6.8)
(1)File:Octagonal prism.png (4.4.8)
(1)File:Hexagonal prism.png (4.4.6)
(1)File:Uniform polyhedron-43-t12.png (4.6.6)
80
464
768
384
[31]
alternated cantitruncated 16-cell snub 24-cell ) (sadi)
File:Snub 24-cell verf.png File:CDel node.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png (1)File:Uniform polyhedron-43-h01.svg (3.3.3.3.3)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
(2)File:Uniform polyhedron-33-s012.png (3.3.3.3.3)
(4)File:Uniform polyhedron-33-t0.png (3.3.3)
144
480
432
96
Nonuniform
Runcic snub rectified 16-cell
File:Runcic snub rectified 16-cell verf.png File:CDel node 1.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png 3 {3,3,4}
(1)File:Rhombicuboctahedron uniform edge coloring.png (3.4.4.4)
(2)File:Triangular prism.png (3.4.4)
(1)File:Tetragonal prism.png (4.4.4)
(1)File:Uniform polyhedron-33-s012.png (3.3.3.3.3)
(2)File:Triangular prism.png (3.4.4)
176
656
672
192
(*) Just as rectifying the tetrahedron produces the octahedron , rectifying the 16-cell produces the 24-cell, the regular member of the following family. The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell , with the half symmetry group [(3,3)+ ,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.
The F4 family This family has diploid icositetrachoric symmetry[ 7] order 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
Coxeter diagram Schläfli Cell counts by location
Element counts
Pos. 3File:CDel node n0.png File:CDel 3.png File:CDel node n1.png File:CDel 4.png File:CDel node n2.png File:CDel 2.png File:CDel 2.png
Pos. 2File:CDel node n0.png File:CDel 3.png File:CDel node n1.png File:CDel 2.png File:CDel 2.png File:CDel node n3.png
Pos. 1File:CDel node n0.png File:CDel 2.png File:CDel 2.png File:CDel 2.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Pos. 0File:CDel 2.png File:CDel 2.png File:CDel node n1.png File:CDel 4.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Cells
Faces
Edges
Vertices
22
24-cell rectified 16-cell )[ 7] File:24 cell verf.svg File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (6)File:Uniform polyhedron-43-t2.png (3.3.3.3)
24
96
96
24
23
rectified 24-cell cantellated 16-cell )File:Rectified 24-cell verf.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-43-t1.png (3.4.3.4)
(2)File:Uniform polyhedron-43-t0.png (4.4.4)
48
240
288
96
24
truncated 24-cell cantitruncated 16-cell )File:Truncated 24-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-43-t12.png (4.6.6)
(1)File:Uniform polyhedron-43-t0.png (4.4.4)
48
240
384
192
25
cantellated 24-cell File:Cantellated 24-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-43-t02.png (3.4.4.4)
(2)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-43-t1.png (3.4.3.4)
144
720
864
288
28
cantitruncated 24-cell File:Cantitruncated 24-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-43-t012.png (4.6.8)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-43-t01.png (3.8.8)
144
720
1152
576
29
runcitruncated 24-cell File:Runcitruncated 24-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,1,3 {3,4,3}
(1)File:Uniform polyhedron-43-t12.png (4.6.6)
(2)File:Hexagonal prism.png (4.4.6)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-43-t02.png (3.4.4.4)
240
1104
1440
576
[3+ ,4,3] uniform 4-polytopes
#
Name
Vertex
Coxeter diagram Schläfli Cell counts by location
Element counts
Pos. 3File:CDel node n0.png File:CDel 3.png File:CDel node n1.png File:CDel 4.png File:CDel node n2.png File:CDel 2.png File:CDel 2.png
Pos. 2File:CDel node n0.png File:CDel 3.png File:CDel node n1.png File:CDel 2.png File:CDel 2.png File:CDel node n3.png
Pos. 1File:CDel node n0.png File:CDel 2.png File:CDel 2.png File:CDel 2.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Pos. 0File:CDel 2.png File:CDel 2.png File:CDel node n1.png File:CDel 4.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Alt
Cells
Faces
Edges
Vertices
31
†snub 24-cell
File:Snub 24-cell verf.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-43-h01.svg (3.3.3.3.3)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
(4)File:Uniform polyhedron-33-t0.png (3.3.3)
144
480
432
96
Nonuniform
runcic snub 24-cell File:Runcic snub 24-cell verf.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 3 {3,4,3}
(1)File:Uniform polyhedron-43-h01.svg (3.3.3.3.3)
(2)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
(3)File:Triangular cupola.png Tricup
240
960
1008
288
[25]
cantic snub 24-cellcantellated 24-cell ) (srico)
File:Cantic snub 24-cell verf.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png 2 {3,4,3}
(2)File:Rhombicuboctahedron uniform edge coloring.png (3.4.4.4)
(1)File:Uniform polyhedron-43-t1.png (3.4.3.4)
(2)File:Triangular prism.png (3.4.4)
144
720
864
288
[29]
runcicantic snub 24-cellruncitruncated 24-cell ) (prico)
File:Runcicantic snub 24-cell verf.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 2,3 {3,4,3}
(1)File:Uniform polyhedron-43-t12.png (4.6.6)
(1)File:Triangular prism.png (3.4.4)
(1)File:Rhombicuboctahedron uniform edge coloring.png (3.4.4.4)
(2)File:Hexagonal prism.png (4.4.6)
240
1104
1440
576
(†) The snub 24-cell here, despite its common name, is not analogous to the snub cube ; rather, it is derived by an alternation of the truncated 24-cell. Its symmetry number is only 576, (the ionic 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
Coxeter diagram File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 4.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png Schläfli Cell counts by location
Element counts
Pos. 3-0File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 2.png File:CDel 2.png File:CDel 2.png File:CDel 2.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png
Pos. 2-1File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel 2.png File:CDel node.png File:CDel node.png File:CDel 2.png File:CDel 2.png File:CDel node.png File:CDel 3.png File:CDel node.png
Cells
Faces
Edges
Vertices
26
runcinated 24-cell File:Runcinated 24-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,3 {3,4,3}
(2)File:Uniform polyhedron-43-t2.png (3.3.3.3)
(6)File:Triangular prism.png (3.4.4)
240
672
576
144
27
bitruncated 24-cell File:Bitruncated 24-cell verf.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (4)File:Uniform polyhedron-43-t01.png (3.8.8)
48
336
576
288
30
omnitruncated 24-cell File:Omnitruncated 24-cell verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,2,3 {3,4,3}
(2)File:Uniform polyhedron-43-t012.png (4.6.8)
(2)File:Hexagonal prism.png (4.4.6)
240
1392
2304
1152
The H4 family This family has diploid hexacosichoric symmetry[ 7] order 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
#
Name
Vertex
Coxeter diagram Schläfli Cell counts by location
Element counts
Pos. 3File:CDel node n0.png File:CDel 5.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 2.png
