Uniform 4-polytope

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

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

History of discovery

  • Convex Regular polytopes:
    • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
  • Regular star 4-polytopes (star polyhedron cells and/or vertex figures)
    • 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
    • 1883: Edmund Hess completed the list of 10 of the nonconvex regular 4-polytopes, in his book (in German) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder .
  • 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.[2]
    • 1910: Alicia Boole Stott, in her publication Geometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowing Archimedean solid and prism cells. This construction enumerated 45 semiregular 4-polytopes.[3]
    • 1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell.
    • 1912: E. L. Elte independently expanded on Gosset's list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.[4]
  • Convex uniform polytopes:
    • 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
    • Convex uniform 4-polytopes:
      • 1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism.
      • 1966 Norman Johnson completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher.
      • 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, and the symmetry of the anomalous grand antiprism.
      • 1998[5]-2000: The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration (used as a basis for this listing). Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots poly ("many") and choros ("room" or "space").[6] The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms called rectified, and bi,tri-prefixes added when the first ring was on the second or third nodes.[7][8]
      • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's naming system in his listing.[9]
      • 2008: The Symmetries of Things[10] was published by John H. Conway contains the first print-published listing of the convex uniform 4-polytopes and higher dimensions by coxeter group family, with general vertex figure diagrams for each ringed Coxeter diagram permutation, snub, grand antiprism, and duoprisms which he called proprisms for product prisms. He used his own ijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, with all of Johnson's names were included in the book index.
  • Nonregular uniform star 4-polytopes: (similar to the nonconvex uniform polyhedra)
    • 2000-2005: In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky[11], with an additional four discovered in 2006 for a total of 1849 known so far.[12]

Regular 4-polytopes

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

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

Existence as a finite 4-polytope is dependent upon an inequality:[13]

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

Convex uniform 4-polytopes

Symmetry of uniform 4-polytopes in four dimensions

Orthogonal subgroups
The 16 mirrors of B4 can be decomposed into 2 orthogonal groups, 4A1 and D4:
  1. = (4 mirrors)
  2. = (12 mirrors)
The 24 mirrors of F4 can be decomposed into 2 orthogonal D4 groups:
  1. = (12 mirrors)
  2. = (12 mirrors)
The 10 mirrors of B3×A1 can be decomposed into orthogonal groups, 4A1 and D3:
  1. = (3+1 mirrors)
  2. = (6 mirrors)

There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4 = , B4 = , D4 = , F4 = , H4 = .[7] There are also 3 prismatic groups A3A1 = , B3A1 = , H3A1 = , and duoprismatic groups: I2(p)×I2(q) = . Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.

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

If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.

Weyl
group
Conway
Quaternion
Abstract
structure
Order Coxeter
diagram
Coxeter
notation
Commutator
subgroup
Coxeter
number

(h)
Mirrors
m=2h
Irreducible
A4 +1/60[I×I].21S5120[3,3,3][3,3,3]+510
D4 ±1/3[T×T].21/2.2S4192[31,1,1][31,1,1]+612
B4 ±1/6[O×O].22S4 = S2≀S4384[4,3,3]8412
F4 ±1/2[O×O].233.2S41152[3,4,3][3+,4,3+]121212
H4 ±[I×I].22.(A5×A5).214400[5,3,3][5,3,3]+3060
Prismatic groups
A3A1 +1/24[O×O].23S4×D148[3,3,2] = [3,3]×[ ][3,3]+-61
B3A1 ±1/24[O×O].2S4×D196[4,3,2] = [4,3]×[ ]-361
H3A1 ±1/60[I×I].2A5×D1240[5,3,2] = [5,3]×[ ][5,3]+-151
Duoprismatic groups (Use 2p,2q for even integers)
I2(p)I2(q) ±1/2[D2p×D2q]Dp×Dq4pq[p,2,q] = [p]×[q][p+,2,q+]-p q
I2(2p)I2(q) ±1/2[D4p×D2q]D2p×Dq8pq[2p,2,q] = [2p]×[q]-p p q
I2(2p)I2(2q) ±1/2[D4p×D4q]D2p×D2q16pq[2p,2,2q] = [2p]×[2q]-p p q q

Enumeration

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

  • 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
  • 13 are polyhedral prisms based on the Archimedean solids
  • 9 are in the self-dual regular A4 [3,3,3] group (5-cell) family.
  • 9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
  • 15 are in the regular B4 [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
  • 15 are in the regular H4 [3,3,5] group (120-cell/600-cell) family.
  • 1 special snub form in the [3,4,3] group (24-cell) family.
  • 1 special non-Wythoffian 4-polytopes, the grand antiprism.
  • TOTAL: 68 4 = 64

These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.

