Uniform polychoron

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Schlegel diagram for the truncated 120-cell with tetrahedral cells visible. This perspective projection makes edges look smaller towards the center of the projection.
orthographic projection of the truncated 120-cell, in the H3 Coxeter plane (D10 symmetry). Only vertices and edges are drawn.

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

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

History of discovery[edit]

  • Regular polytopes: (convex faces)
    • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
  • Regular star-polychora (star polyhedron cells and/or vertex figures)
    • 1852: Ludwig Schläfli also found 4 of the 10 regular star polychora, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
    • 1883: Edmund Hess completed the list of 10 of the nonconvex regular polychora, in his book (in German) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [3].
  • 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 polychora.[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 polychora:
      • 1965: The complete list of convex forms was finally done by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex polychoron, the grand antiprism.
      • 1966 N.W. Johnson completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher
      • 1997: A complete enumeration of the names and elements of the convex uniform polychora is given online by George Olshevsky.[4]
      • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope.[5]
  • Nonregular uniform star polychora: (similar to the nonconvex uniform polyhedra)
    • Ongoing: Thousands of nonconvex uniform polychora are known, but mostly unpublished. The list is presumed not to be complete, and there is no estimate of how long the complete list will be, although 1849 convex and nonconvex uniform polychora are currently known.[6] Participating researchers include Jonathan Bowers,[7] George Olshevsky and Norman Johnson.[8]

Regular polychora[edit]

The uniform polychora include two special subsets, which satisfy additional requirements:

Convex uniform polychora[edit]

Enumeration[edit]

There are 64 convex uniform polychora, including the 6 regular convex polychora, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms.

  • 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
  • 13 are polyhedral prisms based on the Archimedean solids
  • 9 are in the self-dual regular A4 [3,3,3] group (5-cell) family.
  • 9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
  • 15 are in the regular BC4 [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
  • 15 are in the regular H4 [3,3,5] group (120-cell/600-cell) family.
  • 1 special snub form in the [3,4,3] group (24-cell) family.
  • 1 special non-Wythoffian polychoron, the grand antiprism.
  • TOTAL: 68 − 4 = 64

These 64 uniform polychora are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.

In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:

The A4 family[edit]

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

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

[3,3,3] uniform polytopes
# Johnson Name
Bowers name (and acronym)
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 (pen)
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 pentachoron (rap)
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 pentachoron (tip)
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
Small rhombated pentachoron (srip)
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
Great rhombated pentachoron (grip)
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
Prismatotrhombated pentachoron (prip)
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
# Johnson Name
Bowers name (and acronym)
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
Small prismatodecachoron (spid)
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 (deca)
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
Great prismatodecachoron (gippid)
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 (snip)[9] 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 polychora 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.

Graphs[edit]