Pos. 2File:CDel node n0.png File:CDel 5.png File:CDel node n1.png File:CDel 2.png File:CDel 2.png File:CDel node n3.png
Pos. 1File:CDel node n0.png File:CDel 2.png File:CDel 2.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Pos. 0File:CDel 2.png File:CDel node n1.png File:CDel 3.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png
Alt
Cells
Faces
Edges
Vertices
32
120-cell [ 7] File:120-cell verf.svg File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (4)File:Uniform polyhedron-53-t0.png (5.5.5)
120
720
1200
600
33
rectified 120-cell File:Rectified 120-cell verf.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-53-t1.png (3.5.3.5)
(2)File:Uniform polyhedron-33-t0.png (3.3.3)
720
3120
3600
1200
36
truncated 120-cell File:Truncated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (3)File:Uniform polyhedron-53-t01.png (3.10.10)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
720
3120
4800
2400
37
cantellated 120-cell File:Cantellated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-53-t02.png (3.4.5.4)
(2)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-43-t2.png (3.3.3.3)
1920
9120
10800
3600
38
runcinated 120-cell runcinated 600-cell )File:Runcinated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,3 {5,3,3}
(1)File:Uniform polyhedron-53-t0.png (5.5.5)
(3)File:Pentagonal prism.png (4.4.5)
(3)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
2640
7440
7200
2400
39
bitruncated 120-cell bitruncated 600-cell )File:Bitruncated 120-cell verf.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-53-t12.png (5.6.6)
(2)File:Uniform polyhedron-33-t01.png (3.6.6)
720
4320
7200
3600
42
cantitruncated 120-cell File:Cantitruncated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (2)File:Uniform polyhedron-53-t012.png (4.6.10)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
1920
9120
14400
7200
43
runcitruncated 120-cell File:Runcitruncated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,1,3 {5,3,3}
(1)File:Uniform polyhedron-53-t01.png (3.10.10)
(2)File:Decagonal prism.png (4.4.10)
(1)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-43-t1.png (3.4.3.4)
2640
13440
18000
7200
46
omnitruncated 120-cell omnitruncated 600-cell )File:Omnitruncated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,2,3 {5,3,3}
(1)File:Uniform polyhedron-53-t012.png (4.6.10)
(1)File:Decagonal prism.png (4.4.10)
(1)File:Hexagonal prism.png (4.4.6)
(1)File:Uniform polyhedron-43-t12.png (4.6.6)
2640
17040
28800
14400
Nonuniform
omnisnub 120-cell [ 19] omnisnub 600-cell )
File:Snub 120-cell verf.png File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png 0,1,2,3 {5,3,3}
File:Uniform polyhedron-53-s012.png (3.3.3.3.5)
File:Pentagonal antiprism.png (3.3.3.5)
File:Trigonal antiprism.png (3.3.3.3)
File:Uniform polyhedron-33-s012.png (3.3.3.3.3)
File:Uniform polyhedron-33-t0.png (3.3.3)
9840
35040
32400
7200
600-cell truncations
#
Name
Vertex
Coxeter diagram Schläfli Symmetry
Cell counts by location
Element counts
Pos. 3File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png
Pos. 2File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 2.png File:CDel node.png
Pos. 1File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 3.png File:CDel node.png
Pos. 0File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Cells
Faces
Edges
Vertices
35
600-cell [ 7] File:600-cell verf.svg File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png [5,3,3]
(20)File:Uniform polyhedron-33-t0.png (3.3.3)
600
1200
720
120
[47]
20-diminished 600-cell Grand antiprism (gap)
File:Grand antiprism verf.png Nonwythoffian
[[10,2+ ,10]]
(2)File:Pentagonal antiprism.png (3.3.3.5)
(12)File:Uniform polyhedron-33-t0.png (3.3.3)
320
720
500
100
[31]
24-diminished 600-cell Snub 24-cell (sadi)
File:Snub 24-cell verf.png Nonwythoffian
[3+ ,4,3]
(3)File:Uniform polyhedron-53-t2.png (3.3.3.3.3)
(5)File:Uniform polyhedron-33-t0.png (3.3.3)
144
480
432
96
Nonuniform
bi-24-diminished 600-cell File:Biicositetradiminished 600-cell vertex figure.png Nonwythoffian
order 144
(6)File:Tridiminished icosahedron.png tdi
48
192
216
72
34
rectified 600-cell File:Rectified 600-cell verf.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png [5,3,3]
(2)File:Uniform polyhedron-53-t2.png (3.3.3.3.3)
(5)File:Uniform polyhedron-43-t2.png (3.3.3.3)
720
3600
3600
720
Nonuniform
120-diminished rectified 600-cell File:Spidrox-vertex figure.png Nonwythoffian
order 1200
(2)File:Pentagonal antiprism.png 3.3.3.5
(2)File:Pentagonal prism.png 4.4.5
(5)File:Square pyramid.png P4
840
2640
2400
600
41
truncated 600-cell File:Truncated 600-cell verf.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png [5,3,3]
(1)File:Uniform polyhedron-53-t2.png (3.3.3.3.3)
(5)File:Uniform polyhedron-33-t01.png (3.6.6)
720
3600
4320
1440
40
cantellated 600-cell File:Cantellated 600-cell verf.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png [5,3,3]
(1)File:Uniform polyhedron-53-t1.png (3.5.3.5)
(2)File:Pentagonal prism.png (4.4.5)
(1)File:Uniform polyhedron-43-t1.png (3.4.3.4)
1440
8640
10800
3600
[38]
runcinated 600-cell runcinated 120-cell ) (sidpixhi)
File:Runcinated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 0,3 {3,3,5}
[5,3,3]
(1)File:Uniform polyhedron-53-t0.png (5.5.5)
(3)File:Pentagonal prism.png (4.4.5)
(3)File:Triangular prism.png (3.4.4)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
2640
7440
7200
2400
[39]
bitruncated 600-cell bitruncated 120-cell ) (xhi)
File:Bitruncated 120-cell verf.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png [5,3,3]
(2)File:Uniform polyhedron-53-t12.png (5.6.6)
(2)File:Uniform polyhedron-33-t01.png (3.6.6)
720
4320
7200
3600
45
cantitruncated 600-cell File:Cantitruncated 600-cell verf.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png [5,3,3]
(1)File:Uniform polyhedron-53-t12.png (5.6.6)
(1)File:Pentagonal prism.png (4.4.5)
(2)File:Uniform polyhedron-43-t12.png (4.6.6)
1440
8640
14400
7200
44
runcitruncated 600-cell File:Runcitruncated 600-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,3 {3,3,5}
[5,3,3]
(1)File:Uniform polyhedron-53-t02.png (3.4.5.4)
(1)File:Pentagonal prism.png (4.4.5)
(2)File:Hexagonal prism.png (4.4.6)
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
2640
13440
18000
7200
[46]
omnitruncated 600-cell omnitruncated 120-cell ) (gidpixhi)
File:Omnitruncated 120-cell verf.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 0,1,2,3 {3,3,5}
[5,3,3]
(1)File:Uniform polyhedron-53-t012.png (4.6.10)
(1)File:Decagonal prism.png (4.4.10)
(1)File:Hexagonal prism.png (4.4.6)
(1)File:Uniform polyhedron-43-t12.png (4.6.6)
2640
17040
28800
14400
The D4 family This demitesseract family , [31,1,1 ], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family has order 12×16=192: 4!/2=12 permutations of the four axes, half as alternated, 24 =16 for reflection in each axis. There is one small index subgroups that generating uniform 4-polytopes, [31,1,1 ]+ , order 96.