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

The A4 family

The 5-cell has diploid pentachoric [3,3,3] symmetry,[7] of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

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

[3,3,3] uniform polytopes
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(5)
Pos. 2

(10)
Pos. 1

(10)
Pos. 0

(5)
Cells Faces Edges Vertices
1 5-cell
pentachoron[7]

{3,3,3}
(4)

(3.3.3)
5 10 10 5
2 rectified 5-cell
r{3,3,3}
(3)

(3.3.3.3)
(2)

(3.3.3)
10 30 30 10
3 truncated 5-cell
t{3,3,3}
(3)

(3.6.6)
(1)

(3.3.3)
10 30 40 20
4 cantellated 5-cell
rr{3,3,3}
(2)

(3.4.3.4)
(2)

(3.4.4)
(1)

(3.3.3.3)
20 80 90 30
7 cantitruncated 5-cell
tr{3,3,3}
(2)

(4.6.6)
(1)

(3.4.4)
(1)

(3.6.6)
20 80 120 60
8 runcitruncated 5-cell
t0,1,3{3,3,3}
(1)

(3.6.6)
(2)

(4.4.6)
(1)

(3.4.4)
(1)

(3.4.3.4)
30 120 150 60
[[3,3,3]] uniform polytopes
# Name Vertex
figure
Coxeter diagram

and Schläfli
symbols
Cell counts by location Element counts
Pos. 3-0

(10)
Pos. 1-2

(20)
Alt Cells Faces Edges Vertices
5 *runcinated 5-cell
t0,3{3,3,3}
(2)

(3.3.3)
(6)

(3.4.4)
30 70 60 20
6 *bitruncated 5-cell
decachoron

2t{3,3,3}
(4)

(3.6.6)
10 40 60 30
9 *omnitruncated 5-cell
t0,1,2,3{3,3,3}
(2)

(4.6.6)
(2)

(4.4.6)
30 150 240 120
Nonuniformomnisnub 5-cell[14]
ht0,1,2,3{3,3,3}
(2)
(3.3.3.3.3)
(2)
(3.3.3.3)
(4)
(3.3.3)
90 300 270 60

The three uniform 4-polytopes forms marked with an asterisk, *, have the higher extended pentachoric symmetry, of order 240, [[3,3,3]] because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup [3,3,3]+, order 60, or its doubling [[3,3,3]]+, order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform.

The B4 family

This family has diploid hexadecachoric symmetry,[7] [4,3,3], of order 24×16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.

Tesseract truncations

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(8)
Pos. 2

(24)
Pos. 1

(32)
Pos. 0

(16)
Cells Faces Edges Vertices
10 tesseract or
8-cell

{4,3,3}
(4)

(4.4.4)
8 24 32 16
11 Rectified tesseract
r{4,3,3}
(3)

(3.4.3.4)
(2)

(3.3.3)
24 88 96 32
13 Truncated tesseract
t{4,3,3}
(3)

(3.8.8)
(1)

(3.3.3)
24 88 128 64
14 Cantellated tesseract
rr{4,3,3}
(1)

(3.4.4.4)
(2)

(3.4.4)
(1)

(3.3.3.3)
56 248 288 96
15 Runcinated tesseract
(also runcinated 16-cell)

t0,3{4,3,3}
(1)

(4.4.4)
(3)

(4.4.4)
(3)

(3.4.4)
(1)

(3.3.3)
80 208 192 64
16 Bitruncated tesseract
(also bitruncated 16-cell)

2t{4,3,3}
(2)

(4.6.6)
(2)

(3.6.6)
24 120 192 96
18 Cantitruncated tesseract
tr{4,3,3}
(2)

(4.6.8)
(1)

(3.4.4)
(1)

(3.6.6)
56 248 384 192
19 Runcitruncated tesseract
t0,1,3{4,3,3}
(1)

(3.8.8)
(2)

(4.4.8)
(1)

(3.4.4)
(1)

(3.4.3.4)
80 368 480 192
21 Omnitruncated tesseract
(also omnitruncated 16-cell)

t0,1,2,3{3,3,4}
(1)

(4.6.8)
(1)

(4.4.8)
(1)

(4.4.6)
(1)

(4.6.6)
80 464 768 384
Related half tesseract, [1+,4,3,3] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(8)
Pos. 2

(24)
Pos. 1

(32)
Pos. 0

(16)
Alt Cells Faces Edges Vertices
12 Half tesseract
Demitesseract
16-cell
=
h{4,3,3}={3,3,4}
(4)

(3.3.3)
(4)

(3.3.3)
16 32 24 8
[17] Cantic tesseract
(Or truncated 16-cell)
=
h2{4,3,3}=t{4,3,3}
(4)

(6.6.3)
(1)

(3.3.3.3)
24 96 120 48
[11] Runcic tesseract
(Or rectified tesseract)
=
h3{4,3,3}=r{4,3,3}
(3)

(3.4.3.4)
(2)

(3.3.3)
24 88 96 32
[16] Runcicantic tesseract
(Or bitruncated tesseract)
=
h2,3{4,3,3}=2t{4,3,3}
(2)

(3.4.3.4)
(2)

(3.6.6)
24 120 192 96
[11] (rectified tesseract) =
h1{4,3,3}=r{4,3,3}
24 88 96 32
[16] (bitruncated tesseract) =
h1,2{4,3,3}=2t{4,3,3}
24 120 192 96
[23] (rectified 24-cell) =
h1,3{4,3,3}=rr{3,3,4}
48 240 288 96
[24] (truncated 24-cell) =
h1,2,3{4,3,3}=tr{3,3,4}
48 240 384 192
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(8)
Pos. 2

(24)
Pos. 1

(32)
Pos. 0

(16)
Alt Cells Faces Edges Vertices
Nonuniform omnisnub tesseract[15]
(Or omnisnub 16-cell)

ht0,1,2,3{4,3,3}
(1)

(3.3.3.3.4)
(1)

(3.3.3.4)
(1)

(3.3.3.3)
(1)

