Uniform 4-polytope

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Schlegel diagram for the truncated 120-cell with tetrahedral cells visible
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 is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

47 non-prismatic convex uniform 4-polytopes, one finite set of convex prismatic forms, and two infinite sets of convex prismatic forms have been described. There are also an unknown number of non-convex star forms.

History of discovery[edit]

  • Convex Regular polytopes:
    • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
  • Regular star 4-polytopes (star polyhedron cells and/or vertex figures)
    • 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
    • 1883: Edmund Hess completed the list of 10 of the nonconvex regular 4-polytopes, in his book (in German) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [2].
  • Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
    • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[1]
    • 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.[2]
    • 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.[3]
  • 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[4]-2000: The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration (used as a basis for this listing). Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots poly ("many") and choros ("room" or "space").[5] The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, 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.[6][7]
      • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's naming system in his listing.[8]
      • 2008: The Symmetries of Things[9] was published by John H. Conway contains the first print-published listing of the convex uniform 4-polytopes and higher dimensions 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, with all of Johnson's names were included in the book index.
  • Nonregular uniform star 4-polytopes: (similar to the nonconvex uniform polyhedra)
    • 2000-2005: 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.[10]

Regular 4-polytopes[edit]

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:[11]

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

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

Convex uniform 4-polytopes[edit]

Enumeration[edit]

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 hyperprisms.

  • 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
  • 13 are polyhedral prisms based on the Archimedean solids
  • 9 are in the self-dual regular A4 [3,3,3] group (5-cell) family.
  • 9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
  • 15 are in the regular 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-polytopes, 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[edit]

Further information: List of A4 polytopes

The 5-cell has diploid pentachoric [3,3,3] symmetry,[6] of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

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

[3,3,3] uniform polytopes
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.png
(5)
Pos. 2
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node.png
(10)
Pos. 1
CDel node.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(10)
Pos. 0
CDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(5)
Cells Faces Edges Vertices
1 5-cell
pentachoron[6]
5-cell verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,3,3}
(4)
Tetrahedron.png
(3.3.3)
5 10 10 5
2 rectified 5-cell Rectified 5-cell verf.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
r{3,3,3}
(3)
Octahedron.png
(3.3.3.3)
(2)
Tetrahedron.png
(3.3.3)
10 30 30 10
3 truncated 5-cell Truncated 5-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t{3,3,3}
(3)
Truncated tetrahedron.png
(3.6.6)
(1)
Tetrahedron.png
(3.3.3)
10 30 40 20
4 cantellated 5-cell Cantellated 5-cell verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{3,3,3}
(2)
Cuboctahedron.png
(3.4.3.4)
(2)
Triangular prism.png
(3.4.4)
(1)
Octahedron.png
(3.3.3.3)
20 80 90 30
7 cantitruncated 5-cell Cantitruncated 5-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{3,3,3}
(2)
Truncated octahedron.png
(4.6.6)
(1)
Triangular prism.png
(3.4.4)
(1)
Truncated tetrahedron.png
(3.6.6)
20 80 120 60
8 runcitruncated 5-cell Runcitruncated 5-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,3}
(1)
Truncated tetrahedron.png
(3.6.6)
(2)
Hexagonal prism.png
(4.4.6)
(1)
Triangular prism.png
(3.4.4)
(1)
Cuboctahedron.png
(3.4.3.4)
30 120 150 60
[[3,3,3]] uniform polytopes
# Name Vertex
figure
Coxeter diagram
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3-0
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.png
(10)
Pos. 1-2
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node.png
(20)
Alt Cells Faces Edges Vertices
5 *runcinated 5-cell Runcinated 5-cell verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,3}
(2)
Tetrahedron.png
(3.3.3)
(6)
Triangular prism.png
(3.4.4)
30 70 60 20
6 *bitruncated 5-cell
decachoron
Bitruncated 5-cell verf.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,3,3}
(4)
Truncated tetrahedron.png
(3.6.6)
10 40 60 30
9 *omnitruncated 5-cell Omnitruncated 5-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,3}
(2)
Truncated octahedron.png
(4.6.6)
(2)
Hexagonal prism.png
(4.4.6)
30 150 240 120
Nonuniform omnisnub 5-cell[12] Snub 5-cell verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{4,3,3}
Snub tetrahedron.png (2)
(3.3.3.3.3)
Octahedron.png (2)
(3.3.3.3)
Tetrahedron.png (4)
(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 a omnisnub 5-cell which is listed for completeness, but is not uniform.

The B4 family[edit]

Further information: List of B4 polytopes

This family has diploid hexadecachoric symmetry,[6] [4,3,3], of 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[edit]

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel 2.png
(8)
Pos. 2
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.png
(24)
Pos. 1
CDel node n0.pngCDel 2.pngCDel 2.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(32)
Pos. 0
CDel 2.pngCDel 2.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(16)
Cells Faces Edges Vertices
10 tesseract or
8-cell
8-cell verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{4,3,3}
(4)
Hexahedron.png
(4.4.4)
8 24 32 16
11 Rectified tesseract Rectified 8-cell verf.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
r{4,3,3}
(3)
Cuboctahedron.png
(3.4.3.4)
(2)
Tetrahedron.png
(3.3.3)
24 88 96 32
13 Truncated tesseract Truncated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t{4,3,3}
(3)
Truncated hexahedron.png
(3.8.8)
(1)
Tetrahedron.png
(3.3.3)
24 88 128 64
14 Cantellated tesseract Cantellated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{4,3,3}
(1)
Small rhombicuboctahedron.png
(3.4.4.4)
(2)
Triangular prism.png
(3.4.4)
(1)
Octahedron.png
(3.3.3.3)
56 248 288 96
15 Runcinated tesseract
(also runcinated 16-cell)
Runcinated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{4,3,3}
(1)
Hexahedron.png
(4.4.4)
(3)
Hexahedron.png
(4.4.4)
(3)
Triangular prism.png
(3.4.4)
(1)
Tetrahedron.png
(3.3.3)
80 208 192 64
16 Bitruncated tesseract
(also bitruncated 16-cell)
Bitruncated 8-cell verf.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{4,3,3}
(2)
Truncated octahedron.png
(4.6.6)
(2)
Truncated tetrahedron.png
(3.6.6)
24 120 192 96
18 Cantitruncated tesseract Cantitruncated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{4,3,3}
(2)
Great rhombicuboctahedron.png
(4.6.8)
(1)
Triangular prism.png
(3.4.4)
(1)
Truncated tetrahedron.png
(3.6.6)
56 248 384 192
19 Runcitruncated tesseract Runcitruncated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{4,3,3}
(1)
Truncated hexahedron.png
(3.8.8)
(2)
Octagonal prism.png
(4.4.8)
(1)
Triangular prism.png
(3.4.4)
(1)
Cuboctahedron.png
(3.4.3.4)
80 368 480 192
21 Omnitruncated tesseract
(also omnitruncated 16-cell)
Omnitruncated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,4}
(1)
Great rhombicuboctahedron.png
(4.6.8)
(1)
Octagonal prism.png
(4.4.8)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Truncated octahedron.png
(4.6.6)
80 464 768 384
Related half tesseract, [1+,4,3,3] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel 2.png
(8)
Pos. 2
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.png
(24)
Pos. 1
CDel node n0.pngCDel 2.pngCDel 2.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(32)
Pos. 0
CDel 2.pngCDel 2.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(16)
Alt Cells Faces Edges Vertices
12 Half tesseract
Demitesseract
16-cell
16-cell verf.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
h{4,3,3}={3,3,4}
(4)
Tetrahedron.png
(3.3.3)
(4)
Tetrahedron.png
(3.3.3)
16 32 24 8
[17] Cantic tesseract
(Or truncated 16-cell)
Truncated demitesseract verf.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
h2{4,3,3}=t{4,3,3}
(4)
Truncated tetrahedron.png
(6.6.3)
(1)
Octahedron.png
(3.3.3.3)
24 96 120 48
[11] Runcic tesseract
(Or rectified tesseract)
Cantellated demitesseract verf.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
h3{4,3,3}=r{4,3,3}
(3)
Cuboctahedron.png
(3.4.3.4)
(2)
Tetrahedron.png
(3.3.3)
24 88 96 32
[16] Runcicantic tesseract
(Or bitruncated tesseract)
Cantitruncated demitesseract verf.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
h2,3{4,3,3}=2t{4,3,3}
(2)
Truncated octahedron.png
(3.4.3.4)
(2)
Truncated tetrahedron.png
(3.6.6)
24 96 96 24
[11] (rectified tesseract) Cantellated demitesseract verf.png CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
h1{4,3,3}=r{4,3,3}
24 88 96 32
[16] (bitruncated tesseract) Cantitruncated demitesseract verf.png CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
h1,2{4,3,3}=2t{4,3,3}
24 96 96 24
[23] (rectified 24-cell) Runcicantellated demitesseract verf.png CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
h1,3{4,3,3}=rr{3,3,4}
48 240 288 96
[24] (truncated 24-cell) Omnitruncated demitesseract verf.png CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
h1,2,3{4,3,3}=tr{3,3,4}
48 240 384 192
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel 2.png
(8)
Pos. 2
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.png
(24)
Pos. 1
CDel node n0.pngCDel 2.pngCDel 2.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(32)
Pos. 0
CDel 2.pngCDel 2.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(16)
Alt Cells Faces Edges Vertices
Nonuniform omnisnub tesseract[13]
(Or omnisnub 16-cell)
Snub tesseract verf.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{4,3,3}
(1)
Snub hexahedron.png
(3.3.3.3.4)
(1)
Square antiprism.png
(3.3.3.4)
(1)
Octahedron.png
(3.3.3.3)
(1)
Snub tetrahedron.png
(3.3.3.3.3)
(4)
Tetrahedron.png
(3.3.3)
272 944 864 192