[31,1,1 ] uniform 4-polytopes
#
Name (Bowers style acronym)
Vertex
Coxeter diagram File:CD B4 nodes.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node n2.png File:CDel 3.png File:CDel node n3.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png Cell counts by location
Element counts
Pos. 0File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Pos. 2File:CDel nodes.png File:CDel 2.png File:CDel node.png
Pos. 1File:CDel nodes.png File:CDel split2.png File:CDel node.png
Pos. 3File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Pos. Alt
3
2
1
0
[12]
demitesseract16-cell
File:16-cell verf.svg File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
(4)File:Uniform polyhedron-33-t0.png (3.3.3)
(4)File:Uniform polyhedron-33-t0.png (3.3.3)
16
32
24
8
[17]
cantic tesseracttruncated 16-cell
File:Truncated demitesseract verf.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png 2 {4,3,3}
(1)File:Uniform polyhedron-43-t2.png (3.3.3.3)
(2)File:Uniform polyhedron-33-t01.png (3.6.6)
(2)File:Uniform polyhedron-33-t01.png (3.6.6)
24
96
120
48
[11]
runcic tesseractrectified tesseract
File:Cantellated demitesseract verf.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 3 {4,3,3}
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
(3)File:Uniform polyhedron-43-t1.png (3.4.3.4)
24
88
96
32
[16]
runcicantic tesseractbitruncated tesseract
File:Cantitruncated demitesseract verf.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 2,3 {4,3,3}
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
(1)File:Uniform polyhedron-33-t01.png (3.6.6)
(2)File:Uniform polyhedron-43-t12.png (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[31,1,1 ]] = [3,4,3], and thus these polytopes are repeated from the 24-cell family.
[3[31,1,1 ]] uniform 4-polytopes
#
Name (Bowers style acronym)
Vertex
Coxeter diagram File:CDel nodeab c1.png File:CDel split2.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png File:CDel node.png File:CDel 4.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node c2.png File:CDel splitsplit1.png File:CDel branch3 c1.png File:CDel node c1.png Cell counts by location
Element counts
Pos. 0,1,3File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Pos. 2File:CDel nodes.png File:CDel 2.png File:CDel node.png
Pos. Alt
3
2
1
0
[22]
rectified 16-cell24-cell
File:Rectified demitesseract verf.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel splitsplit1.png File:CDel branch3.png File:CDel node.png 1,1,1 } = r{3,3,4} = {3,4,3}
(6)File:Uniform polyhedron-43-t2.png (3.3.3.3)
48
240
288
96
[23]
cantellated 16-cellrectified 24-cell
File:Runcicantellated demitesseract verf.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel splitsplit1.png File:CDel branch3 11.png File:CDel node 1.png 1,1,1 } = rr{3,3,4} = r{3,4,3}
(3)File:Uniform polyhedron-43-t1.png (3.4.3.4)
(2)File:Uniform polyhedron-43-t0.png (4.4.4)
24
120
192
96
[24]
cantitruncated 16-celltruncated 24-cell
File:Omnitruncated demitesseract verf.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel splitsplit1.png File:CDel branch3 11.png File:CDel node 1.png 1,1,1 } = tr{3,3,4} = t{3,4,3}
(3)File:Uniform polyhedron-43-t12.png (4.6.6)
(1)File:Uniform polyhedron-43-t0.png (4.4.4)
48
240
384
192
[31]
snub 24-cell (sadi)
File:Snub 24-cell verf.png File:CDel nodes hh.png File:CDel split2.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel node.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node h.png File:CDel splitsplit1.png File:CDel branch3 hh.png File:CDel node h.png 1,1,1 } = sr{3,3,4} = s{3,4,3}
(3)File:Uniform polyhedron-33-s012.png (3.3.3.3.3)
(1)File:Uniform polyhedron-33-t0.png (3.3.3)
(4)File:Uniform polyhedron-33-t0.png (3.3.3)
144
480
432
96
Here again the snub 24-cell , with the symmetry group [31,1,1 ]+ 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. the snub cube and the snub dodecahedron .
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 ).[citation needed
There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms .[citation needed The symmetry number of a polyhedral prism is twice that of the base polyhedron.
Tetrahedral prisms: A3 × A1 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.