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

The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

# Johnson Name
Bowers name (and acronym)
Coxeter diagram
and Schläfli
symbols
Coxeter plane graphs Schlegel
diagram
A4
[5]
A3
[4]
A2
[3]
Tetrahedron
centered
Dual tetrahedron
centered
1 5-cell
Pentachoron (pen)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{3,3,3}
4-simplex t0.svg 4-simplex t0 A3.svg 4-simplex t0 A2.svg Schlegel wireframe 5-cell.png
2 rectified 5-cell
Rectified pentachoron (rap)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
r{3,3,3}
4-simplex t1.svg 4-simplex t1 A3.svg 4-simplex t1 A2.svg Schlegel half-solid rectified 5-cell.png
3 truncated 5-cell
Truncated pentachoron (tip)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t{3,3,3}
4-simplex t01.svg 4-simplex t01 A3.svg 4-simplex t01 A2.svg Schlegel half-solid truncated pentachoron.png
4 cantellated 5-cell
Small rhombated pentachoron (srip)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
rr{3,3,3}
4-simplex t02.svg 4-simplex t02 A3.svg 4-simplex t02 A2.svg Schlegel half-solid cantellated 5-cell.png
5 *runcinated 5-cell
Small prismatodecachoron (spid)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,3}
4-simplex t03.svg 4-simplex t03 A3.svg 4-simplex t03 A2.svg Schlegel half-solid runcinated 5-cell.png
6 *bitruncated 5-cell
Decachoron (deca)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2t{3,3,3}
4-simplex t12.svg 4-simplex t12 A3.svg 4-simplex t12 A2.svg Schlegel half-solid bitruncated 5-cell.png
7 cantitruncated 5-cell
Great rhombated pentachoron (grip)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{3,3,3}
4-simplex t012.svg 4-simplex t012 A3.svg 4-simplex t012 A2.svg Schlegel half-solid cantitruncated 5-cell.png
8 runcitruncated 5-cell
Prismatotrhombated pentachoron (prip)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,3}
4-simplex t013.svg 4-simplex t013 A3.svg 4-simplex t013 A2.svg Schlegel half-solid runcitruncated 5-cell.png
9 *omnitruncated 5-cell
Great prismatodecachoron (gippid)
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}
4-simplex t0123.svg 4-simplex t0123 A3.svg 4-simplex t0123 A2.svg Schlegel half-solid omnitruncated 5-cell.png

Coordinates[edit]

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

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

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

The BC4 family[edit]

This family has diploid hexadecachoric symmetry, [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 polychora which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.

Tesseract truncations[edit]

# Johnson Name
(Bowers style acronym)
Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.png
(8)
Pos. 2
CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node.png
(24)
Pos. 1
CDel node.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(32)
Pos. 0
CDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(16)
Alt Cells Faces Edges Vertices
10 tesseract or (tes)
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 (rit) 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 (tat) 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 (srit) 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) (sidpith)
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) (tah)
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 (grit) 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 (proh) 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) (gidpith)
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
12 Demitesseract
16-cell (hex)
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}
(4)
Tetrahedron.png
(3.3.3)
(4)
Tetrahedron.png
(3.3.3)
16 32 24 8
[17] Cantic tesseract 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}
(4)
Truncated tetrahedron.png
(6.6.3)
(1)
Octahedron.png
(3.3.3.3)
24 96 120 48
[11] Runcic 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}
(3)
Cuboctahedron.png
(3.4.3.4)
(2)
Tetrahedron.png
(3.3.3)
24 88 96 32
[16] Runcicantic 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}
(2)
Truncated octahedron.png
(3.4.3.4)
(2)
Truncated tetrahedron.png
(3.6.6)
24 96 96 24
Nonuniform omnisnub tesseract (snet)[10]
(Same as the 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]

# Johnson Name (Bowers style acronym) Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(8)
Pos. 2
CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png
(24)
Pos. 1
CDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(32)
Pos. 0
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(16)
Alt Cells Faces Edges Vertices
[12] 16-cell (hex) 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) (ico)
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 (thex) 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) (rico)
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) (sidpith)
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) (tah)
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) (tico)
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 (prit) 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) (gidpith)
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) (sadi)
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.

Graphs[edit]

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.