16-cell truncations[edit]

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.pngCDel 2.png
(8)
Pos. 2
CDel node n0.pngCDel 4.pngCDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.png
(24)
Pos. 1
CDel node n0.pngCDel 2.pngCDel 2.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(32)
Pos. 0
CDel 2.pngCDel 2.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(16)
Alt Cells Faces Edges Vertices
[12] 16-cell, hexadecachoron[6] 16-cell verf.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,3,4}
(8)
Tetrahedron.png
(3.3.3)
16 32 24 8
[22] *rectified 16-cell
(Same as 24-cell)
Rectified 16-cell verf.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
r{3,3,4}
(2)
Octahedron.png
(3.3.3.3)
(4)
Octahedron.png
(3.3.3.3)
24 96 96 24
17 truncated 16-cell Truncated 16-cell verf.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t{3,3,4}
(1)
Octahedron.png
(3.3.3.3)
(4)
Truncated tetrahedron.png
(3.6.6)
24 96 120 48
[23] *cantellated 16-cell
(Same as rectified 24-cell)
Cantellated 16-cell verf.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3,4}
(1)
Cuboctahedron.png
(3.4.3.4)
(2)
Hexahedron.png
(4.4.4)
(2)
Cuboctahedron.png
(3.4.3.4)
48 240 288 96
[15] runcinated 16-cell
(also runcinated 8-cell)
Runcinated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,4}
(1)
Hexahedron.png
(4.4.4)
(3)
Hexahedron.png
(4.4.4)
(3)
Triangular prism.png
(3.4.4)
(1)
Tetrahedron.png
(3.3.3)
80 208 192 64
[16] bitruncated 16-cell
(also bitruncated 8-cell)
Bitruncated 8-cell verf.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,3,4}
(2)
Truncated octahedron.png
(4.6.6)
(2)
Truncated tetrahedron.png
(3.6.6)
24 120 192 96
[24] *cantitruncated 16-cell
(Same as truncated 24-cell)
Cantitruncated 16-cell verf.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3,4}
(1)
Truncated octahedron.png
(4.6.6)
(1)
Hexahedron.png
(4.4.4)
(2)
Truncated octahedron.png
(4.6.6)
48 240 384 192
20 runcitruncated 16-cell Runcitruncated 16-cell verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,4}
(1)
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
Hexahedron.png
(4.4.4)
(2)
Hexagonal prism.png
(4.4.6)
(1)
Truncated tetrahedron.png
(3.6.6)
80 368 480 192
[21] omnitruncated 16-cell
(also omnitruncated 8-cell)
Omnitruncated 8-cell verf.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,4}
(1)
Great rhombicuboctahedron.png
(4.6.8)
(1)
Octagonal prism.png
(4.4.8)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Truncated octahedron.png
(4.6.6)
80 464 768 384
[31] alternated cantitruncated 16-cell
(Same as the snub 24-cell)
Snub 24-cell verf.png CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3,4}
(1)
Snub tetrahedron.png
(3.3.3.3.3)
(1)
Tetrahedron.png
(3.3.3)
(2)
Snub tetrahedron.png
(3.3.3.3.3)
(4)
Tetrahedron.png
(3.3.3)
144 480 432 96
Nonuniform Runcic snub rectified 16-cell Runcic snub rectified 16-cell verf.png CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr3{3,3,4}
(1)
Rhombicuboctahedron uniform edge coloring.png
(3.4.4.4)
(2)
Triangular prism.png
(3.4.4)
(1)
Hexahedron.png
(4.4.4)
(1)
Snub tetrahedron.png
(3.3.3.3.3)
(2)
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[edit]