(3.3.3.3.3)
(4)

(3.3.3)
272 944 864 192

16-cell truncations

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(8)
Pos. 2

(24)
Pos. 1

(32)
Pos. 0

(16)
Alt Cells Faces Edges Vertices
[12] 16-cell, hexadecachoron[7]
{3,3,4}
(8)

(3.3.3)
16 32 24 8
[22] *rectified 16-cell
(Same as 24-cell)
=
r{3,3,4}
(2)

(3.3.3.3)
(4)

(3.3.3.3)
24 96 96 24
17 truncated 16-cell
t{3,3,4}
(1)

(3.3.3.3)
(4)

(3.6.6)
24 96 120 48
[23] *cantellated 16-cell
(Same as rectified 24-cell)
=
rr{3,3,4}
(1)

(3.4.3.4)
(2)

(4.4.4)
(2)

(3.4.3.4)
48 240 288 96
[15] runcinated 16-cell
(also runcinated 8-cell)

t0,3{3,3,4}
(1)

(4.4.4)
(3)

(4.4.4)
(3)

(3.4.4)
(1)

(3.3.3)
80 208 192 64
[16] bitruncated 16-cell
(also bitruncated 8-cell)

2t{3,3,4}
(2)

(4.6.6)
(2)

(3.6.6)
24 120 192 96
[24] *cantitruncated 16-cell
(Same as truncated 24-cell)
=
tr{3,3,4}
(1)

(4.6.6)
(1)

(4.4.4)
(2)

(4.6.6)
48 240 384 192
20 runcitruncated 16-cell
t0,1,3{3,3,4}
(1)

(3.4.4.4)
(1)

(4.4.4)
(2)

(4.4.6)
(1)

(3.6.6)
80 368 480 192
[21] omnitruncated 16-cell
(also omnitruncated 8-cell)

t0,1,2,3{3,3,4}
(1)

(4.6.8)
(1)

(4.4.8)
(1)

(4.4.6)
(1)

(4.6.6)
80 464 768 384
[31] alternated cantitruncated 16-cell
(Same as the snub 24-cell)

sr{3,3,4}
(1)

(3.3.3.3.3)
(1)

(3.3.3)
(2)

(3.3.3.3.3)
(4)

(3.3.3)
144 480 432 96
Nonuniform Runcic snub rectified 16-cell
sr3{3,3,4}
(1)

(3.4.4.4)
(2)

(3.4.4)
(1)

(4.4.4)
(1)

(3.3.3.3.3)
(2)

(3.4.4)
176 656 672 192
(*) Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family.

The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.

The F4 family

This family has diploid icositetrachoric symmetry,[7] [3,4,3], of order 24×48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform 4-polytopes which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576.

[3,4,3] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(24)
Pos. 2

(96)
Pos. 1

(96)
Pos. 0

(24)
Cells Faces Edges Vertices
22 24-cell, icositetrachoron[7]
(Same as rectified 16-cell)

{3,4,3}
(6)

(3.3.3.3)
24 96 96 24
23 rectified 24-cell
(Same as cantellated 16-cell)

r{3,4,3}
(3)

(3.4.3.4)
(2)

(4.4.4)
48 240 288 96
24 truncated 24-cell
(Same as cantitruncated 16-cell)

t{3,4,3}
(3)

(4.6.6)
(1)

(4.4.4)
48 240 384 192
25 cantellated 24-cell
rr{3,4,3}
(2)

(3.4.4.4)
(2)

(3.4.4)
(1)

(3.4.3.4)
144 720 864 288
28 cantitruncated 24-cell
tr{3,4,3}
(2)

(4.6.8)
(1)

(3.4.4)
(1)

(3.8.8)
144 720 1152 576
29 runcitruncated 24-cell
t0,1,3{3,4,3}
(1)

(4.6.6)
(2)

(4.4.6)
(1)

(3.4.4)
(1)

(3.4.4.4)
240 1104 1440 576
[3+,4,3] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(24)
Pos. 2

(96)
Pos. 1

(96)
Pos. 0

(24)
Alt Cells Faces Edges Vertices
31 snub 24-cell
s{3,4,3}
(3)

(3.3.3.3.3)
(1)

(3.3.3)
(4)

(3.3.3)
144 480 432 96
Nonuniform runcic snub 24-cell
s3{3,4,3}
(1)

(3.3.3.3.3)
(2)

(3.4.4)
(1)

(3.6.6)
(3)

Tricup
240 960 1008 288
[25] cantic snub 24-cell
(Same as cantellated 24-cell)

s2{3,4,3}
(2)

(3.4.4.4)
(1)

(3.4.3.4)
(2)

(3.4.4)
144 720 864 288
[29] runcicantic snub 24-cell
(Same as runcitruncated 24-cell)

s2,3{3,4,3}
(1)

(4.6.6)
(1)

(3.4.4)
(1)

(3.4.4.4)
(2)

(4.4.6)
240 1104 1440 576
(†) The snub 24-cell here, despite its common name, is not analogous to the snub cube; rather, is derived by an alternation of the truncated 24-cell. Its symmetry number is only 576, (the ionic diminished icositetrachoric group, [3+,4,3]).

