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Uniform 4-polytope

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Uniform 4-polytope

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

In geometry, a uniform 4-polytope is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

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

Contents

  • History of discovery 1
  • Regular 4-polytopes 2
  • Convex uniform 4-polytopes 3
    • Enumeration 3.1
    • The A4 family 3.2
    • The B4 family 3.3
      • Tesseract truncations 3.3.1
      • 16-cell truncations 3.3.2
    • The F4 family 3.4
    • The H4 family 3.5
      • 120-cell truncations 3.5.1
      • 600-cell truncations 3.5.2
    • The D4 family 3.6
    • The grand antiprism 3.7
    • Prismatic uniform 4-polytopes 3.8
      • Convex polyhedral prisms 3.8.1
      • Tetrahedral prisms: A3 × A1 3.8.2
      • Octahedral prisms: B3 × A1 3.8.3
      • Icosahedral prisms: H3 × A1 3.8.4
      • Duoprisms: [p] × [q] 3.8.5
      • Polygonal prismatic prisms: [p] × [ ] × [ ] 3.8.6
    • Nonuniform alternations 3.9
    • Geometric derivations for 46 nonprismatic Wythoffian uniform polychora 3.10
      • Summary of constructions by extended symmetry 3.10.1
  • Symmetries in four dimensions 4
  • See also 5
  • Notes 6
  • References 7
  • External links 8

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 [7].
  • Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
    • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[1]
    • 1910: Alicia Boole Stott, in her publication Geometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowing Archimedean solid and prism cells. This construction enumerated 45 semiregular 4-polytopes.[2]
    • 1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell.
    • 1912: E. L. Elte independently expanded on Gosset's list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.[3]
  • Convex uniform polytopes:
    • 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
    • Convex uniform 4-polytopes:
      • 1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism.
      • 1966 Norman Johnson completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher.
      • 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, and the symmetry of the anomalous grand antiprism.
      • 1998[4]-2000: The 4-polytopes were systematically named by Norman Johnson, and given by Greek roots poly ("many") and choros ("room" or "space").[5] The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms called rectified, and bi,tri-prefixes added when the first ring was on the second or third nodes.[6][7]
      • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's naming system in his listing.[8]
      • 2008: The Symmetries of Things[9] was published by John H. Conway contains the first print-published listing of the convex uniform 4-polytopes and higher dimensions by coxeter group family, with general vertex figure diagrams for each ringed Coxeter diagram permutation, snub, grand antiprism, and duoprisms which he called proprisms for product prisms. He used his own ijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, with all of Johnson's names were included in the book index.
  • Nonregular uniform star 4-polytopes: (similar to the nonconvex uniform polyhedra)
    • 2000-2005: In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky.[10]

Regular 4-polytopes

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

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

Existence as a finite 4-polytope is dependent upon an inequality:[11]

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

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

Convex uniform 4-polytopes

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,[6] of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

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

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

(5)
Pos. 2

(10)
Pos. 1

(10)
Pos. 0

(5)
Cells Faces Edges Vertices
1 5-cell
pentachoron[6]

{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
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
Nonuniform omnisnub 5-cell[12]
ht0,1,2,3{4,3,3}
(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, 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 +, order 120, defining a omnisnub 5-cell which is listed for completeness, but is not uniform.

The B4 family

This family has diploid hexadecachoric symmetry,[6] [4,3,3], of order 24*16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.

Tesseract truncations

# 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 96 96 24
[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 96 96 24
[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[13]
(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[6]
{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,[6] [3,4,3], of order 24*48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform 4-polytopes which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576.

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

(24)
Pos. 2

(96)
Pos. 1

(96)
Pos. 0

(24)
Cells Faces Edges Vertices
22 24-cell, icositetrachoron[6]
(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 three asterisked forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry ).

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
+ 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[14]
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,[6] [5,3,3], of order 120*120=24*600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]+, all order 7200.

120-cell truncations

# 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[6]
{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[15]
(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
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[6]
{3,3,5}
(20)

(3.3.3)
600 1200 720 120
[47] 20-diminished 600-cell
(grand antiprism)
Nonwythoffian
construction
order 400
(2)

(3.3.3.5)
(12)

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

(3.3.3.3.3)
(5)

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

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

3.3.3.5
(2)

4.4.5
(5)

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

(3.3.3.3.3)
(5)

(3.6.6)
720 3600 4320 1440
40 cantellated 600-cell
rr{3,3,5}
(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}
(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}
(2)

(5.6.6)
(2)

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


(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)
==[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
==[2+[8,2,8]]
=
(order 512)
(1) [(2+,4)[4,2,4]]+=[2+[8,2,8]]+
(order 256)
(1)

Symmetries in four dimensions

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

See also

Notes

  1. ^ T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  2. ^ http://dissertations.ub.rug.nl/FILES/faculties/science/2007/i.polo.blanco/c5.pdf
  3. ^ Elte (1912)
  4. ^ https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html December 6, 1998 oldest archive
  5. ^ The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes By David Darling, (2004) ASIN: B00SB4TU58
  6. ^ a b c d e f g h i j k Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups
  7. ^ Uniform Polytopes in Four Dimensions, George Olshevsky.
  8. ^ 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope (German)
  9. ^ Conway (2008)
  10. ^ [8] Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract
  11. ^ Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135
  12. ^ http://www.bendwavy.org/klitzing/incmats/s3s3s3s.htm
  13. ^ http://www.bendwavy.org/klitzing/incmats/s3s3s4s.htm
  14. ^ http://www.bendwavy.org/klitzing/incmats/s3s4s3s.htm
  15. ^ http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm
  16. ^ H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) p. 582-588 2.7 The four-dimensional analogues of the snub cube

References

  • 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, [9] [10]  
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
  •   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) [11]

External links

  • Convex uniform 4-polytopes
    • Uniform, convex polytopes in four dimensions:, Marco Möller (German)
      • 2004 Dissertation Four-dimensional Archimedean polytopes (German)
    • Uniform Polytopes in Four Dimensions, George Olshevsky.
    • Convex uniform polychora based on the pentachoron, George Olshevsky.
    • Convex uniform polychora based on the tesseract/16-cell, George Olshevsky.
    • Convex uniform polychora based on the 24-cell, George Olshevsky.
    • Convex uniform polychora based on the 120-cell/600-cell, George Olshevsky.
    • Anomalous convex uniform polychoron: (grand antiprism), George Olshevsky.
    • Convex uniform prismatic polychora, George Olshevsky.
    • Uniform polychora derived from glomeric tetrahedron B4, George Olshevsky.
    • Regular and semi-regular convex polytopes a short historical overview
    • Java3D Applets with sources
  • Nonconvex uniform 4-polytopes
    • Uniform polychora by Jonathan Bowers
    • Stella4D Stella (software) produces interactive views of known uniform polychora including the 64 convex forms and the infinite prismatic families.
  • Richard Klitzing, 4D, uniform polytopes
  • 4D-Polytopes and Their Dual Polytopes of the Coxeter Group W(A4) Represented by Quaternions International Journal of Geometric Methods in Modern Physics,Vol. 9, No. 4 (2012) Mehmet Koca, Nazife Ozdes Koca, Mudhahir Al-Ajmi (2012) [12]
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