Further information: List of F4 polytopes

This family has diploid icositetrachoric symmetry,[6] [3,4,3], of 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
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 4.pngCDel node n2.pngCDel 2.pngCDel 2.png
(24)
Pos. 2
CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.png
(96)
Pos. 1
CDel node n0.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(96)
Pos. 0
CDel 2.pngCDel 2.pngCDel node n1.pngCDel 4.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(24)
Cells Faces Edges Vertices
22 24-cell, icositetrachoron[6]
(Same as rectified 16-cell)
24 cell verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{3,4,3}
(6)
Octahedron.png
(3.3.3.3)
24 96 96 24
23 rectified 24-cell
(Same as cantellated 16-cell)
Rectified 24-cell verf.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
r{3,4,3}
(3)
Cuboctahedron.png
(3.4.3.4)
(2)
Hexahedron.png
(4.4.4)
48 240 288 96
24 truncated 24-cell
(Same as cantitruncated 16-cell)
Truncated 24-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
t{3,4,3}
(3)
Truncated octahedron.png
(4.6.6)
(1)
Hexahedron.png
(4.4.4)
48 240 384 192
25 cantellated 24-cell Cantellated 24-cell verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{3,4,3}
(2)
Small rhombicuboctahedron.png
(3.4.4.4)
(2)
Triangular prism.png
(3.4.4)
(1)
Cuboctahedron.png
(3.4.3.4)
144 720 864 288
28 cantitruncated 24-cell Cantitruncated 24-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{3,4,3}
(2)
Great rhombicuboctahedron.png
(4.6.8)
(1)
Triangular prism.png
(3.4.4)
(1)
Truncated hexahedron.png
(3.8.8)
144 720 1152 576
29 runcitruncated 24-cell Runcitruncated 24-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,4,3}
(1)
Truncated octahedron.png
(4.6.6)
(2)
Hexagonal prism.png
(4.4.6)
(1)
Triangular prism.png
(3.4.4)
(1)
Small rhombicuboctahedron.png
(3.4.4.4)
240 1104 1440 576
[3+,4,3] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 4.pngCDel node n2.pngCDel 2.pngCDel 2.png
(24)
Pos. 2
CDel node n0.pngCDel 3.pngCDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.png
(96)
Pos. 1
CDel node n0.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(96)
Pos. 0
CDel 2.pngCDel 2.pngCDel node n1.pngCDel 4.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(24)
Alt Cells Faces Edges Vertices
31 snub 24-cell Snub 24-cell verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
s{3,4,3}
(3)
Uniform polyhedron-43-h01.svg
(3.3.3.3.3)
(1)
Tetrahedron.png
(3.3.3)
(4)
Tetrahedron.png
(3.3.3)
144 480 432 96
Nonuniform runcic snub 24-cell Runcic snub 24-cell verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
s3{3,4,3}
(1)
Uniform polyhedron-43-h01.svg
(3.3.3.3.3)
(2)
Triangular prism.png
(3.4.4)
(1)
Truncated tetrahedron.png
(3.6.6)
(3)
Triangular cupola.png
Tricup
240 960 1008 288
[25] cantic snub 24-cell
(Same as cantellated 24-cell)
Cantic snub 24-cell verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
s2{3,4,3}
(2)
Rhombicuboctahedron uniform edge coloring.png
(3.4.4.4)
(1)
Cuboctahedron.png
(3.4.3.4)
(2)
Triangular prism.png
(3.4.4)
144 720 864 288
[29] runcicantic snub 24-cell
(Same as runcitruncated 24-cell)
Runcicantic snub 24-cell verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
s2,3{3,4,3}
(1)
Truncated octahedron.png
(4.6.6)
(1)
Triangular prism.png
(3.4.4)
(1)
Rhombicuboctahedron uniform edge coloring.png
(3.4.4.4)
(2)
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, 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 three asterisked forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry [[3,4,3]]).

[[3,4,3]] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c1.png
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3-0
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel 2.png
CDel 2.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(48)
Pos. 2-1
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.pngCDel node.png
CDel node.pngCDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(192)
Cells Faces Edges Vertices
26 runcinated 24-cell Runcinated 24-cell verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,4,3}
(2)
Octahedron.png
(3.3.3.3)
(6)
Triangular prism.png
(3.4.4)
240 672 576 144
27 bitruncated 24-cell
tetracontoctachoron
Bitruncated 24-cell verf.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,4,3}
(4)
Truncated hexahedron.png
(3.8.8)
48 336 576 288
30 omnitruncated 24-cell Omnitruncated 24-cell verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,4,3}
(2)
Great rhombicuboctahedron.png
(4.6.8)
(2)
Hexagonal prism.png
(4.4.6)
240 1392 2304 1152
[[3,4,3]]+ isogonal 4-polytope
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3-0
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel 2.png
CDel 2.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(48)
Pos. 2-1
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.pngCDel node.png
CDel node.pngCDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(192)
Alt Cells Faces Edges Vertices
Nonuniform omnisnub 24-cell[14] Full snub 24-cell verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{3,4,3}
(2)
Snub hexahedron.png
(3.3.3.3.4)
(2)
Octahedron.png
(3.3.3.3)
(4)
Tetrahedron.png
(3.3.3)
816 2832 2592 576

The H4 family[edit]

Further information: List of H4 polytopes

This family has diploid hexacosichoric symmetry,[6] [5,3,3], of 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[edit]

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node n0.pngCDel 5.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 2.png
(120)
Pos. 2
CDel node n0.pngCDel 5.pngCDel node n1.pngCDel 2.pngCDel 2.pngCDel node n3.png
(720)
Pos. 1
CDel node n0.pngCDel 2.pngCDel 2.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(1200)
Pos. 0
CDel 2.pngCDel node n1.pngCDel 3.pngCDel node n2.pngCDel 3.pngCDel node n3.png
(600)
Alt Cells Faces Edges Vertices
32 120-cell, hecatonicosachoron or dodecacontachoron[6] 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{5,3,3}
(4)
Dodecahedron.png
(5.5.5)
120 720 1200 600
33 rectified 120-cell Rectified 120-cell verf.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
r{5,3,3}
(3)
Icosidodecahedron.png
(3.5.3.5)
(2)
Tetrahedron.png
(3.3.3)
720 3120 3600 1200
36 truncated 120-cell Truncated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t{5,3,3}
(3)
Truncated dodecahedron.png
(3.10.10)
(1)
Tetrahedron.png
(3.3.3)
720 3120 4800 2400
37 cantellated 120-cell Cantellated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{5,3,3}
(1)
Small rhombicosidodecahedron.png
(3.4.5.4)
(2)
Triangular prism.png
(3.4.4)
(1)
Octahedron.png
(3.3.3.3)
1920 9120 10800 3600
38 runcinated 120-cell
(also runcinated 600-cell)
Runcinated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{5,3,3}
(1)
Dodecahedron.png
(5.5.5)
(3)
Pentagonal prism.png
(4.4.5)
(3)
Triangular prism.png
(3.4.4)
(1)
Tetrahedron.png
(3.3.3)
2640 7440 7200 2400
39 bitruncated 120-cell
(also bitruncated 600-cell)
Bitruncated 120-cell verf.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{5,3,3}
(2)
Truncated icosahedron.png
(5.6.6)
(2)
Truncated tetrahedron.png
(3.6.6)
720 4320 7200 3600
42 cantitruncated 120-cell Cantitruncated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{5,3,3}
(2)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Triangular prism.png
(3.4.4)
(1)
Truncated tetrahedron.png
(3.6.6)
1920 9120 14400 7200
43 runcitruncated 120-cell Runcitruncated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{5,3,3}
(1)
Truncated dodecahedron.png
(3.10.10)
(2)
Decagonal prism.png
(4.4.10)
(1)
Triangular prism.png
(3.4.4)
(1)
Cuboctahedron.png
(3.4.3.4)
2640 13440 18000 7200
46 omnitruncated 120-cell
(also omnitruncated 600-cell)
Omnitruncated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{5,3,3}
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Decagonal prism.png
(4.4.10)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Truncated octahedron.png
(4.6.6)
2640 17040 28800 14400
Nonuniform omnisnub 120-cell[15]
(Same as the omnisnub 600-cell)
Snub 120-cell verf.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
ht0,1,2,3{5,3,3}
Snub dodecahedron cw.png (1)
(3.3.3.3.5)
Pentagonal antiprism.png (1)
(3.3.3.5)
Octahedron.png (1)
(3.3.3.3)
Snub tetrahedron.png (1)
(3.3.3.3.3)
Tetrahedron.png (4)
(3.3.3)
9840 35040 32400 7200

600-cell truncations[edit]