# Johnson Name
(Bowers style acronym)
Coxeter plane projections Schlegel
diagrams
F4
[12/3]
B4
[8]
B3
[6]
B2
[4]
A3
[4]
Cube
centered
Tetrahedron
centered
10 8-cell
or tesseract (tes)
4-cube t0 F4.svg 4-cube t0.svg 4-cube t0 B3.svg 4-cube t0 B2.svg 4-cube t0 A3.svg Schlegel wireframe 8-cell.png
11 rectified 8-cell (rit) 4-cube t1 F4.svg 4-cube t1.svg 4-cube t1 B3.svg 4-cube t1 B2.svg 4-cube t1 A3.svg Schlegel half-solid rectified 8-cell.png
12 16-cell (hex) 4-cube t3 F4.svg 4-cube t3.svg 4-cube t3 B3.svg 4-cube t3 B2.svg 4-cube t3 A3.svg Schlegel wireframe 16-cell.png
13 truncated 8-cell (tat) 4-cube t01 F4.svg 4-cube t01.svg 4-cube t01 B3.svg 4-cube t01 B2.svg 4-cube t01 A3.svg Schlegel half-solid truncated tesseract.png
14 cantellated 8-cell (srit) 4-cube t02 F4.svg 4-cube t02.svg 4-cube t02 B3.svg 4-cube t02 B2.svg 4-cube t02 A3.svg Schlegel half-solid cantellated 8-cell.png
15 runcinated 8-cell
(also runcinated 16-cell) (sidpith)
4-cube t03 F4.svg 4-cube t03.svg 4-cube t03 B3.svg 4-cube t03 B2.svg 4-cube t03 A3.svg Schlegel half-solid runcinated 8-cell.png Schlegel half-solid runcinated 16-cell.png
16 bitruncated 8-cell
(also bitruncated 16-cell) (tah)
4-cube t12 F4.svg 4-cube t12.svg 4-cube t12 B3.svg 4-cube t12 B2.svg 4-cube t12 A3.svg Schlegel half-solid bitruncated 8-cell.png Schlegel half-solid bitruncated 16-cell.png
17 truncated 16-cell (thex) 4-cube t23 F4.svg 4-cube t23.svg 4-cube t23 B3.svg 4-cube t23 B2.svg 4-cube t23 A3.svg Schlegel half-solid truncated 16-cell.png
18 cantitruncated 8-cell (grit) 4-cube t012 F4.svg 4-cube t012.svg 4-cube t012 B3.svg 4-cube t012 B2.svg 4-cube t012 A3.svg Schlegel half-solid cantitruncated 8-cell.png
19 runcitruncated 8-cell (proh) 4-cube t013 F4.svg 4-cube t013.svg 4-cube t013 B3.svg 4-cube t013 B2.svg 4-cube t013 A3.svg Schlegel half-solid runcitruncated 8-cell.png
20 runcitruncated 16-cell (prit) 4-cube t023 F4.svg 4-cube t023.svg 4-cube t023 B3.svg 4-cube t023 B2.svg 4-cube t023 A3.svg Schlegel half-solid runcitruncated 16-cell.png
21 omnitruncated 8-cell
(also omnitruncated 16-cell) (gidpith)
4-cube t0123 F4.svg 4-cube t0123.svg 4-cube t0123 B3.svg 4-cube t0123 B2.svg 4-cube t0123 A3.svg Schlegel half-solid omnitruncated 8-cell.png Schlegel half-solid omnitruncated 16-cell.png
[22] *rectified 16-cell
(Same as 24-cell) (ico)
4-cube t2 F4.svg 4-cube t2.svg 4-cube t2 B3.svg 4-cube t2 B2.svg 4-cube t2 A3.svg Schlegel half-solid rectified 16-cell.png
[23] *cantellated 16-cell
(Same as rectified 24-cell) (rico)
4-cube t13 F4.svg 4-cube t13.svg 4-cube t13 B3.svg 4-cube t13 B2.svg 4-cube t13 A3.svg Schlegel half-solid cantellated 16-cell.png
[24] *cantitruncated 16-cell
(Same as truncated 24-cell) (tico)
4-cube t123 F4.svg 4-cube t123.svg 4-cube t123 B3.svg 4-cube t123 B2.svg 4-cube t123 A3.svg Schlegel half-solid cantitruncated 16-cell.png
[31] alternated cantitruncated 16-cell
(Same as the snub 24-cell) (sadi)
24-cell h01 F4.svg 24-cell h01 B4.svg 24-cell h01 B3.svg 24-cell h01 B2.svg Schlegel half-solid alternated cantitruncated 16-cell.png

Coordinates[edit]

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

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

The F4 family[edit]

This family has diploid icositetrachoric symmetry, [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 polychora which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576.