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

[[3,4,3]] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram

and Schläfli
symbols
Cell counts by location Element counts
Pos. 3-0


(48)
Pos. 2-1


(192)
Cells Faces Edges Vertices
26 runcinated 24-cell
t0,3{3,4,3}
(2)

(3.3.3.3)
(6)

(3.4.4)
240 672 576 144
27 bitruncated 24-cell
tetracontoctachoron

2t{3,4,3}
(4)

(3.8.8)
48 336 576 288
30 omnitruncated 24-cell
t0,1,2,3{3,4,3}
(2)

(4.6.8)
(2)

(4.4.6)
240 1392 2304 1152
[[3,4,3]]+ isogonal 4-polytope
# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3-0


(48)
Pos. 2-1


(192)
Alt Cells Faces Edges Vertices
Nonuniform omnisnub 24-cell[16]
ht0,1,2,3{3,4,3}
(2)

(3.3.3.3.4)
(2)

(3.3.3.3)
(4)

(3.3.3)
816 2832 2592 576

The H4 family

This family has diploid hexacosichoric symmetry,[7] [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

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cell counts by location Element counts
Pos. 3

(120)
Pos. 2

(720)
Pos. 1

(1200)
Pos. 0

(600)
Alt Cells Faces Edges Vertices
32 120-cell
(hecatonicosachoron or dodecacontachoron)[7]

{5,3,3}
(4)

(5.5.5)
120 720 1200 600
33 rectified 120-cell
r{5,3,3}
(3)

(3.5.3.5)
(2)

(3.3.3)
720 3120 3600 1200
36 truncated 120-cell
t{5,3,3}
(3)

(3.10.10)
(1)

(3.3.3)
720 3120 4800 2400
37 cantellated 120-cell
rr{5,3,3}
(1)

(3.4.5.4)
(2)

(3.4.4)
(1)

(3.3.3.3)
1920 9120 10800 3600
38 runcinated 120-cell
(also runcinated 600-cell)

t0,3{5,3,3}
(1)

(5.5.5)
(3)

(4.4.5)
(3)

(3.4.4)
(1)

(3.3.3)
2640 7440 7200 2400
39 bitruncated 120-cell
(also bitruncated 600-cell)

2t{5,3,3}
(2)

(5.6.6)
(2)

(3.6.6)
720 4320 7200 3600
42 cantitruncated 120-cell
tr{5,3,3}
(2)

(4.6.10)
(1)

(3.4.4)
(1)

(3.6.6)
1920 9120 14400 7200
43 runcitruncated 120-cell
t0,1,3{5,3,3}
(1)

(3.10.10)
(2)

(4.4.10)
(1)

(3.4.4)
(1)

(3.4.3.4)
2640 13440 18000 7200
46 omnitruncated 120-cell
(also omnitruncated 600-cell)

t0,1,2,3{5,3,3}
(1)

(4.6.10)
(1)

(4.4.10)
(1)

(4.4.6)
(1)

(4.6.6)
2640 17040 28800 14400
Nonuniform omnisnub 120-cell[17]
(Same as the omnisnub 600-cell)

ht0,1,2,3{5,3,3}
(1)
(3.3.3.3.5)
(1)
(3.3.3.5)
(1)
(3.3.3.3)
(1)
(3.3.3.3.3)
(4)
(3.3.3)
9840 35040 32400 7200

600-cell truncations

# Name Vertex
figure
Coxeter diagram
and Schläfli
symbols
Symmetry Cell counts by location Element counts
Pos. 3

(120)
Pos. 2

(720)
Pos. 1

(1200)
Pos. 0

(600)
Cells Faces Edges Vertices
35 600-cell, hexacosichoron[7]
{3,3,5}
[5,3,3]
order 14400
(20)

(3.3.3)
600 1200 720 120
[47] 20-diminished 600-cell
(grand antiprism)
Nonwythoffian
construction
[[10,2<sup>+</sup>,10]]
order 400
Index 36
(2)

(3.3.3.5)
(12)

(3.3.3)
320 720 500 100
[31] 24-diminished 600-cell
(snub 24-cell)
Nonwythoffian
construction
[3+,4,3]
order 576
index 25
(3)

(3.3.3.3.3)
(5)

(3.3.3)
144 480 432 96
Nonuniform bi-24-diminished 600-cell Nonwythoffian
construction
order 144
index 100
(6)

tdi
48 192 216 72
34 rectified 600-cell
r{3,3,5}
[5,3,3] (2)

(3.3.3.3.3)
(5)

(3.3.3.3)
720 3600 3600 720
Nonuniform 120-diminished rectified 600-cell Nonwythoffian
construction
order 1200
index 12
(2)

3.3.3.5
(2)

4.4.5
(5)

P4
840 2640 2400 600
41 truncated 600-cell
t{3,3,5}
[5,3,3] (1)

(3.3.3.3.3)
(5)

(3.6.6)
720 3600 4320 1440
40 cantellated 600-cell
rr{3,3,5}
[5,3,3] (1)

(3.5.3.5)
(2)

(4.4.5)
(1)

(3.4.3.4)
1440 8640 10800 3600
[38] runcinated 600-cell
(also runcinated 120-cell)

t0,3{3,3,5}
[5,3,3] (1)

(5.5.5)
(3)

(4.4.5)
(3)

(3.4.4)
(1)

(3.3.3)
2640 7440 7200 2400
[39] bitruncated 600-cell
(also bitruncated 120-cell)

2t{3,3,5}
[5,3,3] (2)

(5.6.6)
(2)