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
(120)
Pos. 2
CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png
(720)
Pos. 1
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(1200)
Pos. 0
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(600)
Cells Faces Edges Vertices
35 600-cell, hexacosichoron[6] 600-cell verf.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
{3,3,5}
(20)
Tetrahedron.png
(3.3.3)
600 1200 720 120
[47] 20-diminished 600-cell
(grand antiprism)
Grand antiprism verf.png Nonwythoffian
construction
order 400
(2)
Pentagonal antiprism.png
(3.3.3.5)
(12)
Tetrahedron.png
(3.3.3)
320 720 500 100
[31] 24-diminished 600-cell
(snub 24-cell)
Snub 24-cell verf.png Nonwythoffian
construction
order 576
(3)
Icosahedron.png
(3.3.3.3.3)
(5)
Tetrahedron.png
(3.3.3)
144 480 432 96
Nonuniform bi-24-diminished 600-cell Biicositetradiminished 600-cell vertex figure.png Nonwythoffian
construction
order 144
(6)
Tridiminished icosahedron.png
tdi
48 192 216 72
34 rectified 600-cell Rectified 600-cell verf.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
r{3,3,5}
(2)
Icosahedron.png
(3.3.3.3.3)
(5)
Octahedron.png
(3.3.3.3)
720 3600 3600 720
Nonuniform 120-diminished rectified 600-cell Spidrox-vertex figure.png Nonwythoffian
construction
order 1200
(2)
Pentagonal antiprism.png
3.3.3.5
(2)
Pentagonal prism.png
4.4.5
(5)
Square pyramid.png
P4
840 2640 2400 600
41 truncated 600-cell Truncated 600-cell verf.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t{3,3,5}
(1)
Icosahedron.png
(3.3.3.3.3)
(5)
Truncated tetrahedron.png
(3.6.6)
720 3600 4320 1440
40 cantellated 600-cell Cantellated 600-cell verf.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3,5}
(1)
Icosidodecahedron.png
(3.5.3.5)
(2)
Pentagonal prism.png
(4.4.5)
(1)
Cuboctahedron.png
(3.4.3.4)
1440 8640 10800 3600
[38] runcinated 600-cell
(also runcinated 120-cell)
Runcinated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,5}
(1)
Dodecahedron.png
(5.5.5)
(3)
Pentagonal prism.png
(4.4.5)
(3)
Triangular prism.png
(3.4.4)
(1)
Tetrahedron.png
(3.3.3)
2640 7440 7200 2400
[39] bitruncated 600-cell
(also bitruncated 120-cell)
Bitruncated 120-cell verf.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,3,5}
(2)
Truncated icosahedron.png
(5.6.6)
(2)
Truncated tetrahedron.png
(3.6.6)
720 4320 7200 3600
45 cantitruncated 600-cell Cantitruncated 600-cell verf.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3,5}
(1)
Truncated icosahedron.png
(5.6.6)
(1)
Pentagonal prism.png
(4.4.5)
(2)
Truncated octahedron.png
(4.6.6)
1440 8640 14400 7200
44 runcitruncated 600-cell Runcitruncated 600-cell verf.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,5}
(1)
Small rhombicosidodecahedron.png
(3.4.5.4)
(1)
Pentagonal prism.png
(4.4.5)
(2)
Hexagonal prism.png
(4.4.6)
(1)
Truncated tetrahedron.png
(3.6.6)
2640 13440 18000 7200
[46] omnitruncated 600-cell
(also omnitruncated 120-cell)
Omnitruncated 120-cell verf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,5}
(1)
Great rhombicosidodecahedron.png
(4.6.10)
(1)
Decagonal prism.png
(4.4.10)
(1)
Hexagonal prism.png
(4.4.6)
(1)
Truncated octahedron.png
(4.6.6)
2640 17040 28800 14400

The D4 family[edit]

Further information: List of D4 polytopes

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 Vertex
figure
Coxeter diagram
CD B4 nodes.png
CDel nodes 10ru.pngCDel split2.pngCDel node n2.pngCDel 3.pngCDel node n3.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node n2.pngCDel 3.pngCDel node n3.png
CDel nodes 10ru.pngCDel split2.pngCDel node c1.pngCDel 3.pngCDel node c2.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c2.png
Cell counts by location Element counts
Pos. 0
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(8)
Pos. 2
CDel nodes.pngCDel 2.pngCDel node.png
(24)
Pos. 1
CDel nodes.pngCDel split2.pngCDel node.png
(8)
Pos. 3
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(8)
Pos. Alt
(96)
3 2 1 0
[12] demitesseract
half tesseract
(Same as 16-cell)
16-cell verf.png CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
h{4,3,3}
(4)
Tetrahedron.png
(3.3.3)
(4)
Tetrahedron.png
(3.3.3)
16 32 24 8
[17] cantic tesseract
(Same as truncated 16-cell)
Truncated demitesseract verf.png CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
h2{4,3,3}
(1)
Octahedron.png
(3.3.3.3)
(2)
Truncated tetrahedron.png
(3.6.6)
(2)
Truncated tetrahedron.png
(3.6.6)
24 96 120 48
[11] runcic tesseract
(Same as rectified tesseract)
Cantellated demitesseract verf.png CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
h3{4,3,3}
(1)
Tetrahedron.png
(3.3.3)
(1)
Tetrahedron.png
(3.3.3)
(3)
Cuboctahedron.png
(3.4.3.4)
24 88 96 32
[16] runcicantic tesseract
(Same as bitruncated tesseract)
Cantitruncated demitesseract verf.png CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
h2,3{4,3,3}
(1)
Truncated tetrahedron.png
(3.6.6)
(1)
Truncated tetrahedron.png
(3.6.6)
(2)
Truncated octahedron.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 Vertex
figure
Coxeter diagram
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 3.pngCDel node c1.png = CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png
CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node c2.pngCDel splitsplit1.pngCDel branch3 c1.pngCDel node c1.png
Cell counts by location Element counts
Pos. 0,1,3
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(24)
Pos. 2
CDel nodes.pngCDel 2.pngCDel node.png
(24)
Pos. Alt
(96)
3 2 1 0
[22] rectified 16-cell)
(Same as 24-cell)
Rectified demitesseract verf.png CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png
{31,1,1} = r{3,3,4} = {3,4,3}
(6)
Octahedron.png
(3.3.3.3)
48 240 288 96
[23] cantellated 16-cell
(Same as rectified 24-cell)
Runcicantellated demitesseract verf.png CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
r{31,1,1} = rr{3,3,4} = r{3,4,3}
(3)
Cuboctahedron.png
(3.4.3.4)
(2)
Hexahedron.png
(4.4.4)
24 120 192 96
[24] cantitruncated 16-cell
(Same as truncated 24-cell)
Omnitruncated demitesseract verf.png CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel splitsplit1.pngCDel branch3 11.pngCDel node 1.png
t{31,1,1} = tr{3,3,4} = t{3,4,3}
(3)
Truncated octahedron.png
(4.6.6)
(1)
Hexahedron.png
(4.4.4)
48 240 384 192
[31] snub 24-cell Snub 24-cell verf.png CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h.pngCDel splitsplit1.pngCDel branch3 hh.pngCDel node h.png
s{31,1,1} = sr{3,3,4} = s{3,4,3}
(3)
Snub tetrahedron.png
(3.3.3.3.3)
(1)
Tetrahedron.png
(3.3.3)
(4)
Tetrahedron.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[edit]

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.