[3,4,3] uniform polychora
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel 2.png
(24)
Pos. 2
CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel 2.pngCDel node.png
(96)
Pos. 1
CDel node.pngCDel 2.pngCDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(96)
Pos. 0
CDel 2.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(24)
Alt Cells Faces Edges Vertices
22 24-cell
(Same as rectified 16-cell) (ico)
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) (rico)
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) (tico)
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 (srico) 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 (grico) 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 (prico) 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
31 snub 24-cell (sadi) 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)
Icosahedron.png
(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 (prissi) 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)
Icosahedron.png
(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 24-cell
(Same as cantellated 24-cell) (srico)
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 24-cell
(Same as runcitruncated 24-cell) (prico)
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]).
[[3,4,3]] uniform polychora
# 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)
Alt Cells Faces Edges Vertices
26 *runcinated 24-cell (spic) 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 (cont) 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 (gippic) 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
Nonuniform omnisnub 24-cell (snico)[11] 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
(*) 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]]).

Graphs[edit]

# Name
Coxeter diagram
Schläfli symbol
Graph
Schlegel
diagram
Orthogonal
Projection
F4
[12]
B4
[8]
B3
[6]
B2
[4]
Octahedron
centered
Dual octahedron
centered
Octahedron
centered
22 24-cell (ico)
(rectified 16-cell)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{3,4,3}
24-cell t0 F4.svg 24-cell t0 B4.svg 24-cell t0 B3.svg 24-cell t0 B2.svg Schlegel wireframe 24-cell.png
23 rectified 24-cell (rico)
(cantellated 16-cell)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
t1{3,4,3}
24-cell t1 F4.svg 24-cell t1 B4.svg 24-cell t1 B3.svg 24-cell t1 B2.svg Schlegel half-solid cantellated 16-cell.png
24 truncated 24-cell (tico)
(cantitruncated 16-cell)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
t0,1{3,4,3}
24-cell t01 F4.svg 24-cell t01 B4.svg 24-cell t01 B3.svg 24-cell t01 B2.svg Schlegel half-solid truncated 24-cell.png
25 cantellated 24-cell (srico)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,2{3,4,3}
24-cell t02 F4.svg 24-cell t02 B4.svg 24-cell t02 B3.svg 24-cell t02 B2.svg Cantel 24cell1.png
26 *runcinated 24-cell (spic)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,4,3}
24-cell t03 F4.svg 24-cell t03 B4.svg 24-cell t03 B3.svg 24-cell t03 B2.svg Runcinated 24-cell Schlegel halfsolid.png
27 *bitruncated 24-cell (cont)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2{3,4,3}
24-cell t12 F4.svg 24-cell t12 B4.svg 24-cell t12 B3.svg 24-cell t12 B2.svg Bitruncated 24-cell Schlegel halfsolid.png
28 cantitruncated 24-cell (grico)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,2{3,4,3}
24-cell t012 F4.svg 24-cell t012 B4.svg 24-cell t012 B3.svg 24-cell t012 B2.svg Cantitruncated 24-cell schlegel halfsolid.png
29 runcitruncated 24-cell (prico)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,3{3,4,3}
24-cell t013 F4.svg 24-cell t013 B4.svg 24-cell t013 B3.svg 24-cell t013 B2.svg Runcitruncated 24-cell.png
30 *omnitruncated 24-cell (gippic)
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}
24-cell t0123 F4.svg 24-cell t0123 B4.svg 24-cell t0123 B3.svg 24-cell t0123 B2.svg Omnitruncated 24-cell.png
31 snub 24-cell (sadi)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
s{3,4,3}
24-cell h01 F4.svg 24-cell h01 B4.svg 24-cell h01 B3.svg 24-cell h01 B2.svg Schlegel half-solid alternated cantitruncated 16-cell.png Ortho solid 969-uniform polychoron 343-snub.png
- Runcic snub 24-cell (prissi)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
s3{3,4,3}
Runcic snub 24-cell.png

Coordinates[edit]