[3,3,2] uniform 4-polytopes
#
Name (Bowers style acronym)
Picture
Vertex
Coxeter diagram Schläfli Cells by type
Element counts
Net
Cells
Faces
Edges
Vertices
48
Tetrahedral prism (tepe)
File:Tetrahedral prism.png File:Tetrahedral prism verf.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 0,3 {3,3,2}
2 File:Uniform polyhedron-33-t0.png 3.3.3
4 File:Triangular prism.png 3.4.4
6
8 {3}
16
8
File:Tetrahedron prism net.png
49
Truncated tetrahedral prism (tuttip)
File:Truncated tetrahedral prism.png File:Truncated tetrahedral prism verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 0,1,3 {3,3,2}
2 File:Uniform polyhedron-33-t01.png 3.6.6
4 File:Triangular prism.png 3.4.4
4 File:Hexagonal prism.png 4.4.6
10
8 {3}
48
24
File:Truncated tetrahedral prism net.png
[[3,3],2] uniform 4-polytopes
#
Name (Bowers style acronym)
Picture
Vertex
Coxeter diagram Schläfli Cells by type
Element counts
Net
Cells
Faces
Edges
Vertices
[51]
Rectified tetrahedral prism octahedral prism ) (ope)
File:Octahedral prism.png File:Tetratetrahedral prism verf.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 1,3 {3,3,2}
2 File:Uniform polyhedron-43-t2.png 3.3.3.3
4 File:Triangular prism.png 3.4.4
6
16 {3}
30
12
File:Octahedron prism net.png
[50]
Cantellated tetrahedral prism cuboctahedral prism ) (cope)
File:Cuboctahedral prism.png File:Cuboctahedral prism verf.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 0,2,3 {3,3,2}
2 File:Uniform polyhedron-43-t1.png 3.4.3.4
8 File:Triangular prism.png 3.4.4
6 File:Uniform polyhedron-43-t0.png 4.4.4
16
16 {3}
60
24
File:Cuboctahedral prism net.png
[54]
Cantitruncated tetrahedral prism truncated octahedral prism ) (tope)
File:Truncated octahedral prism.png File:Truncated octahedral prism verf.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 0,1,2,3 {3,3,2}
2 File:Uniform polyhedron-43-t12.png 4.6.6
8 File:Hexagonal prism.png 6.4.4
6 File:Uniform polyhedron-43-t0.png 4.4.4
16
48 {4}
96
48
File:Truncated octahedral prism net.png
[59]
Snub tetrahedral prism icosahedral prism ) (ipe)
File:Icosahedral prism.png File:Snub tetrahedral prism verf.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png 2 File:Uniform polyhedron-53-t2.png 3.3.3.3.3
20 File:Triangular prism.png 3.4.4
22
40 {3}
72
24
File:Icosahedral prism net.png
Nonuniform
omnisnub tetrahedral antiprism
File:Snub 332 verf.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png s { 3 3 2 } 2 File:Uniform polyhedron-33-s012.png 3.3.3.3.3
8 File:Trigonal antiprism.png 3.3.3.3
6+24 File:Uniform polyhedron-33-t0.png 3.3.3
40
16+96 {3}
96
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]+ .
#
Name (Bowers style acronym)
Picture
Vertex
Coxeter diagram Schläfli Cells by type
Element counts
Net
Cells
Faces
Edges
Vertices
[10]
Cubic prism tesseract 4-4 duoprism ) (tes)
File:Schlegel wireframe 8-cell.png File:Cubic prism verf.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 0,3 {4,3,2}
2 File:Uniform polyhedron-43-t0.png 4.4.4
6 File:Uniform polyhedron-43-t0.png 4.4.4
8
24 {4}
32
16
File:8-cell net.png
50
Cuboctahedral prism cantellated tetrahedral prism ) (cope)
File:Cuboctahedral prism.png File:Cuboctahedral prism verf.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 1,3 {4,3,2}
2 File:Uniform polyhedron-43-t1.png 3.4.3.4
8 File:Triangular prism.png 3.4.4
6 File:Uniform polyhedron-43-t0.png 4.4.4
16
16 {3}
60
24
File:Cuboctahedral prism net.png
51
Octahedral prism rectified tetrahedral prism )triangular antiprismatic prism ) (ope)
File:Octahedral prism.png File:Tetratetrahedral prism verf.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 2,3 {4,3,2}
2 File:Uniform polyhedron-43-t2.png 3.3.3.3
8 File:Triangular prism.png 3.4.4
10
16 {3}
30
12
File:Octahedron prism net.png
52
Rhombicuboctahedral prism (sircope)
File:Rhombicuboctahedral prism.png File:Rhombicuboctahedron prism verf.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 0,2,3 {4,3,2}
2 File:Uniform polyhedron-43-t02.png 3.4.4.4
8 File:Triangular prism.png 3.4.4
18 File:Uniform polyhedron-43-t0.png 4.4.4
28
16 {3}
120
48
File:Small rhombicuboctahedral prism net.png
53
Truncated cubic prism (ticcup)
File:Truncated cubic prism.png File:Truncated cubic prism verf.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 0,1,3 {4,3,2}
2 File:Uniform polyhedron-43-t01.png 3.8.8
8 File:Triangular prism.png 3.4.4
6 File:Octagonal prism.png 4.4.8
16
16 {3}
96
48
File:Truncated cubic prism net.png
54
Truncated octahedral prism cantitruncated tetrahedral prism ) (tope)
File:Truncated octahedral prism.png File:Truncated octahedral prism verf.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 1,2,3 {4,3,2}
2 File:Uniform polyhedron-43-t12.png 4.6.6
6 File:Uniform polyhedron-43-t0.png 4.4.4
8 File:Hexagonal prism.png 4.4.6
16
48 {4}
96
48
File:Truncated octahedral prism net.png
55
Truncated cuboctahedral prism (gircope)
File:Truncated cuboctahedral prism.png File:Truncated cuboctahedral prism verf.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 0,1,2,3 {4,3,2}
2 File:Uniform polyhedron-43-t012.png 4.6.8
12 File:Uniform polyhedron-43-t0.png 4.4.4
8 File:Hexagonal prism.png 4.4.6
6 File:Octagonal prism.png 4.4.8
28
96 {4}
192
96
File:Great rhombicuboctahedral prism net.png
56
Snub cubic prism (sniccup)
File:Snub cubic prism.png File:Snub cubic prism verf.png File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png 2 File:Snub hexahedron.png 3.3.3.3.4
32 File:Triangular prism.png 3.4.4
6 File:Uniform polyhedron-43-t0.png 4.4.4
40
64 {3}
144
48
File:Snub cuboctahedral prism net.png
[48]
Tetrahedral prism (tepe)
File:Tetrahedral prism.png File:Tetrahedral prism verf.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 2 File:Uniform polyhedron-33-t0.png 3.3.3
4 File:Triangular prism.png 3.4.4
6
8 {3}
16
8
File:Tetrahedron prism net.png
[49]
Truncated tetrahedral prism (tuttip)
File:Truncated tetrahedral prism.png File:Truncated tetrahedral prism verf.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 2 {4,3}×{ }
2 File:Uniform polyhedron-33-t01.png 3.3.6