# Name Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
47 grand antiprism Grand antiprism.png Grand antiprism verf.png No symbol 300 Tetrahedron.png
(3.3.3)
20 Pentagonal antiprism.png
(3.3.3.5)
320 20 {5}
700 {3}
500 100 Pentagonal double antiprismoid net.png

Prismatic uniform 4-polytopes[edit]

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[edit]

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[edit]

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 Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
48 Tetrahedral prism Tetrahedral prism.png Tetrahedral prism verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{3,3}×{ }
t0,3{3,3,2}
2 Tetrahedron.png
3.3.3
4 Triangular prism.png
3.4.4
6 8 {3}
6 {4}
16 8 Tetrahedron prism net.png
49 Truncated tetrahedral prism Truncated tetrahedral prism.png Truncated tetrahedral prism verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{3,3}×{ }
t0,1,3{3,3,2}
2 Truncated tetrahedron.png
3.6.6
4 Triangular prism.png
3.4.4
4 Hexagonal prism.png
4.4.6
10 8 {3}
18 {4}
8 {6}
48 24 Truncated tetrahedral prism net.png
[[3,3],2] uniform 4-polytopes
# Name Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
[51] Rectified tetrahedral prism
(Same as octahedral prism)
Octahedral prism.png Tetratetrahedral prism verf.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{3,3}×{ }
t1,3{3,3,2}
2 Octahedron.png
3.3.3.3
4 Triangular prism.png
3.4.4
6 16 {3}
12 {4}
30 12 Octahedron prism net.png
[50] Cantellated tetrahedral prism
(Same as cuboctahedral prism)
Cuboctahedral prism.png Cuboctahedral prism verf.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{3,3}×{ }
t0,2,3{3,3,2}
2 Cuboctahedron.png
3.4.3.4
8 Triangular prism.png
3.4.4
6 Hexahedron.png
4.4.4
16 16 {3}
36 {4}
60 24 Cuboctahedral prism net.png
[54] Cantitruncated tetrahedral prism
(Same as truncated octahedral prism)
Truncated octahedral prism.png Truncated octahedral prism verf.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{3,3}×{ }
t0,1,2,3{3,3,2}
2 Truncated octahedron.png
4.6.6
8 Hexagonal prism.png
6.4.4
6 Hexahedron.png
4.4.4
16 48 {4}
16 {6}
96 48 Truncated octahedral prism net.png
[59] Snub tetrahedral prism
(Same as icosahedral prism)
Icosahedral prism.png Snub tetrahedral prism verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
sr{3,3}×{ }
2 Icosahedron.png
3.3.3.3.3
20 Triangular prism.png
3.4.4
22 40 {3}
30 {4}
72 24 Icosahedral prism net.png
Nonuniform omnisnub tetrahedral antiprism Snub 332 verf.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}
2 Icosahedron.png
3.3.3.3.3
8 Octahedron.png
3.3.3.3
6+24 Tetrahedron.png
3.3.3
40 16+96 {3} 96 24

Octahedral prisms: B3 × A1[edit]

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 Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
[10] Cubic prism
(Same as tesseract)
(Same as 4-4 duoprism)
Schlegel wireframe 8-cell.png Cubic prism verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{4,3}×{ }
t0,3{4,3,2}
2 Hexahedron.png
4.4.4
6 Hexahedron.png
4.4.4
8 24 {4} 32 16 8-cell net.png
50 Cuboctahedral prism
(Same as cantellated tetrahedral prism)
Cuboctahedral prism.png Cuboctahedral prism verf.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{4,3}×{ }
t1,3{4,3,2}
2 Cuboctahedron.png
3.4.3.4
8 Triangular prism.png
3.4.4
6 Hexahedron.png
4.4.4
16 16 {3}
36 {4}
60 24 Cuboctahedral prism net.png
51 Octahedral prism
(Same as rectified tetrahedral prism)
(Same as triangular antiprismatic prism)
Octahedral prism.png Tetratetrahedral prism verf.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{3,4}×{ }
t2,3{4,3,2}
2 Octahedron.png
3.3.3.3
8 Triangular prism.png
3.4.4
10 16 {3}
12 {4}
30 12 Octahedron prism net.png
52 Rhombicuboctahedral prism Rhombicuboctahedral prism.png Rhombicuboctahedron prism verf.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{4,3}×{ }
t0,2,3{4,3,2}
2 Small rhombicuboctahedron.png
3.4.4.4
8 Triangular prism.png
3.4.4
18 Hexahedron.png
4.4.4
28 16 {3}
84 {4}
120 48 Small rhombicuboctahedral prism net.png
53 Truncated cubic prism Truncated cubic prism.png Truncated cubic prism verf.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{4,3}×{ }
t0,1,3{4,3,2}
2 Truncated hexahedron.png
3.8.8
8 Triangular prism.png
3.4.4
6 Octagonal prism.png
4.4.8
16 16 {3}
36 {4}
12 {8}
96 48 Truncated cubic prism net.png
54 Truncated octahedral prism
(Same as cantitruncated tetrahedral prism)
Truncated octahedral prism.png Truncated octahedral prism verf.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{3,4}×{ }
t1,2,3{4,3,2}
2 Truncated octahedron.png
4.6.6
6 Hexahedron.png
4.4.4
8 Hexagonal prism.png
4.4.6
16 48 {4}
16 {6}
96 48 Truncated octahedral prism net.png
55 Truncated cuboctahedral prism Truncated cuboctahedral prism.png Truncated cuboctahedral prism verf.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{4,3}×{ }
t0,1,2,3{4,3,2}
2 Great rhombicuboctahedron.png
4.6.8
12 Hexahedron.png
4.4.4
8 Hexagonal prism.png
4.4.6
6 Octagonal prism.png
4.4.8
28 96 {4}
16 {6}
12 {8}
192 96 Great rhombicuboctahedral prism net.png
56 Snub cubic prism Snub cubic prism.png Snub cubic prism verf.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
sr{4,3}×{ }
2 Snub hexahedron.png
3.3.3.3.4
32 Triangular prism.png
3.4.4
6 Hexahedron.png
4.4.4
40 64 {3}
72 {4}
144 48 Snub cuboctahedral prism net.png
[48] Tetrahedral prism Tetrahedral prism.png Tetrahedral prism verf.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
h{4,3}×{ }
2 Tetrahedron.png
3.3.3
4 Triangular prism.png
3.4.4
6 8 {3}
6 {4}
16 8 Tetrahedron prism net.png
[49] Truncated tetrahedral prism Truncated tetrahedral prism.png Truncated tetrahedral prism verf.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
h2{4,3}×{ }
2 Truncated tetrahedron.png
3.3.6
4 Triangular prism.png
3.4.4
4 Hexagonal prism.png
4.4.6
6 8 {3}
6 {4}
16 8 Truncated tetrahedral prism net.png
[50] Cuboctahedral prism Cuboctahedral prism.png Cuboctahedral prism verf.png CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
rr{3,3}×{ }
2 Cuboctahedron.png
3.4.3.4
8 Triangular prism.png
3.4.4
6 Hexahedron.png
4.4.4
16 16 {3}
36 {4}
60 24 Cuboctahedral prism net.png
[52] Rhombicuboctahedral prism Rhombicuboctahedral prism.png Rhombicuboctahedron prism verf.png CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
s2{3,4}×{ }
2 Rhombicuboctahedron uniform edge coloring.png
3.4.4.4
8 Triangular prism.png
3.4.4
18 Hexahedron.png
4.4.4
28 16 {3}
84 {4}
120 48 Small rhombicuboctahedral prism net.png
[54] Truncated octahedral prism Truncated octahedral prism.png Truncated octahedral prism verf.png CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{3,3}×{ }
2 Truncated octahedron.png
4.6.6
6 Hexahedron.png
4.4.4
8 Hexagonal prism.png
4.4.6
16 48 {4}
16 {6}
96 48 Truncated octahedral prism net.png
[59] Icosahedral prism Icosahedral prism.png Snub tetrahedral prism verf.png CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
s{3,4}×{ }
2 Icosahedron.png
3.3.3.3.3
20 Triangular prism.png
3.4.4
22 40 {3}
30 {4}
72 24 Icosahedral prism net.png
[12] 16-cell Schlegel wireframe 16-cell.png 16-cell verf.png CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
s{2,4,3}
2+6+8 Tetrahedron.png
3.3.3.3
16 32 {3} 24 8 16-cell net.png
Nonuniform Omnisnub tetrahedral antiprism Snub 332 verf.png CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
sr{2,3,4}
2 Icosahedron.png
3.3.3.3.3
8 Octahedron.png
3.3.3.3
6+24 Tetrahedron.png
3.3.3
40 16+96 {3} 96 24
Nonuniform Omnisnub cubic antiprism Snub 432 verf.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}
2 Snub hexahedron.png
3.3.3.3.4
12+48 Tetrahedron.png
3.3.3
8 Octahedron.png
3.3.3.3
6 Square antiprism.png
3.3.3.4
76 16+192 {3}
12 {4}
192 48
Nonuniform Runcic snub cubic hosochoron Runcic snub cubic hosochoron.png Runcic snub 243 verf.png CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
s3{2,4,3}
2 Truncated tetrahedron.png
3.6.6
6 Tetrahedron.png
3.3.3
8 Triangular cupola.png
triangular cupola
16 52 60 24 Truncated tetrahedral cupoliprism net.png