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

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

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

The H4 family[edit]

This family has diploid hexacosichoric symmetry, [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]

# Johnson Name
(Bowers style Acronym)
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.pngCDel 2.png
(120)
Pos. 2
CDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node.png
(720)
Pos. 1
CDel node.pngCDel 2.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
(1200)
Pos. 0
CDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
(600)
Alt Cells Faces Edges Vertices
32 120-cell (hi) 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 (rahi) 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 (thi) 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 (srahi) 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) (sidpixhi)
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) (xhi)
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 (grahi) 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 (prix) 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) (gidpixhi)
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 (snahi)[12]
(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]

# Johnson Name
(Bowers style acronym)
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 (ex) 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
34 rectified 600-cell (rox) 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
41 truncated 600-cell (tex) 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 (srix) 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) (sidpixhi)
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) (xhi)
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 (grix) 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 (prahi) 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) (gidpixhi)
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

Graphs[edit]

# Johnson Name
(Bowers style Acronym)
Coxeter plane projections Schlegel diagrams
F4
[12]
[20] H4
[30]
H3
[10]
A3
[4]
A2
[3]
Dodecahedron
centered
Tetrahedron
centered
32 120-cell (hi) 120-cell t0 F4.svg 120-cell t0 p20.svg 120-cell graph H4.svg 120-cell t0 H3.svg 120-cell t0 A3.svg 120-cell t0 A2.svg Schlegel wireframe 120-cell.png
33 rectified 120-cell (rahi) 120-cell t1 F4.svg 120-cell t1 p20.svg 120-cell t1 H4.svg 120-cell t1 H3.svg 120-cell t1 A3.svg 120-cell t1 A2.svg Rectified 120-cell schlegel halfsolid.png
34 rectified 600-cell (rox) 600-cell t1 F4.svg 600-cell t1 p20.svg 600-cell t1 H4.svg 600-cell t1 H3.svg 600-cell t1.svg 600-cell t1 A2.svg Rectified 600-cell schlegel halfsolid.png
35 600-cell (ex) 600-cell t0 F4.svg 600-cell t0 p20.svg 600-cell graph H4.svg 600-cell t0 H3.svg 600-cell t0.svg 600-cell t0 A2.svg Schlegel wireframe 600-cell vertex-centered.png Stereographic polytope 600cell.png
36 truncated 120-cell (thi) 120-cell t01 F4.svg 120-cell t01 p20.svg 120-cell t01 H4.svg 120-cell t01 H3.svg 120-cell t01 A3.svg 120-cell t01 A2.svg Schlegel half-solid truncated 120-cell.png
37 cantellated 120-cell (srahi) 120-cell t02 H3.png 120-cell t02 B3.png Cantellated 120 cell center.png
38 runcinated 120-cell
(also runcinated 600-cell) (sidpixhi)
120-cell t03 H3.png 120-cell t03 B3.png Runcinated 120-cell.png
39 bitruncated 120-cell
(also bitruncated 600-cell) (xhi)
120-cell t12 H3.png 120-cell t12 A3.png 120-cell t12 B3.png Bitruncated 120-cell schlegel halfsolid.png
40 cantellated 600-cell (srix) 600-cell t02 F4.svg 600-cell t02 p20.svg 600-cell t02 H4.svg 600-cell t02 H3.svg 600-cell t02 B2.svg 600-cell t02 B3.svg Cantellated 600 cell center.png
41 truncated 600-cell (tex) 600-cell t01 F4.svg 600-cell t01 p20.svg 600-cell t01 H4.svg 600-cell t01 H3.svg 600-cell t01.svg 600-cell t01 A2.svg Schlegel half-solid truncated 600-cell.png
42 cantitruncated 120-cell (grahi) 120-cell t012 H3.png 120-cell t012 B3.png Cantitruncated 120-cell.png
43 runcitruncated 120-cell (prix) 120-cell t013 H3.png 120-cell t013 B3.png Runcitruncated 120-cell.png
44 runcitruncated 600-cell (prahi) 120-cell t023 H3.png 120-cell t023 B3.png Runcitruncated 600-cell.png
45 cantitruncated 600-cell (grix) 120-cell t123 H3.png 120-cell t123 B3.png Cantitruncated 600-cell.png
46 omnitruncated 120-cell
(also omnitruncated 600-cell) (gidpixhi)
120-cell t0123 H3.png 120-cell t0123 B3.png Omnitruncated 120-cell wireframe.png