4 File:Triangular prism.png 3.4.4
4 File:Hexagonal prism.png 4.4.6
6
8 {3}
16
8
File:Truncated tetrahedral prism net.png
[50]
Cuboctahedral prism (cope)
File:Cuboctahedral prism.png File:Cuboctahedral prism verf.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 2 File:Uniform polyhedron-43-t1.png 3.4.3.4
8 File:Triangular prism.png 3.4.4
6 File:Uniform polyhedron-43-t0.png 4.4.4
16
16 {3}
60
24
File:Cuboctahedral prism net.png
[52]
Rhombicuboctahedral prism (sircope)
File:Rhombicuboctahedral prism.png File:Rhombicuboctahedron prism verf.png File:CDel node 1.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png 2 {3,4}×{ }
2 File:Rhombicuboctahedron uniform edge coloring.png 3.4.4.4
8 File:Triangular prism.png 3.4.4
18 File:Uniform polyhedron-43-t0.png 4.4.4
28
16 {3}
120
48
File:Small rhombicuboctahedral prism net.png
[54]
Truncated octahedral prism (tope)
File:Truncated octahedral prism.png File:Truncated octahedral prism verf.png File:CDel node h0.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 2 File:Uniform polyhedron-43-t12.png 4.6.6
6 File:Uniform polyhedron-43-t0.png 4.4.4
8 File:Hexagonal prism.png 4.4.6
16
48 {4}
96
48
File:Truncated octahedral prism net.png
[59]
Icosahedral prism (ipe)
File:Icosahedral prism.png File:Snub tetrahedral prism verf.png File:CDel node.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png 2 File:Uniform polyhedron-53-t2.png 3.3.3.3.3
20 File:Triangular prism.png 3.4.4
22
40 {3}
72
24
File:Icosahedral prism net.png
[12]
16-cell (hex)
File:Schlegel wireframe 16-cell.png File:16-cell verf.svg File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png 2+6+8 File:Uniform polyhedron-33-t0.png 3.3.3.3
16
32 {3}
24
8
File:16-cell net.png
Nonuniform
Omnisnub tetrahedral antiprism
File:Snub 332 verf.png File:CDel node.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png 2 File:Uniform polyhedron-53-t2.png 3.3.3.3.3
8 File:Trigonal antiprism.png 3.3.3.3
6+24 File:Uniform polyhedron-33-t0.png 3.3.3
40
16+96 {3}
96
24
Nonuniform
Edge-snub octahedral hosochoron
File:Bialternatosnub octahedral hosochoron vertex figure.png File:CDel node 1.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png 3 {2,3,4}
2 File:Rhombicuboctahedron uniform edge coloring.png 3.4.4.4
6 File:Cube rotorotational symmetry.png 4.4.4
8 File:Trigonal antiprism.png 3.3.3.3
24 File:Triangular prism.png 3.4.4
40
16+48 {3}
144
48
Nonuniform
Omnisnub cubic antiprism
File:Snub 432 verf.png File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png s { 4 3 2 } 2 File:Snub hexahedron.png 3.3.3.3.4
12+48 File:Uniform polyhedron-33-t0.png 3.3.3
8 File:Trigonal antiprism.png 3.3.3.3
6 File:Square antiprism.png 3.3.3.4
76
16+192 {3}
192
48
Nonuniform
Runcic snub cubic hosochoron File:Runcic snub cubic hosochoron.png File:Runcic snub 243 verf.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png 3 {2,4,3}
2 File:Uniform polyhedron-33-t01.png 3.6.6
6 File:Uniform polyhedron-33-t0.png 3.3.3
8 File:Triangular cupola.png triangular cupola
16
52
60
24
File:Truncated tetrahedral cupoliprism net.png
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.
#
Name (Bowers name and acronym)
Picture
Vertex
Coxeter diagram Schläfli Cells by type
Element counts
Net
Cells
Faces
Edges
Vertices
57
Dodecahedral prism (dope)
File:Dodecahedral prism.png File:Dodecahedral prism verf.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 0,3 {5,3,2}
2 File:Uniform polyhedron-53-t0.png 5.5.5
12 File:Pentagonal prism.png 4.4.5
14
30 {4}
80
40
File:Dodecahedral prism net.png
58
Icosidodecahedral prism (iddip)
File:Icosidodecahedral prism.png File:Icosidodecahedral prism verf.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 1,3 {5,3,2}
2 File:Uniform polyhedron-53-t1.png 3.5.3.5
20 File:Triangular prism.png 3.4.4
12 File:Pentagonal prism.png 4.4.5
34
40 {3}
150
60
File:Icosidodecahedral prism net.png
59
Icosahedral prism snub tetrahedral prism ) (ipe)
File:Icosahedral prism.png File:Snub tetrahedral prism verf.png File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 2,3 {5,3,2}
2 File:Uniform polyhedron-53-t2.png 3.3.3.3.3
20 File:Triangular prism.png 3.4.4
22
40 {3}
72
24
File:Icosahedral prism net.png
60
Truncated dodecahedral prism (tiddip)
File:Truncated dodecahedral prism.png File:Truncated dodecahedral prism verf.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node 1.png 0,1,3 {5,3,2}
2 File:Uniform polyhedron-53-t01.png 3.10.10
20 File:Triangular prism.png 3.4.4
12 File:Decagonal prism.png 4.4.10
34
40 {3}
240
120
File:Truncated dodecahedral prism net.png
61
Rhombicosidodecahedral prism (sriddip)
File:Rhombicosidodecahedral prism.png File:Rhombicosidodecahedron prism verf.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 0,2,3 {5,3,2}
2 File:Uniform polyhedron-53-t02.png 3.4.5.4
20 File:Triangular prism.png 3.4.4
30 File:Uniform polyhedron-43-t0.png 4.4.4
12 File:Pentagonal prism.png 4.4.5
64
40 {3}
300
120
File:Small rhombicosidodecahedral prism net.png
62
Truncated icosahedral prism (tipe)
File:Truncated icosahedral prism.png File:Truncated icosahedral prism verf.png File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 1,2,3 {5,3,2}
2 File:Uniform polyhedron-53-t12.png 5.6.6
12 File:Pentagonal prism.png 4.4.5
20 File:Hexagonal prism.png 4.4.6
34
90 {4}
240
120
File:Truncated icosahedral prism net.png
63
Truncated icosidodecahedral prism (griddip)
File:Truncated icosidodecahedral prism.png File:Truncated icosidodecahedral prism verf.png File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 0,1,2,3 {5,3,2}
2 File:Uniform polyhedron-53-t012.png 4.6.10
30 File:Uniform polyhedron-43-t0.png 4.4.4
20 File:Hexagonal prism.png 4.4.6
12 File:Decagonal prism.png 4.4.10
64
240 {4}
480
240
File:Great rhombicosidodecahedral prism net.png
64
Snub dodecahedral prism (sniddip)
File:Snub dodecahedral prism.png File:Snub dodecahedral prism verf.png File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png 2 File:Snub dodecahedron ccw.png 3.3.3.3.5
80 File:Triangular prism.png 3.4.4