(3.6.6)
720 4320 7200 3600
45 cantitruncated 600-cell
tr{3,3,5}
[5,3,3] (1)

(5.6.6)
(1)

(4.4.5)
(2)

(4.6.6)
1440 8640 14400 7200
44 runcitruncated 600-cell
t0,1,3{3,3,5}
[5,3,3] (1)

(3.4.5.4)
(1)

(4.4.5)
(2)

(4.4.6)
(1)

(3.6.6)
2640 13440 18000 7200
[46] omnitruncated 600-cell
(also omnitruncated 120-cell)

t0,1,2,3{3,3,5}
[5,3,3] (1)

(4.6.10)
(1)

(4.4.10)
(1)

(4.4.6)
(1)

(4.6.6)
2640 17040 28800 14400

The D4 family

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

[31,1,1] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram

=
=
Cell counts by location Element counts
Pos. 0

(8)
Pos. 2

(24)
Pos. 1

(8)
Pos. 3

(8)
Pos. Alt
(96)
3 2 1 0
[12] demitesseract
half tesseract
(Same as 16-cell)
=
h{4,3,3}
(4)

(3.3.3)
(4)

(3.3.3)
16 32 24 8
[17] cantic tesseract
(Same as truncated 16-cell)
=
h2{4,3,3}
(1)

(3.3.3.3)
(2)

(3.6.6)
(2)

(3.6.6)
24 96 120 48
[11] runcic tesseract
(Same as rectified tesseract)
=
h3{4,3,3}
(1)

(3.3.3)
(1)

(3.3.3)
(3)

(3.4.3.4)
24 88 96 32
[16] runcicantic tesseract
(Same as bitruncated tesseract)
=
h2,3{4,3,3}
(1)

(3.6.6)
(1)

(3.6.6)
(2)

(4.6.6)
24 96 96 24

When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as [3[31,1,1]] = [3,4,3], and thus these polytopes are repeated from the 24-cell family.

[3[31,1,1]] uniform 4-polytopes
# Name Vertex
figure
Coxeter diagram
=
=
Cell counts by location Element counts
Pos. 0,1,3

(24)
Pos. 2

(24)
Pos. Alt
(96)
3 2 1 0
[22] rectified 16-cell)
(Same as 24-cell)
= = =
{31,1,1} = r{3,3,4} = {3,4,3}
(6)

(3.3.3.3)
48 240 288 96
[23] cantellated 16-cell
(Same as rectified 24-cell)
= = =
r{31,1,1} = rr{3,3,4} = r{3,4,3}
(3)

(3.4.3.4)
(2)

(4.4.4)
24 120 192 96
[24] cantitruncated 16-cell
(Same as truncated 24-cell)
= = =
t{31,1,1} = tr{3,3,4} = t{3,4,3}
(3)

(4.6.6)
(1)

(4.4.4)
48 240 384 192
[31] snub 24-cell = = =
s{31,1,1} = sr{3,3,4} = s{3,4,3}
(3)

(3.3.3.3.3)
(1)

(3.3.3)
(4)

(3.3.3)
144 480 432 96

Here again the snub 24-cell, with the symmetry group [31,1,1]+ this time, represents an alternated truncation of the truncated 24-cell creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed 4-polytope, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron.

The grand antiprism

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

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

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

Prismatic uniform 4-polytopes

A prismatic polytope is a Cartesian product of two polytopes of lower dimension; familiar examples are the 3-dimensional prisms, which are products of a polygon and a line segment. The prismatic uniform 4-polytopes consist of two infinite families:

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

Convex polyhedral prisms

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

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

Tetrahedral prisms: A3 × A1

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

[3,3,2] uniform 4-polytopes
# Name Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
48 Tetrahedral prism
{3,3}×{ }
t0,3{3,3,2}
2
3.3.3
4
3.4.4
6 8 {3}
6 {4}
16 8
49 Truncated tetrahedral prism
t{3,3}×{ }
t0,1,3{3,3,2}
2
3.6.6
4
3.4.4
4
4.4.6
10 8 {3}
18 {4}
8 {6}
48 24
[[3,3],2] uniform 4-polytopes
# Name Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
[51] Rectified tetrahedral prism
(Same as octahedral prism)

r{3,3}×{ }
t1,3{3,3,2}
2
3.3.3.3
4
3.4.4
6 16 {3}
12 {4}
30 12
[50] Cantellated tetrahedral prism
(Same as cuboctahedral prism)

rr{3,3}×{ }
t0,2,3{3,3,2}
2
3.4.3.4
8
3.4.4
6
4.4.4
16 16 {3}
36 {4}
60 24
[54] Cantitruncated tetrahedral prism
(Same as truncated octahedral prism)

tr{3,3}×{ }
t0,1,2,3{3,3,2}
2
4.6.6
8
6.4.4
6
4.4.4
16 48 {4}
16 {6}
96 48
[59] Snub tetrahedral prism
(Same as icosahedral prism)

sr{3,3}×{ }
2
3.3.3.3.3
20
3.4.4
22 40 {3}
30 {4}
72 24
Nonuniform omnisnub tetrahedral antiprism
2
3.3.3.3.3
8
3.3.3.3
6+24
3.3.3
40 16+96 {3} 96 24