The D4 family[edit]

This demitesseract family, [31,1,1], introduces no new uniform polychora, but it is worthy to repeat these alternative constructions. This family has order 12*16=192: 4!/2=12 permutations of the four axes, half as alternated, 24=16 for reflection in each axis. There is one small index subgroups that generating uniform polychora, [31,1,1]+, order 96.

[31,1,1] uniform polychora
# Johnson Name (Bowers style acronym) Vertex
figure
Coxeter diagram
CD B4 nodes.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) (hex)
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{4,3,4}
(4)
Tetrahedron.png
(3.3.3)
(4)
Tetrahedron.png
(3.3.3)
16 32 24 8
[17] Cantic tesseract
(Same as truncated 16-cell) (thex)
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) (rit)
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) (tah)
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 polychora
# Johnson Name (Bowers style acronym) 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) (ico)
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) (rico)
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) (tico)
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 (sadi) 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 polychoron, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron.

Graphs[edit]

# Johnson Name (Bowers style acronym)
Coxeter diagram
Coxeter plane projections Schlegel diagrams Parallel
3D
B4
[8/2]
D4
[6]
D3
[2]
Cube
centered
Tetrahedron
centered
D4
[6]
[12] demitesseract
(Same as 16-cell) (hex)
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
h{4,3,3} = {3,31,1}
4-demicube t0 B4.svg 4-demicube t0 D4.svg 4-demicube t0 D3.svg Schlegel wireframe 16-cell.png
[17] truncated demitesseract (thex)
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
h2{4,3,3} = t{3,31,1}
4-demicube t01 B4.svg 4-demicube t01 D4.svg 4-demicube t01 D3.svg Schlegel half-solid truncated 16-cell.png
[11] birectified demitesseract
(Same as rectified tesseract) (rit)
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png or CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.png
h3{4,3,3} = 2r{3,31,1}
4-demicube t02 B4.svg 4-demicube t02 D4.svg 4-demicube t02 D3.svg Schlegel half-solid rectified 8-cell.png
[16] bitruncated demitesseract
(Same as bitruncated tesseract) (tah)
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png or CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png
h2,3{4,3,3} = 2t{3,31,1}
4-demicube t012 B4.svg 4-demicube t012 D4.svg 4-demicube t012 D3.svg Schlegel half-solid bitruncated 16-cell.png
[22] rectified demitesseract
(Same as 24-cell) (ico)
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
{31,1,1}
4-cube t2.svg 4-demicube t1 D4.svg 4-demicube t1 D3.svg Schlegel wireframe 24-cell.png
[23] Cantellated demitesseract
(Same as rectified 24-cell) (rico)
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png
r{31,1,1}
4-cube t02.svg 4-demicube t023 D4.svg 4-demicube t023 D3.svg Schlegel half-solid cantellated 16-cell.png
[24] cantitruncated demitesseract
(Same as truncated 24-cell) (tico)
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t{31,1,1} = tr{3,31,1}
4-cube t012.svg 4-demicube t0123 D4.svg 4-demicube t0123 D3.svg Schlegel half-solid truncated 24-cell.png
[31] Snub demitesseract
(snub 24-cell) (sadi)
CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 3.pngCDel node h.png
s{31,1,1} = sr{3,31,1}
24-cell h01 F4.svg 24-cell h01 B3.svg 24-cell h01 B2.svg Ortho solid 969-uniform polychoron 343-snub.png