12 File:Pentagonal prism.png 4.4.5
94
160 {3}
360
120
File:Snub icosidodecahedral prism net.png
Nonuniform
Omnisnub dodecahedral antiprism
File:Snub 532 verf.png File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png s { 5 3 2 } 2 File:Snub dodecahedron ccw.png 3.3.3.3.5
30+120 File:Uniform polyhedron-33-t0.png 3.3.3
20 File:Uniform polyhedron-43-t2.png 3.3.3.3
12 File:Pentagonal antiprism.png 3.3.3.5
184
20+240 {3}
220
120
Duoprisms: [p] × [q] File:3-3 duoprism.png The simplest of the duoprisms, the 3,3-duoprism, in Schlegel diagram , one of 6 triangular prism cells shown. The second is the infinite family of uniform duoprisms , products of two regular polygons . A duoprism's Coxeter-Dynkin diagram is File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node 1.png File:CDel q.png File:CDel node.png vertex figure is a disphenoid tetrahedron , File:Pq-duoprism verf.png 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 8p 2 . The tesseract can also be considered a 4,4-duoprism.
The extended f-vector of {p }×{q } 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 - File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node 1.png File:CDel q.png File:CDel node.png p q -gonal prisms, q p -gonal prisms:
Alternations are possible. File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel q.png File:CDel node h.png File:CDel node h.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 2x.png File:CDel node h.png File:CDel 2x.png File:CDel q.png File:CDel node.png 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 . File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2x.png File:CDel q.png File:CDel node 1.png prismantiprismoid (an edge-alternation of the 2p-4q duoprism), but this cannot be made uniform in any cases.[ 20]
Polygonal prismatic prisms: [p] × [ ] × [ ] The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png 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 , {4}×{4}.
Polygonal antiprismatic prisms: [p] × [ ] × [ ] The infinite sets of uniform antiprismatic prisms antiprisms ): (p≥2) - File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png p -gonal antiprisms, connected by 2 p -gonal prisms and 2p triangular prisms.
Convex p -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
File:CDel node.png File:CDel 4.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node.png File:CDel 6.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node.png File:CDel 8.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node.png File:CDel 10.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node.png File:CDel 12.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel 6.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node.png File:CDel 14.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel 7.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node.png File:CDel 16.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel 8.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png
Image
File:Digonal antiprismatic prism.png File:Triangular antiprismatic prism.png File:Square antiprismatic prism.png File:Pentagonal antiprismatic prism.png File:Hexagonal antiprismatic prism.png File:Heptagonal antiprismatic prism.png File:Octagonal antiprismatic prism.png File:15-gonal antiprismatic prism.png
Vertex
File:Tetrahedral prism verf.png File:Tetratetrahedral prism verf.png File:Square antiprismatic prism verf2.png File:Pentagonal antiprismatic prism verf.png File:Hexagonal antiprismatic prism verf.png File:Heptagonal antiprismatic prism verf.png File:Octagonal antiprismatic prism verf.png File:Uniform antiprismatic prism verf.png
Cells
2 s{2,2} {4} {3}×{}
2 s{2,3} {3}×{} {3}×{}
2 s{2,4} {4}×{} {3}×{}
2 s{2,5} {5}×{} {3}×{}
2 s{2,6} {6}×{} {3}×{}
2 s{2,7} {7}×{} {3}×{}
2 s{2,8} {8}×{} {3}×{}
2 s{2,p}p {3}×{}
Net
File:Tetrahedron prism net.png File:Octahedron prism net.png File:4-antiprismatic prism net.png File:5-antiprismatic prism net.png File:6-antiprismatic prism net.png File:7-antiprismatic prism net.png File:8-antiprismatic prism net.png File:15-gonal antiprismatic prism verf.png
A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices.
Nonuniform alternations File:Snubcubes in grCO.svg Like the 3-dimensional snub cube , File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png alternation removes half the vertices, in two chiral sets of vertices from the ringed form File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png uniform 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 rings alternated (shown with empty circle nodes). The first is File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png 1,1,1 } which represented an index 24 subgroup (symmetry [2,2,2]+ , order 8) form of the demitesseract , File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png + ,4,3,3] = [31,1,1 ], order 192). The second is File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel split1.png File:CDel nodes hh.png 1,1,1 }, which is an index 6 subgroup (symmetry [31,1,1 ]+ , order 96) form of the snub 24-cell , File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png + ,4,3], order 576).
Other alternations, such as File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png omnitruncated tesseract File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png 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 .[ 21] 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 .[ 22] 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 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
Coxeter diagram
Description
Parent
t0 {p,q,r}
[p,q,r]
File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node.png Original regular form {p,q,r}
Rectification
t1 {p,q,r}
File:CDel node.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node.png Truncation operation applied until the original edges are degenerated into points.
Birectification
t2 {p,q,r}
File:CDel node.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node.png Face are fully truncated to points. Same as rectified dual.