Icosahedral prisms: H3 × A1[edit]

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 Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
57 Dodecahedral prism Dodecahedral prism.png Dodecahedral prism verf.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
{5,3}×{ }
t0,3{5,3,2}
2 Dodecahedron.png
5.5.5
12 Pentagonal prism.png
4.4.5
14 30 {4}
24 {5}
80 40 Dodecahedral prism net.png
58 Icosidodecahedral prism Icosidodecahedral prism.png Icosidodecahedral prism verf.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
r{5,3}×{ }
t1,3{5,3,2}
2 Icosidodecahedron.png
3.5.3.5
20 Triangular prism.png
3.4.4
12 Pentagonal prism.png
4.4.5
34 40 {3}
60 {4}
24 {5}
150 60 Icosidodecahedral prism net.png
59 Icosahedral prism
(same as snub tetrahedral prism)
Icosahedral prism.png Snub tetrahedral prism verf.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
{3,5}×{ }
t2,3{5,3,2}
2 Icosahedron.png
3.3.3.3.3
20 Triangular prism.png
3.4.4
22 40 {3}
30 {4}
72 24 Icosahedral prism net.png
60 Truncated dodecahedral prism Truncated dodecahedral prism.png Truncated dodecahedral prism verf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
t{5,3}×{ }
t0,1,3{5,3,2}
2 Truncated dodecahedron.png
3.10.10
20 Triangular prism.png
3.4.4
12 Decagonal prism.png
4.4.5
34 40 {3}
90 {4}
24 {10}
240 120 Truncated dodecahedral prism net.png
61 Rhombicosidodecahedral prism Rhombicosidodecahedral prism.png Rhombicosidodecahedron prism verf.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
rr{5,3}×{ }
t0,2,3{5,3,2}
2 Small rhombicosidodecahedron.png
3.4.5.4
20 Triangular prism.png
3.4.4
30 Hexahedron.png
4.4.4
12 Pentagonal prism.png
4.4.5
64 40 {3}
180 {4}
24 {5}
300 120 Small rhombicosidodecahedral prism net.png
62 Truncated icosahedral prism Truncated icosahedral prism.png Truncated icosahedral prism verf.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{3,5}×{ }
t1,2,3{5,3,2}
2 Truncated icosahedron.png
5.6.6
12 Pentagonal prism.png
4.4.5
20 Hexagonal prism.png
4.4.6
34 90 {4}
24 {5}
40 {6}
240 120 Truncated icosahedral prism net.png
63 Truncated icosidodecahedral prism Truncated icosidodecahedral prism.png Truncated icosidodecahedral prism verf.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{5,3}×{ }
t0,1,2,3{5,3,2}
2 Great rhombicosidodecahedron.png
4.6.10
30 Hexahedron.png
4.4.4
20 Hexagonal prism.png
4.4.6
12 Decagonal prism.png
4.4.10
64 240 {4}
40 {6}
24 {10}
480 240 Great rhombicosidodecahedral prism net.png
64 Snub dodecahedral prism Snub dodecahedral prism.png Snub dodecahedral prism verf.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
sr{5,3}×{ }
2 Snub dodecahedron ccw.png
3.3.3.3.5
80 Triangular prism.png
3.4.4
12 Pentagonal prism.png
4.4.5
94 240 {4}
40 {6}
24 {5}
360 120 Snub icosidodecahedral prism net.png
Nonuniform Omnisnub dodecahedral antiprism Snub 532 verf.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}
2 Snub dodecahedron ccw.png
3.3.3.3.5
30+120 Tetrahedron.png
3.3.3
20 Octahedron.png
3.3.3.3
12 Pentagonal antiprism.png
3.3.3.5
184 20+240 {3}
24 {5}
220 120

Duoprisms: [p] × [q][edit]

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 CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png. Its vertex figure is an disphenoid tetrahedron, Pq-duoprism verf.png.

This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if pq; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.

The elements of a p,q-duoprism (p ≥ 3, q ≥ 3) are:

  • Cells: p q-gonal prisms, q p-gonal prisms
  • Faces: pq squares, p q-gons, q p-gons
  • Edges: 2pq
  • Vertices: pq

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

Infinite set of p-q duoprism - CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png - p q-gonal prisms, q p-gonal prisms:

Name Coxeter graph Cells Images Net
3-3 duoprism CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png 3+3 triangular prisms 3-3 duoprism.png 3-3 duoprism net.png
3-4 duoprism CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png 3 cubes
4 triangular prisms
3-4 duoprism.png 4-3 duoprism.png 4-3 duoprism net.png
4-4 duoprism
(same as tesseract)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png 4+4 cubes 4-4 duoprism.png 8-cell net.png
3-5 duoprism CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png 3 pentagonal prisms
5 triangular prisms
5-3 duoprism.png 3-5 duoprism.png 5-3 duoprism net.png
4-5 duoprism CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png 4 pentagonal prisms
5 cubes
4-5 duoprism.png 5-4 duoprism.png 5-4 duoprism net.png
5-5 duoprism CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png 5+5 pentagonal prisms 5-5 duoprism.png 5-5 duoprism net.png
3-6 duoprism CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png 3 hexagonal prisms
6 triangular prisms
3-6 duoprism.png 6-3 duoprism.png 6-3 duoprism net.png
4-6 duoprism CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png 4 hexagonal prisms
6 cubes
4-6 duoprism.png 6-4 duoprism.png 6-4 duoprism net.png
5-6 duoprism CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png 5 hexagonal prisms
6 pentagonal prisms
5-6 duoprism.png 6-5 duoprism.png 6-5 duoprism net.png
6-6 duoprism CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png 6+6 hexagonal prisms 6-6 duoprism.png 6-6 duoprism net.png
3-3 duoprism.png
3-3
3-4 duoprism.png
3-4
3-5 duoprism.png
3-5
3-6 duoprism.png
3-6
3-7 duoprism.png
3-7
3-8 duoprism.png
3-8
4-3 duoprism.png
4-3
4-4 duoprism.png
4-4
4-5 duoprism.png
4-5
4-6 duoprism.png
4-6
4-7 duoprism.png
4-7
4-8 duoprism.png
4-8
5-3 duoprism.png
5-3
5-4 duoprism.png
5-4
5-5 duoprism.png
5-5
5-6 duoprism.png
5-6
5-7 duoprism.png
5-7
5-8 duoprism.png
5-8
6-3 duoprism.png
6-3
6-4 duoprism.png
6-4
6-5 duoprism.png
6-5
6-6 duoprism.png
6-6
6-7 duoprism.png
6-7
6-8 duoprism.png
6-8
7-3 duoprism.png
7-3
7-4 duoprism.png
7-4
7-5 duoprism.png
7-5
7-6 duoprism.png
7-6
7-7 duoprism.png
7-7
7-8 duoprism.png
7-8
8-3 duoprism.png
8-3
8-4 duoprism.png
8-4
8-5 duoprism.png
8-5
8-6 duoprism.png
8-6
8-7 duoprism.png
8-7
8-8 duoprism.png
8-8

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

The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png - p cubes and 4 p-gonal prisms - (All are the same as 4-p duoprism)

Convex p-gonal prismatic prisms
Name {3}×{}×{} {4}×{}×{} {5}×{}×{} {6}×{}×{} {7}×{}×{} {8}×{}×{} {p}×{}×{}
Coxeter
diagrams
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Image 3-4 duoprism.png
4-3 duoprism.png
4-4 duoprism.png 4-5 duoprism.png
5-4 duoprism.png
4-6 duoprism.png
6-4 duoprism.png
4-7 duoprism.png
7-4 duoprism.png
4-8 duoprism.png
8-4 duoprism.png
Cells 3 {4}×{} Hexahedron.png
4 {3}×{} Triangular prism.png
4 {4}×{} Hexahedron.png
4 {4}×{} Tetragonal prism.png
5 {4}×{} Hexahedron.png
4 {5}×{} Pentagonal prism.png
6 {4}×{} Hexahedron.png
4 {6}×{} Hexagonal prism.png
7 {4}×{} Hexahedron.png
4 {7}×{} Prism 7.png
8 {4}×{} Hexahedron.png
4 {8}×{} Octagonal prism.png
p {4}×{} Hexahedron.png
4 {p}×{}
Net 4-3 duoprism net.png 8-cell net.png 5-4 duoprism net.png 6-4 duoprism net.png 7-4 duoprism net.png 8-4 duoprism net.png

The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥2) - CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png - 2 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
diagram
CDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 10.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 12.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 14.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 7.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 16.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 2x.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Image Digonal antiprismatic prism.png Triangular antiprismatic prism.png Square antiprismatic prism.png Pentagonal antiprismatic prism.png Hexagonal antiprismatic prism.png Heptagonal antiprismatic prism.png Octagonal antiprismatic prism.png 15-gonal antiprismatic prism.png
Vertex
figure
Tetrahedral prism verf.png Tetratetrahedral prism verf.png Square antiprismatic prism verf2.png Pentagonal antiprismatic prism verf.png Hexagonal antiprismatic prism verf.png Heptagonal antiprismatic prism verf.png Octagonal antiprismatic prism verf.png Uniform antiprismatic prism verf.png
Cells 2 s{2,2}
(2) {2}×{}={4}
4 {3}×{}
2 s{2,3}
2 {3}×{}
6 {3}×{}
2 s{2,4}
2 {4}×{}
8 {3}×{}
2 s{2,5}
2 {5}×{}
10 {3}×{}
2 s{2,6}
2 {6}×{}
12 {3}×{}
2 s{2,7}
2 {7}×{}
14 {3}×{}
2 s{2,8}
2 {8}×{}
16 {3}×{}
2 s{2,p}
2 {p}×{}
2p {3}×{}
Net Tetrahedron prism net.png Octahedron prism net.png 4-antiprismatic prism net.png 5-antiprismatic prism net.png 6-antiprismatic prism net.png 7-antiprismatic prism net.png 8-antiprismatic prism net.png 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[edit]

Like the 3-dimensional snub cube, CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png, an alternation removes half the vertices, in two chiral sets of vertices from the ringed form CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png, however the 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 noted only two uniform solutions for rank 4 Coxeter groups with all rings alternated. The first is CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png, s{21,1,1} which represented an index 24 subgroup (symmetry [2,2,2]+, order 8) form of the demitesseract, CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, h{4,3,3} (symmetry [1+,4,3,3] = [31,1,1], order 192). The second is CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png, s{31,1,1}, which is an index 6 subgroup (symmetry [31,1,1]+, order 96) form of the snub 24-cell, CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, s{3,4,3}, (symmetry [3+,4,3], order 576).

Other alternations, such as CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png, as an alternation from the omnitruncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png, can not be made uniform as solving for equal edge lengths are in general overdetermined (there are six equations but only four variables). Such nonuniform alternated figures can be constructed as vertex-transitive 4-polytopes by the removal of one of two chiral half set 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.[16]

Geometric derivations for 46 nonprismatic Wythoffian uniform polychora[edit]

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.

Polychoron truncation chart.png
Summary chart of truncation operations
Uniform honeycomb truncations.png
Example locations of kaleidoscopic generator point on fundamental domain.

The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes are truncating operations. A 4-polytope may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.

The Coxeter-Dynkin diagram shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors (π/n radians or 180/n degrees). Circled nodes show which mirrors are active for each form; a mirror is active with respect to a vertex that does not lie on it.

Operation Schläfli symbol Symmetry Coxeter diagram Description
Parent t0{p,q,r} [p,q,r] CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Original regular form {p,q,r}
Rectification t1{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Truncation operation applied until the original edges are degenerated into points.
Birectification
(Rectified dual)
t2{p,q,r} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Face are fully truncated to points. Same as rectified dual.
Trirectification
(dual)
t3{p,q,r} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Cells are truncated to points. Regular dual {r,q,p}
Truncation t0,1{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel 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} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png A truncation between a rectified form and the dual rectified form.
Tritruncation t2,3{p,q,r} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Truncated dual {r,q,p}.
Cantellation t0,2{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel 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} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Cantellated dual {r,q,p}.
Runcination
(or expansion)
t0,3{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel 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} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Both the cantellation and truncation operations applied together.
Bicantitruncation t1,2,3{p,q,r} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Cantitruncated dual {r,q,p}.
Runcitruncation t0,1,3{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Both the runcination and truncation operations applied together.
Runcicantellation t0,1,3{p,q,r} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Runcitruncated dual {r,q,p}.
Omnitruncation
(runcicantitruncation)
t0,1,2,3{p,q,r} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Application of all three operators.
Half h{2p,3,q} [1+,2p,3,q]
=[(3,p,3),q]
CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel q.pngCDel node.png Alternation of CDel node 1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel q.pngCDel node.png, same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel q.pngCDel node.png
Cantic h2{2p,3,q} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel q.pngCDel node.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel q.pngCDel node.png
Runcic h3{2p,3,q} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel q.pngCDel node 1.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel q.pngCDel node 1.png
Runcicantic h2,3{2p,3,q} CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel q.pngCDel node 1.png Same as CDel labelp.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel q.pngCDel node 1.png
Quarter q{2p,3,2q} [1+,2p,3,2r,1+] CDel node h1.pngCDel 2x.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2x.pngCDel q.pngCDel node h1.png Same as CDel labelp.pngCDel branch 10r.pngCDel splitcross.pngCDel branch 01l.pngCDel labelq.png
Snub s{p,2q,r} [p+,2q,r] CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node.pngCDel r.pngCDel node.png Alternated truncation
Cantic snub s2{p,2q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.png Cantellated alternated truncation
Runcic snub s3{p,2q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.png Runcinated alternated truncation
Runcicantic snub s2,3{p,2q,r} CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.png Runcicantellated alternated truncation
Snub rectified sr{p,q,2r} [(p,q)+,2r] CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel 2x.pngCDel r.pngCDel node.png Alternated truncated rectification
ht0,3{2p,q,2r} [(2p,q,2r,2+)] CDel node h.pngCDel 2x.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel 2x.pngCDel r.pngCDel node h.png Alternated runcination
Bisnub 2s{2p,q,2r} [2p,q+,2r] CDel node.pngCDel 2x.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel 2x.pngCDel r.pngCDel node.png Alternated bitruncation
Omnisnub ht0,1,2,3{p,q,r} [p,q,r]+ CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel r.pngCDel 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[edit]