Octahedral prisms: B3 × A1

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

# Name Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
[10] Cubic prism
(Same as tesseract)
(Same as 4-4 duoprism)

{4,3}×{ }
t0,3{4,3,2}
2
4.4.4
6
4.4.4
824 {4}3216
50 Cuboctahedral prism
(Same as cantellated tetrahedral prism)

r{4,3}×{ }
t1,3{4,3,2}
2
3.4.3.4
8
3.4.4
6
4.4.4
1616 {3}
36 {4}
6024
51 Octahedral prism
(Same as rectified tetrahedral prism)
(Same as triangular antiprismatic prism)

{3,4}×{ }
t2,3{4,3,2}
2
3.3.3.3
8
3.4.4
1016 {3}
12 {4}
3012
52 Rhombicuboctahedral prism
rr{4,3}×{ }
t0,2,3{4,3,2}
2
3.4.4.4
8
3.4.4
18
4.4.4
2816 {3}
84 {4}
12048
53 Truncated cubic prism
t{4,3}×{ }
t0,1,3{4,3,2}
2
3.8.8
8
3.4.4
6
4.4.8
1616 {3}
36 {4}
12 {8}
9648
54 Truncated octahedral prism
(Same as cantitruncated tetrahedral prism)

t{3,4}×{ }
t1,2,3{4,3,2}
2
4.6.6
6
4.4.4
8
4.4.6
1648 {4}
16 {6}
9648
55 Truncated cuboctahedral prism
tr{4,3}×{ }
t0,1,2,3{4,3,2}
2
4.6.8
12
4.4.4
8
4.4.6
6
4.4.8
2896 {4}
16 {6}
12 {8}
19296
56 Snub cubic prism
sr{4,3}×{ }
2
3.3.3.3.4
32
3.4.4
6
4.4.4
4064 {3}
72 {4}
14448
[48] Tetrahedral prism
h{4,3}×{ }
2
3.3.3
4
3.4.4
68 {3}
6 {4}
168
[49] Truncated tetrahedral prism
h2{4,3}×{ }
2
3.3.6
4
3.4.4
4
4.4.6
68 {3}
6 {4}
168
[50] Cuboctahedral prism
rr{3,3}×{ }
2
3.4.3.4
8
3.4.4
6
4.4.4
1616 {3}
36 {4}
6024
[52] Rhombicuboctahedral prism
s2{3,4}×{ }
2
3.4.4.4
8
3.4.4
18
4.4.4
2816 {3}
84 {4}
12048
[54] Truncated octahedral prism
tr{3,3}×{ }
2
4.6.6
6
4.4.4
8
4.4.6
1648 {4}
16 {6}
9648
[59] Icosahedral prism
s{3,4}×{ }
2
3.3.3.3.3
20
3.4.4
2240 {3}
30 {4}
7224
[12] 16-cell
s{2,4,3}
2+6+8
3.3.3.3
1632 {3}248
Nonuniform Omnisnub tetrahedral antiprism
sr{2,3,4}
2
3.3.3.3.3
8
3.3.3.3
6+24
3.3.3
4016+96 {3}9624
Nonuniform Omnisnub cubic antiprism
2
3.3.3.3.4
12+48
3.3.3
8
3.3.3.3
6
3.3.3.4
7616+192 {3}
12 {4}
19248
Nonuniform Runcic snub cubic hosochoron
s3{2,4,3}
2
3.6.6
6
3.3.3
8
triangular cupola
16526024

Icosahedral prisms: H3 × A1

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

# Name Picture Vertex
figure
Coxeter diagram
and Schläfli
symbols
Cells by type Element counts Net
Cells Faces Edges Vertices
57 Dodecahedral prism
{5,3}×{ }
t0,3{5,3,2}
2
5.5.5
12
4.4.5
14 30 {4}
24 {5}
80 40
58 Icosidodecahedral prism
r{5,3}×{ }
t1,3{5,3,2}
2
3.5.3.5
20
3.4.4
12
4.4.5
34 40 {3}
60 {4}
24 {5}
150 60
59 Icosahedral prism
(same as snub tetrahedral prism)

{3,5}×{ }
t2,3{5,3,2}
2
3.3.3.3.3
20
3.4.4
22 40 {3}
30 {4}
72 24
60 Truncated dodecahedral prism
t{5,3}×{ }
t0,1,3{5,3,2}
2
3.10.10
20
3.4.4
12
4.4.10
34 40 {3}
90 {4}
24 {10}
240 120
61 Rhombicosidodecahedral prism
rr{5,3}×{ }
t0,2,3{5,3,2}
2
3.4.5.4
20
3.4.4
30
4.4.4
12
4.4.5
64 40 {3}
180 {4}
24 {5}
300 120
62 Truncated icosahedral prism
t{3,5}×{ }
t1,2,3{5,3,2}
2
5.6.6
12
4.4.5
20
4.4.6
34 90 {4}
24 {5}
40 {6}
240 120
63 Truncated icosidodecahedral prism
tr{5,3}×{ }
t0,1,2,3{5,3,2}
2
4.6.10
30
4.4.4
20
4.4.6
12
4.4.10
64 240 {4}
40 {6}
24 {10}
480 240
64 Snub dodecahedral prism
sr{5,3}×{ }
2
3.3.3.3.5
80
3.4.4
12
4.4.5
94 160 {3}
150 {4}
24 {5}
360 120
Nonuniform Omnisnub dodecahedral antiprism
2
3.3.3.3.5
30+120
3.3.3
20
3.3.3.3
12
3.3.3.5
18420+240 {3}
24 {5}
220120

Duoprisms: [p] × [q]

The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is . Its vertex figure is a disphenoid tetrahedron, .