Trirectificationdual )
t3 {p,q,r}
File:CDel node.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node 1.png Cells are truncated to points. Regular dual {r,q,p}
Truncation
t0,1 {p,q,r}
File:CDel node 1.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node.png Each vertex is cut off so that the middle of each original edge remains. Where the vertex was, there appears a new cell, the parent's vertex figure . Each original cell is likewise truncated.
Bitruncation
t1,2 {p,q,r}
File:CDel node.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node.png A truncation between a rectified form and the dual rectified form.
Tritruncation
t2,3 {p,q,r}
File:CDel node.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node 1.png Truncated dual {r,q,p}.
Cantellation
t0,2 {p,q,r}
File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node.png A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
Bicantellation
t1,3 {p,q,r}
File:CDel node.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node 1.png Cantellated dual {r,q,p}.
Runcination expansion )
t0,3 {p,q,r}
File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node 1.png A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual.
Cantitruncation
t0,1,2 {p,q,r}
File:CDel node 1.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node.png Both the cantellation and truncation operations applied together.
Bicantitruncation
t1,2,3 {p,q,r}
File:CDel node.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node 1.png Cantitruncated dual {r,q,p}.
Runcitruncation
t0,1,3 {p,q,r}
File:CDel node 1.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node 1.png Both the runcination and truncation operations applied together.
Runcicantellation
t0,2,3 {p,q,r}
File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node 1.png Runcitruncated dual {r,q,p}.
Omnitruncation t0,1,2,3 {p,q,r}
File:CDel node 1.png File:CDel p.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node 1.png Application of all three operators.
Half
h{2p,3,q}
[1+ ,2p,3,q]
File:CDel node h1.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel q.png File:CDel node.png Alternation of File:CDel node 1.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel labelp.png File:CDel branch 10ru.png File:CDel split2.png File:CDel node.png File:CDel q.png File:CDel node.png
Cantic
h2 {2p,3,q}
File:CDel node h1.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel q.png File:CDel node.png Same as File:CDel labelp.png File:CDel branch 10ru.png File:CDel split2.png File:CDel node 1.png File:CDel q.png File:CDel node.png
Runcic
h3 {2p,3,q}
File:CDel node h1.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel q.png File:CDel node 1.png Same as File:CDel labelp.png File:CDel branch 10ru.png File:CDel split2.png File:CDel node.png File:CDel q.png File:CDel node 1.png
Runcicantic
h2,3 {2p,3,q}
File:CDel node h1.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png Same as File:CDel labelp.png File:CDel branch 10ru.png File:CDel split2.png File:CDel node 1.png File:CDel q.png File:CDel node 1.png
Quarter
q{2p,3,2q}
[1+ ,2p,3,2q,1+ ]
File:CDel node h1.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2x.png File:CDel q.png File:CDel node h1.png Same as File:CDel labelp.png File:CDel branch 10r.png File:CDel splitcross.png File:CDel branch 01l.png File:CDel labelq.png
Snub
s{p,2q,r}
[p+ ,2q,r]
File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node.png Alternated truncation
Cantic snub
s2 {p,2q,r}
File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node.png Cantellated alternated truncation
Runcic snub
s3 {p,2q,r}
File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel q.png File:CDel node.png File:CDel r.png File:CDel node 1.png Runcinated alternated truncation
Runcicantic snub
s2,3 {p,2q,r}
File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel 2x.png File:CDel q.png File:CDel node 1.png File:CDel r.png File:CDel node 1.png Runcicantellated alternated truncation
Snub rectified
sr{p,q,2r}
[(p,q)+ ,2r]
File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel q.png File:CDel node h.png File:CDel 2x.png File:CDel r.png File:CDel node.png Alternated truncated rectification
ht0,3 {2p,q,2r}
[(2p,q,2r,2+ )]
File:CDel node h.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel q.png File:CDel node.png File:CDel 2x.png File:CDel r.png File:CDel node h.png Alternated runcination
Bisnub
2s{2p,q,2r}
[2p,q+ ,2r]
File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node h.png File:CDel q.png File:CDel node h.png File:CDel 2x.png File:CDel r.png File:CDel node.png Alternated bitruncation
Omnisnub
ht0,1,2,3 {p,q,r}
[p,q,r]+
File:CDel node h.png File:CDel p.png File:CDel node h.png File:CDel q.png File:CDel node h.png File:CDel r.png File:CDel node h.png 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
Polychora
Chiral
Alternation honeycombs
[3,3,3]File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
[3,3,3]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 3.png File:CDel node c4.png
6
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (1) | File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (2) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (3) File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (4) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (7) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (8)
[2+ [3,3,3]]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png
3
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (5) | File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (6) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (9)
[2+ [3,3,3]]+
(1)
File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png (−)
[3,31,1 ]File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png
[3,31,1 ]File:CDel node c3.png File:CDel 3.png File:CDel node c4.png File:CDel split1.png File:CDel nodeab c1-2.png
0
(none)
[1[3,31,1 ]]=[4,3,3]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel split1.png File:CDel nodeab c3.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 4.png File:CDel node.png
(4)
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png (12) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png (17) | File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes 11.png (11) | File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png (16)
[3[31,1,1 ]]=[3,4,3]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel split1.png File:CDel nodeab c1.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png
(3)
File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png (22) | File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes 11.png (23) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png (24)
[3[3,31,1 ]]+ +
(1)
File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel split1.png File:CDel nodes hh.png (31) (= File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (−)
[4,3,3]File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
[3[1+ ,4,3,3]]=[3,4,3]File:CDel node.png File:CDel 4.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png
(3)
File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (22) | File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (23) | File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (24)
[4,3,3]File:CDel node c1.png File:CDel 4.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 3.png File:CDel node c4.png
12
File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (10) | File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (11) | File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (12) | File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (13) | File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (14) File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (15) | File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (16) | File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (17) | File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (18) | File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (19) File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (20) | File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (21)
[1+ ,4,3,3]+
(2)
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (12) (= File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png (31) File:CDel node 1.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png (−)