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. Alternations are grouped by their chiral symmetry. All alternations are given, although the snub 24-cell, with its 3 family of constructions is the only one that is uniform. Counts in parenthesis are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46. The 3-3 and 4-4 duoprismatic family is included, the second for its relation to the B4 family.

Coxeter group Extended
symmetry
Polychora Chiral
extended
symmetry
Alternation honeycombs
[3,3,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3,3,3]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png
(order 120)
6 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png1 | CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png2 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png4 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png7 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png8
[2+[3,3,3]]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png
(order 240)
3 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png5| CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png6 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png9 [2+[3,3,3]]+
(order 120)
(1) CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png-
[3,31,1]
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
[3,31,1]
CDel node c3.pngCDel 3.pngCDel node c4.pngCDel split1.pngCDel nodeab c1-2.png
(order 192)
0 (none)
[1[3,31,1]]=[4,3,3]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png = CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node.png
(order 384)
(4) CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png12 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png17 | CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png11 | CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png16
[3[31,1,1]]=[3,4,3]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png = CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(order 1152)
(3) CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png22 | CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png23 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png24 [3[3,31,1]]+
=[3,4,3]+
(order 576)
(1) CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png31 (= CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png-
[4,3,3]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[3[1+,4,3,3]]=[3,4,3]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c1.png = CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(order 1152)
(3) CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png22 | CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png23 | CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png24
[4,3,3]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png
(order 384)
12 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png10 | CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png11 | CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png12 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png13 | CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png14
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png15 | CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png16 | CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png17 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png18 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png19
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png20 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png21
[1+,4,3,3]+
(order 96)
(2) CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png12 (= CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png)
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png31
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png-
[4,3,3]+
(order 192)
(1) CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png-
[3,4,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[3,4,3]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel 3.pngCDel node c4.png
(order 1152)
6 CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png22 | CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png23 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png24
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png25 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png28 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png29
[2+[3+,4,3+]]
(order 576)
1 CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png31
[2+[3,4,3]]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c1.png
(order 2304)
3 CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png26 | CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png27 | CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png30 [2+[3,4,3]]+
(order 1152)
(1) CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png-
[5,3,3]
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[5,3,3]
CDel node c1.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 3.pngCDel node c4.png
(order 14400)
15 CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png32 | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png33 | CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png34 | CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png35 | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png36
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png37 | CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png38 | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png39 | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png40 | CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png41
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png42 | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png43 | CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png44 | CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png45 | CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png46
[5,3,3]+
(order 7200)
(1) CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png-
[3,2,3]
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
[3,2,3]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 3.pngCDel node c3.png
(order 36)
0 (none) [3,2,3]+
(order 18)
0 (none)
[2+[3,2,3]]
CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel 3.pngCDel node c1.png
(order 72)
0 CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png [2+[3,2,3]]+
(order 36)
0 (none)
[[3],2,3]=[6,2,3]
CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 3.pngCDel node c3.png = CDel node c1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 3.pngCDel node c3.png
(order 72)
1 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png [1[3,2,3]]=[[3],2,3]+=[6,2,3]+
(order 36)
(1) CDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
[(2+,4)[3,2,3]]=[2+[6,2,6]]
CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 3.pngCDel node c1.png = CDel node c1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node c1.pngCDel 6.pngCDel node.png
(order 288)
1 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png [(2+,4)[3,2,3]]+=[2+[6,2,6]]+
(order 144)
(1) CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
[4,2,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
[4,2,4]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c3.pngCDel 4.pngCDel node c4.png
(order 64)
0 (none) [4,2,4]+
(order 32)
0 (none)
[2+[4,2,4]]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node c1.png
(order 128)
0 (none) [2+[(4,2+,4,2+)]]
(order 64)
0 (none)
[(3,3)[4,2*,4]]=[4,3,3]
CDel node c1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(order 384)
(1) CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png10 [(3,3)[4,2*,4]]+=[4,3,3]+
(order 192)
(1) CDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png12
[[4],2,4]=[8,2,4]
CDel node c1.pngCDel 4.pngCDel node c1.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node c3.png = CDel node c1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node c2.pngCDel 4.pngCDel node c3.png
(order 128)
(1) CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png [1[4,2,4]]=[[4],2,4]+=[8,2,4]+
(order 64)
(1) CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
[(2+,4)[4,2,4]]=[2+[8,2,8]]
CDel node c1.pngCDel 4.pngCDel node c1.pngCDel 2.pngCDel node c1.pngCDel 4.pngCDel node c1.png = CDel node c1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node c1.pngCDel 8.pngCDel node.png
(order 512)
(1) CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png [(2+,4)[4,2,4]]+=[2+[8,2,8]]+
(order 256)
(1) CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png

Symmetries in four dimensions[edit]

There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, BC4: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, D4: CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, F4: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, H4: CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, and I2(p)×I2(q) as CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.[6]

See also[edit]

Notes[edit]

  1. ^ T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  2. ^ http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf
  3. ^ Elte (1912)
  4. ^ https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html December 6, 1998 oldest archive
  5. ^ The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes By David Darling, (2004) ASIN: B00SB4TU58
  6. ^ a b c d e f g h i j k Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups
  7. ^ Uniform Polytopes in Four Dimensions, George Olshevsky.
  8. ^ 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope (German)
  9. ^ Conway (2008)
  10. ^ [1] Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract
  11. ^ Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135
  12. ^ http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm
  13. ^ http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm
  14. ^ http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm
  15. ^ http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm
  16. ^ 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

References[edit]

  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 1-4181-7968-X  [3] [4]
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
  • Schoute, Pieter Hendrik (1911), "Analytic treatment of the polytopes regularly derived from the regular polytopes", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam 11 (3): 87 pp.  Googlebook, 370-381
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980 p92, p122.
  • J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • N.W. Johnson: Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups
  • B. Grünbaum Convex polytopes, New York ; London : Springer, c2003. ISBN 0-387-00424-6.
    Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
  • Richard Klitzing, Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010) [5]

External links[edit]