This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if 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 - - p q-gonal prisms, q p-gonal prisms:

Name Coxeter graph Cells Images Net
3-3 duoprism 3+3 triangular prisms
3-4 duoprism 3 cubes
4 triangular prisms
4-4 duoprism
(same as tesseract)
4+4 cubes
3-5 duoprism 3 pentagonal prisms
5 triangular prisms
4-5 duoprism 4 pentagonal prisms
5 cubes
5-5 duoprism 5+5 pentagonal prisms
3-6 duoprism 3 hexagonal prisms
6 triangular prisms
4-6 duoprism 4 hexagonal prisms
6 cubes
5-6 duoprism 5 hexagonal prisms
6 pentagonal prisms
6-6 duoprism 6+6 hexagonal prisms

3-3

3-4

3-5

3-6

3-7

3-8

4-3

4-4

4-5

4-6

4-7

4-8

5-3

5-4

5-5

5-6

5-7

5-8

6-3

6-4

6-5

6-6

6-7

6-8

7-3

7-4

7-5

7-6

7-7

7-8

8-3

8-4

8-5

8-6

8-7

8-8

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

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

Convex p-gonal prismatic prisms
Name {3}×{4} {4}×{4} {5}×{4} {6}×{4} {7}×{4} {8}×{4} {p}×{4}
Coxeter
diagrams



Image




Cells 3 {4}×{}
4 {3}×{}
4 {4}×{}
4 {4}×{}
5 {4}×{}
4 {5}×{}
6 {4}×{}
4 {6}×{}
7 {4}×{}
4 {7}×{}
8 {4}×{}
4 {8}×{}
p {4}×{}
4 {p}×{}
Net

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

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

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








Image
Vertex
figure
Cells 2 s{2,2}
(2) {2}×{}={4}
4 {3}×{}
2 s{2,3}
2 {3}×{}
6 {3}×{}
2 s{2,4}
2 {4}×{}
8 {3}×{}
2 s{2,5}
2 {5}×{}
10 {3}×{}
2 s{2,6}
2 {6}×{}
12 {3}×{}
2 s{2,7}
2 {7}×{}
14 {3}×{}
2 s{2,8}
2 {8}×{}
16 {3}×{}
2 s{2,p}
2 {p}×{}
2p {3}×{}
Net

A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices.

Nonuniform alternations

Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings alternated (shown with empty circle nodes). The first is , s{21,1,1} which represented an index 24 subgroup (symmetry [2,2,2]+, order 8) form of the demitesseract, , h{4,3,3} (symmetry [1+,4,3,3] = [31,1,1], order 192). The second is , s{31,1,1}, which is an index 6 subgroup (symmetry [31,1,1]+, order 96) form of the snub 24-cell, , s{3,4,3}, (symmetry [3+,4,3], order 576).

Other alternations, such as , as an alternation from the omnitruncated tesseract , can not be made uniform as solving for equal edge lengths are in general overdetermined (there are six equations but only four variables). Such nonuniform alternated figures can be constructed as vertex-transitive 4-polytopes by the removal of one of two half sets of the vertices of the full ringed figure, but will have unequal edge lengths. Just like uniform alternations, they will have half of the symmetry of uniform figure, like [4,3,3]+, order 192, is the symmetry of the alternated omnitruncated tesseract.[18]

Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

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


Summary chart of truncation operations

Example locations of kaleidoscopic generator point on fundamental domain.

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

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

Operation Schläfli symbol Symmetry Coxeter diagram Description
Parent t0{p,q,r} [p,q,r] Original regular form {p,q,r}
Rectification t1{p,q,r} Truncation operation applied until the original edges are degenerated into points.
Birectification
(Rectified dual)
t2{p,q,r} Face are fully truncated to points. Same as rectified dual.
Trirectification
(dual)
t3{p,q,r} Cells are truncated to points. Regular dual {r,q,p}
Truncation t0,1{p,q,r} Each vertex is cut off so that the middle of each original edge remains. Where the vertex was, there appears a new cell, the parent's vertex figure. Each original cell is likewise truncated.
Bitruncation t1,2{p,q,r} A truncation between a rectified form and the dual rectified form.
Tritruncation t2,3{p,q,r} Truncated dual {r,q,p}.
Cantellation t0,2{p,q,r} A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
Bicantellation t1,3{p,q,r} Cantellated dual {r,q,p}.
Runcination
(or expansion)
t0,3{p,q,r} 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} Both the cantellation and truncation operations applied together.
Bicantitruncation t1,2,3{p,q,r} Cantitruncated dual {r,q,p}.
Runcitruncation t0,1,3{p,q,r} Both the runcination and truncation operations applied together.
Runcicantellation t0,1,3{p,q,r} Runcitruncated dual {r,q,p}.
Omnitruncation
(runcicantitruncation)
t0,1,2,3{p,q,r} Application of all three operators.
Half h{2p,3,q} [1+,2p,3,q]
=[(3,p,3),q]
Alternation of , same as
Cantic h2{2p,3,q} Same as
Runcic h3{2p,3,q} Same as
Runcicantic h2,3{2p,3,q} Same as
Quarter q{2p,3,2q} [1+,2p,3,2q,1+] Same as
Snub s{p,2q,r} [p+,2q,r] Alternated truncation
Cantic snub s2{p,2q,r} Cantellated alternated truncation
Runcic snub s3{p,2q,r} Runcinated alternated truncation
Runcicantic snub s2,3{p,2q,r} Runcicantellated alternated truncation
Snub rectified sr{p,q,2r} [(p,q)+,2r] Alternated truncated rectification
ht0,3{2p,q,2r} [(2p,q,2r,2+)] Alternated runcination
Bisnub 2s{2p,q,2r} [2p,q+,2r] Alternated bitruncation
Omnisnub ht0,1,2,3{p,q,r} [p,q,r]+ Alternated omnitruncation