[4,3,3]+
(1)
File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png (−)
[3,4,3]File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png
[3,4,3]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 4.png File:CDel node c3.png File:CDel 3.png File:CDel node c4.png
6
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (22) | File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (23) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (24) File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (25) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (28) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (29)
[2+ [3+ ,4,3+ ]]
1
File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png (31)
[2+ [3,4,3]]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 4.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png
3
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (26) | File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (27) | File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (30)
[2+ [3,4,3]]+
(1)
File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 3.png File:CDel node h.png (−)
[5,3,3]File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
[5,3,3]File:CDel node c1.png File:CDel 5.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 3.png File:CDel node c4.png
15
File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (32) | File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (33) | File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (34) | File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (35) | File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png (36) File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (37) | File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (38) | File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (39) | File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (40) | File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (41) File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png (42) | File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png (43) | File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (44) | File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (45) | File:CDel node 1.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png (46)
[5,3,3]+
(1)
File:CDel node h.png File:CDel 5.png File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 3.png File:CDel node h.png (−)
[3,2,3]File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 3.png File:CDel node.png
[3,2,3]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 2.png File:CDel node c3.png File:CDel 3.png File:CDel node c3.png
0
(none)
[3,2,3]+
0
(none)
[2+ [3,2,3]]File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 2.png File:CDel node c2.png File:CDel 3.png File:CDel node c1.png
0
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png [2+ [3,2,3]]+
0
(none)
[[3],2,3]=[6,2,3]File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 2.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel node c1.png File:CDel 6.png File:CDel node.png File:CDel 2.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png
1
File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png [1[3,2,3]]=[[3],2,3]+ =[6,2,3]+
(1)
File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png
[(2+ ,4)[3,2,3]]=[2+ [6,2,6]]File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel 2.png File:CDel node c1.png File:CDel 3.png File:CDel node c1.png File:CDel node c1.png File:CDel 6.png File:CDel node.png File:CDel 2.png File:CDel node c1.png File:CDel 6.png File:CDel node.png
1
File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png [(2+ ,4)[3,2,3]]+ =[2+ [6,2,6]]+
(1)
File:CDel node h.png File:CDel 3.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 3.png File:CDel node h.png
[4,2,4]File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 4.png File:CDel node.png
[4,2,4]File:CDel node c1.png File:CDel 4.png File:CDel node c2.png File:CDel 2.png File:CDel node c3.png File:CDel 4.png File:CDel node c4.png
0
(none)
[4,2,4]+
0
(none)
[2+ [4,2,4]]File:CDel node c1.png File:CDel 4.png File:CDel node c2.png File:CDel 2.png File:CDel node c2.png File:CDel 4.png File:CDel node c1.png
0
(none)
[2+ [(4,2+ ,4,2+ )]]
0
(none)
[(3,3)[4,2*,4]]=[4,3,3]File:CDel node c1.png File:CDel 4.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 4.png File:CDel node c1.png File:CDel node c1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
(1)
File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 4.png File:CDel node.png (10)
[(3,3)[4,2*,4]]+ =[4,3,3]+
(1)
File:CDel node.png File:CDel 4.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 4.png File:CDel node.png (12)
[[4],2,4]=[8,2,4]File:CDel node c1.png File:CDel 4.png File:CDel node c1.png File:CDel 2.png File:CDel node c2.png File:CDel 4.png File:CDel node c3.png File:CDel node c1.png File:CDel 8.png File:CDel node.png File:CDel 2.png File:CDel node c2.png File:CDel 4.png File:CDel node c3.png
(1)
File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 4.png File:CDel node.png [1[4,2,4]]=[[4],2,4]+ =[8,2,4]+
(1)
File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 4.png File:CDel node.png
[(2+ ,4)[4,2,4]]=[2+ [8,2,8]]File:CDel node c1.png File:CDel 4.png File:CDel node c1.png File:CDel 2.png File:CDel node c1.png File:CDel 4.png File:CDel node c1.png File:CDel node c1.png File:CDel 8.png File:CDel node.png File:CDel 2.png File:CDel node c1.png File:CDel 8.png File:CDel node.png
(1)
File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png [(2+ ,4)[4,2,4]]+ =[2+ [8,2,8]]+
(1)
File:CDel node h.png File:CDel 4.png File:CDel node h.png File:CDel 2x.png File:CDel node h.png File:CDel 4.png File:CDel node h.png
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: {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 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.
See also References
↑ N.W. Johnson : Geometries and Transformations , (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups , 11.1 Polytopes and Honeycombs , p.224↑ T. Gosset : On the Regular and Semi-Regular Figures in Space of n Dimensions ↑ "Archived copy" (PDF) . Archived from the original (PDF) on 2009-12-29. Retrieved 2010-08-13 .{{cite web }}: CS1 maint: archived copy as title (link )↑ 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↑ 7.00 7.01 7.02 7.03 7.04 7.05 7.06 7.07 7.08 7.09 7.10 Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups
↑ Uniform Polytopes in Four Dimensions , George Olshevsky.↑ Möller, Marco (2004). Vierdimensionale Archimedische Polytope (PDF) (Doctoral thesis) (in Deutsch). University of Hamburg. ↑ Conway (2008)
↑ Multidimensional Glossary , George Olshevsky↑ https://www.mit.edu/~hlb/Associahedron/program.pdf Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract↑ 13.0 13.1 "Uniform Polychora" . www.polytope.net . Retrieved February 20, 2020 .↑ "Uniform polytope" . Polytope Wiki . 6 November 2023. Retrieved 11 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 2.7 The four-dimensional analogues of the snub cube
↑ "Polytope-tree" .↑ "tuta" .↑ Category S1: Simple Scaliforms tutcup↑ "Prissi" .↑ Category S3: Special Scaliforms prissi↑ "bidex" . bendwavy.org . Retrieved 11 November 2023 .↑ Category S3: Special Scaliforms bidex↑ The Bi-icositetradiminished 600-cell ↑ "spidrox" . bendwavy.org . Retrieved 11 November 2023 .↑ Category S4: Scaliform Swirlprisms spidrox
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, 1910B. Grünbaum Convex Polytopes ISBN 0-387-00424-6 . Victor Klee , and Günter M. Ziegler.Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces , Groningen: University of Groningen, ISBN 1-4181-7968-X 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 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 p. 92, p. 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 and M.J.T. Guy : Four-Dimensional Archimedean Polytopes , Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965N.W. Johnson : The Theory of Uniform Polytopes and Honeycombs , Ph.D. Dissertation, University of Toronto, 1966N.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) [1]
Schoute, Pieter Hendrik (1911), "Analytic treatment of the polytopes regularly derived from the regular polytopes", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam , 11 (3): 87 ppGooglebook, 370-381 External links