Coordinates[edit]

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

# Name(s) Base point Johnson and Bowers Names Coxeter diagrams
D4 B4 F4
[12] t3γ4 = β4 (0,0,0,2) 16-cell CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
[22] t2γ4 = t1β4 (0,0,2,2) Rectified 16-cell 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
[17] t2,3γ4 = t0,1β4 (0,0,2,4) Truncated 16-cell CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
[11] t1γ4 = t2β4 (0,2,2,2) Cantellated 16-cell CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
[23] t1,3γ4 = t0,2β4 (0,2,2,4) Cantellated 16-cell 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
[16] t1,2γ4 = t1,2β4 (0,2,4,4) Bitruncated 16-cell CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
[24] t1,2,3γ = t0,1,2β4 (0,2,4,6) Cantitruncated 16-cell 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
[12] 4 Even (1,1,1,1) 16-cell 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
[17] h2γ4 Even (1,1,3,3) Truncated 16-cell 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
[11] h3γ4 Even (1,1,1,3) Cantellated 16-cell 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
[16] h2,3γ4 Even (1,3,3,3) Cantitruncated 16-cell 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
[31] s{31,1,1} (0,1,φ,φ+1)/√2 Snub 24-cell 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

The grand antiprism[edit]

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

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

# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
Cells Faces Edges Vertices
47 grand antiprism (gap) 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

Prismatic uniform polychora[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 polychora consist of two infinite families:

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

Convex polyhedral prisms[edit]

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

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

Tetrahedral prisms: A3 × A1[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 polychoron.

[3,3,2] uniform polychora
# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
Cells Faces Edges Vertices
48 Tetrahedral prism (tepe) 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
49 Truncated tetrahedral prism (tuttip) 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
[[3,3],2] uniform polychora
# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
Cells Faces Edges Vertices
[51] Rectified tetrahedral prism
(Same as octahedral prism) (ope)
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
[50] Cantellated tetrahedral prism
(Same as cuboctahedral prism) (cope)
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
[54] Cantitruncated tetrahedral prism
(Same as truncated octahedral prism) (tope)
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
[59] Snub tetrahedral prism
(Same as icosahedral prism) (ipe)
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
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: BC3 × 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 polychora below. Symmetries are [(4,3)+,2], [1+,4,3,2], [4,3,2+], [4,3+,2], [4,(3,2)+], and [4,3,2]+.

# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
Cells Faces Edges Vertices
[10] Cubic prism
(Same as tesseract)
(Same as 4-4 duoprism) (tes)
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
50 Cuboctahedral prism
(Same as cantellated tetrahedral prism) (cope)
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
51 Octahedral prism
(Same as rectified tetrahedral prism)
(Same as triangular antiprismatic prism) (ope)
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
52 Rhombicuboctahedral prism (sircope) 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
53 Truncated cubic prism (ticcup) 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
54 Truncated octahedral prism
(Same as cantitruncated tetrahedral prism) (tope)
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
55 Truncated cuboctahedral prism (gircope) 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
56 Snub cubic prism (sniccup) 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
[48] Tetrahedral prism (tepe) 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
[59] Icosahedral prism (ipe) 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
[12] 16-cell (hex) 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
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
(Scaliform)
Runcic snub cubic hosochoron
Truncated tetrahedral cupoliprism (tutcup)
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

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.

# Johnson Name (Bowers style acronym) Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts
Cells Faces Edges Vertices
57 Dodecahedral prism (dope) 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
58 Icosidodecahedral prism (iddip) 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
59 Icosahedral prism
(same as snub tetrahedral prism) (ipe)
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
60 Truncated dodecahedral prism (tiddip) 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
61 Rhombicosidodecahedral prism (sriddip) 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
62 Truncated icosahedral prism (tipe) 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
63 Truncated icosidodecahedral prism (griddip) 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
64 Snub dodecahedral prism (sniddip) 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
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 cw.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
3-3 duoprism (triddip) 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
3-4 duoprism (tisdip) 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
4-4 duoprism (tes)
(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
3-5 duoprism (trapedip) 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
4-5 duoprism (squipdip) CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png 4 pentagonal prisms
5 cubes
3-4 duoprism.png 4-3 duoprism.png
5-5 duoprism (pedip) 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
3-6 duoprism (thiddip) 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
4-6 duoprism (shiddip) 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
5-6 duoprism (phiddip) 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
6-6 duoprism (hiddip) 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
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}×{}