See also convex uniform honeycombs, some of which illustrate these operations as applied to the regular cubic honeycomb.

If two polytopes are duals of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent.

Summary of constructions by extended symmetry

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

Coxeter group Extended
symmetry
Polychora Chiral
extended
symmetry
Alternation honeycombs
[3,3,3]
[3,3,3]

(order 120)
6 (2) | (3)
(4) | (7) | (8)
[2+[3,3,3]]

(order 240)
3 (6) | (9) [2+[3,3,3]]+
(order 120)
(1) (−)
[3,31,1]
[3,31,1]

(order 192)
0 (none)
[1[3,31,1]]=[4,3,3]
=
(order 384)
(4) (17) | (11) | (16)
[3[31,1,1]]=[3,4,3]
=
(order 1152)
(3) (23) | (24) [3[3,31,1]]+
=[3,4,3]+
(order 576)
(1) (31) (= )
(−)
[4,3,3]
[3[1+,4,3,3]]=[3,4,3]
=
(order 1152)
(3)(22) | (23) | (24)
[4,3,3]

(order 384)
12 (10) | (11) | (12) | (13) | (14)
(15) | (16) | (17) | (18) | (19)
(20) | (21)
[1+,4,3,3]+
(order 96)
(2) (12) (= )
(31)
(−)
[4,3,3]+
(order 192)
(1) (−)
[3,4,3]
[3,4,3]

(order 1152)
6 (23) | (24)
(25) | (28) | (29)
[2+[3+,4,3+]]
(order 576)
1 (31)
[2+[3,4,3]]

(order 2304)
3 (27) | (30) [2+[3,4,3]]+
(order 1152)
(1) (−)
[5,3,3]
[5,3,3]

(order 14400)
15 (33) | (34) | (35) | (36)
(37) | (38) | (39) | (40) | (41)
(42) | (43) | (44) | (45) | (46)
[5,3,3]+
(order 7200)
(1) (−)
[3,2,3]
[3,2,3]

(order 36)
0 (none) [3,2,3]+
(order 18)
0 (none)
[2+[3,2,3]]

(order 72)
0 [2+[3,2,3]]+
(order 36)
0 (none)
[[3],2,3]=[6,2,3]
=
(order 72)
1 [1[3,2,3]]=[[3],2,3]+=[6,2,3]+
(order 36)
(1)
[(2+,4)[3,2,3]]=[2+[6,2,6]]
=
(order 288)
1 [(2+,4)[3,2,3]]+=[2+[6,2,6]]+
(order 144)
(1)
[4,2,4]
[4,2,4]

(order 64)
0 (none) [4,2,4]+
(order 32)
0 (none)
[2+[4,2,4]]

(order 128)
0 (none) [2+[(4,2+,4,2+)]]
(order 64)
0 (none)
[(3,3)[4,2*,4]]=[4,3,3]
=
(order 384)
(1) (10) [(3,3)[4,2*,4]]+=[4,3,3]+
(order 192)
(1) (12)
[[4],2,4]=[8,2,4]
=
(order 128)
(1) [1[4,2,4]]=[[4],2,4]+=[8,2,4]+
(order 64)
(1)
[(2+,4)[4,2,4]]=[2+[8,2,8]]
=
(order 512)
(1) [(2+,4)[4,2,4]]+=[2+[8,2,8]]+
(order 256)
(1)

See also

Notes

  1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.1 Polytopes and Honeycombs, p.224
  2. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  3. "Archived copy" (PDF). Archived from the original (PDF) on 2009-12-29. Retrieved 2010-08-13.CS1 maint: archived copy as title (link)
  4. Elte (1912)
  5. https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html December 6, 1998 oldest archive
  6. The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes By David Darling, (2004) ASIN: B00SB4TU58
  7. Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups
  8. Uniform Polytopes in Four Dimensions, George Olshevsky.
  9. 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope (in German)
  10. Conway (2008)
  11. Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract
  12. "Uniform Polychora". www.polytope.net. Retrieved 2019-12-07.
  13. Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135
  14. http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm
  15. http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm
  16. http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm
  17. http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm
  18. H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) p. 582-588 2.7 The four-dimensional analogues of the snub cube

References

  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 1-4181-7968-X
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londen, 1954
  • Schoute, Pieter Hendrik (1911), "Analytic treatment of the polytopes regularly derived from the regular polytopes", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, 11 (3): 87 pp. Googlebook, 370-381
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980 p92, p122.
  • J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • N.W. Johnson: Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups
  • B. Grünbaum Convex polytopes, New York ; London : Springer, c2003. ISBN 0-387-00424-6.
    Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
  • Richard Klitzing, Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010)
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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