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.pngCDel 6.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.pngCDel 10.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.pngCDel 14.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.pngCDel 2x.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}×{}

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]

Vertex figures
Snub 5-cell verf.png
ht0,1,2,3{3,3,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub tesseract verf.png
ht0,1,2,3{4,3,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub 120-cell verf.png
ht0,1,2,3{5,3,3}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
16-cell verf.png
s\left\{\begin{array}{l}2\\2\\2\end{array}\right\}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png
Snub p2q verf.png
ht0,1,2,3{p,2,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel q.pngCDel node h.png
Full snub 24-cell verf.png
ht0,1,2,3{3,4,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Snub 332 verf.png
s\left\{\begin{array}{l}3\\3\\2\end{array}\right\}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
Snub 432 verf.png
s\left\{\begin{array}{l}4\\3\\2\end{array}\right\}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
Snub 532 verf.png
s\left\{\begin{array}{l}5\\3\\2\end{array}\right\}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png
Runcic snub 243 verf.png
s3{2,4,3}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcic snub 24-cell verf.png
s3{3,4,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcic snub rectified 16-cell verf.png
sr3{3,3,4}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png

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

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

The polyhedral prisms CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel 2.pngCDel node 1.png, can be alternated into CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.pngCDel 2x.pngCDel node h.png, but do not generate uniform solutions.

  1. Omnisnub tetrahedral antiprism, s\left\{\begin{array}{l}3\\3\\2\end{array}\right\} CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, 2 icosahedrons connected by 6 tetrahedrons, and 8 octahedrons, with 24 tetrahedra in the alternated gaps.
  2. Omnisnub cubic antiprism, s\left\{\begin{array}{l}4\\3\\2\end{array}\right\} CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, 2 snub cubes connected by 12 tetrahedrons, 6 square antiprisms, and 8 octahedrons, with 48 tetrahedra in the alternated gaps.
  3. Omnisnub dodecahedral antiprism, s\left\{\begin{array}{l}5\\3\\2\end{array}\right\} CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, 2 snub dodecahedrons connected by 30 tetrahedrons, 12 pentagonal antiprisms, and 20 octahedrons, with 120 tetrahedra in the alternated gaps.
  4. Truncated tetrahedral cupoliprism (tutcup), s3{2,4,3}, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png, from 2 truncated tetrahedra, 6 tetrahedra, and 8 triangular cupolae in the gaps, for a total of 16 cells, 52 faces, 60 edges, and 24 vertices. It is vertex-transitive, and equilateral, but not uniform, due to the cupolae. It has symmetry [2+,4,3], order 48.[14]

The duoprisms CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node 1.png, t0,1,2,3{p,2,q}, can be alternated into CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel q.pngCDel node h.png, ht0,1,2,3{p,2,q}, called duoantiprisms, which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the tesseract CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.png, t0,1,2,3{2,2,2} = t{21,1,1}, with its alternation as the 16-cell, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png, s{21,1,1} = s\left\{\begin{array}{l}2\\2\\2\end{array}\right\}.

Geometric derivations for 46 nonprismatic Wythoffian uniform polychora[edit]

The 46 Wythoffian polychora include the six convex regular polychora. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their symmetries, and therefore may be classified by the symmetry groups that they have in common.

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 polychora from the regular polychora are truncating operations. A polychoron may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.

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

Operation Schläfli symbol 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] 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 ht1,2{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, BC4, 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 BC4 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 3.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.png