Uniform polychoron
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In geometry, a uniform polychoron (plural: uniform polychora) is a polychoron or 4-polytope which is vertex-uniform and whose cells are uniform polyhedra.
This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms.
Contents
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[edit] History of discovery
- 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. He also found 4 of the 10 nonconvex regular polychora, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
- Nonconvex regular polychora:
- 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 [1].
- Semiregular polytopes: (convex polytopes)
- 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.
- Convex uniform polytopes:
- 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.
- 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 64 nonprismatic 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.
- 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedishe Polytope (in German).
- Nonconvex uniform 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- Ongoing: Thousands of nonconvex uniform polychora are known, but mostly unpublished.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. Participating researchers include Jonathan Bowers, George Olshevsky and Norman Johnson.
[edit] Regular polychora
The uniform polychora include two special subsets, which satisfy additional requirements:
- The 16 regular polychora, with the property that all cells, faces, edges, and vertices are congruent:
[edit] Convex uniform polychora
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 hyperprisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
- 13 are hyperprisms based on the Archimedean solids
- 9 are in the self-dual regular {3,3,3} (5-cell) family.
- 9 are in the self-dual regular {3,4,3} (24-cell) family. (Excluding snub 24-cell)
- 15 are in the regular {3,3,4} (tesseract/16-cell) family (3 overlap with 24-cell family)
- 15 are in the regular {3,3,5} (120-cell/600-cell) family.
- 1 special snub form in the {3,4,3} (24-cell) family.
- 1 special non-Wythoffian polychoron, the grand antiprism.
- TOTAL: 68 - 4 = 64
[edit] The {3,3,3} (self-dual 5-cell) family
| Name | Extended Schläfli symbol |
Cell counts by location | Element counts | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Cells (5) |
Cells (10) |
Cells (10) |
Cells (5) |
Cells | Faces | Edges | Vertices | ||
| 5-cell | {3,3,3} |  (3.3.3) |
5 | 10 | 10 | 5 | |||
| truncated 5-cell | t0,1{3,3,3} |  (3.6.6) |
(3.3.3) |
10 | 30 | 40 | 20 | ||
| rectified 5-cell | t1{3,3,3} |  (3.3.3.3) |
(3.3.3) |
10 | 30 | 30 | 10 | ||
| cantellated 5-cell | t0,2{3,3,3} |  (3.4.3.4) |
 (3.4.4) |
(3.3.3.3) |
20 | 80 | 90 | 30 | |
| cantitruncated 5-cell | t0,1,2{3,3,3} |  (4.6.6) |
(3.4.4) |
(3.6.6) |
20 | 80 | 120 | 60 | |
| runcitruncated 5-cell | t0,1,3{3,3,3} | (3.6.6) |
 (4.4.6) |
(3.4.4) |
(3.4.3.4) |
30 | 120 | 150 | 60 |
| *bitruncated 5-cell | t1,2{3,3,3} | (3.6.6) |
(3.6.6) |
10 | 40 | 60 | 30 | ||
| *runcinated 5-cell | t0,3{3,3,3} | (3.3.3) |
(3.4.4) |
(3.4.4) |
(3.3.3) |
30 | 70 | 60 | 20 |
| *omnitruncated 5-cell | t0,1,2,3{3,3,3} | (4.6.6) |
(4.4.6) |
(4.4.6) |
(4.6.6) |
30 | 150 | 240 | 120 |
The 5-cell has diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.
The three 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.
[edit] The {3,3,4} (tesseract/16-cell) family
| Name | Extended Schläfli symbol |
Cell counts by location | Element counts | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Cells (8) |
Cells (24) |
Cells (32) |
Cells (16) |
Cells | Faces | Edges | Vertices | ||
| 8-cell or tesseract |
{4,3,3} |  (4.4.4) |
8 | 24 | 32 | 16 | |||
| 16-cell | {3,3,4} | (3.3.3) |
16 | 32 | 24 | 8 | |||
| truncated 8-cell | t0,1{4,3,3} |  (3.8.8) |
(3.3.3) |
24 | 88 | 128 | 64 | ||
| truncated 16-cell | t0,1{3,3,4} | (3.3.3.3) |
(3.6.6) |
24 | 96 | 120 | 48 | ||
| rectified 8-cell | t1{4,3,3} | (3.4.3.4) |
(3.3.3) |
24 | 88 | 96 | 32 | ||
| *rectified 16-cell Same as 24-cell (see below) |
t1{3,3,4} | (3.3.3.3) |
(3.3.3.3) |
24 | 96 | 96 | 24 | ||
| cantellated 8-cell | t0,2{4,3,3} |  (3.4.4.4) |
(3.4.4) |
(3.3.3.3) |
56 | 248 | 288 | 96 | |
| *cantellated 16-cell Same as rectified 24-cell (see below) |
t0,2{3,3,4} | (3.4.3.4) |
(4.4.4) |
(3.4.3.4) |
48 | 240 | 288 | 96 | |
| cantitruncated 8-cell | t0,1,2{4,3,3} |  (4.6.8) |
(3.4.4) |
(3.6.6) |
56 | 248 | 384 | 192 | |
| *cantitruncated 16-cell Same as truncated 24-cell (see below) |
t0,1,2{3,3,4} | (4.6.6) |
(4.4.4) |
(4.6.6) |
48 | 240 | 384 | 192 | |
| runcitruncated 8-cell | t0,1,3{4,3,3} | (3.8.8) |
 (4.4.8) |
(3.4.4) |
(3.4.3.4) |
80 | 368 | 480 | 192 |
| runcitruncated 16-cell | t0,1,3{3,3,4} | (3.4.4.4) |
(4.4.4) |
(4.4.6) |
(3.6.6) |
80 | 368 | 480 | 192 |
| bitruncated 8-cell also bitruncated 16-cell |
t1,2{4,3,3} | (4.6.6) |
(3.6.6) |
24 | 120 | 192 | 96 | ||
| runcinated 8-cell also runcinated 16-cell |
t0,3{4,3,3} | (4.4.4) |
(4.4.4) |
(3.4.4) |
(3.3.3) |
80 | 208 | 192 | 64 |
| omnitruncated 8-cell also omnitruncated 16-cell |
t0,1,2,3{3,3,4} | (4.6.8) |
(4.4.8) |
(4.4.6) |
(4.6.6) |
80 | 464 | 768 | 384 |
This family has diploid hexadecachoric symmetry, of order 24*16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis.
Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family.
[edit] The {3,4,3} (self-dual 24-cell) family
| Name | Extended Schläfli symbol |
Cell counts by location | Element counts | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Cells (24) |
Cells (96) |
Cells (96) |
Cells (24) |
Cells | Faces | Edges | Vertices | ||
| 24-cell | {3,4,3} | (3.3.3.3) |
24 | 96 | 96 | 24 | |||
| truncated 24-cell | t0,1{3,4,3} | (4.6.6) |
(4.4.4) |
48 | 240 | 384 | 192 | ||
| rectified 24-cell | t1{3,4,3} | (3.4.3.4) |
(4.4.4) |
48 | 240 | 288 | 96 | ||
| cantellated 24-cell | t0,2{3,4,3} | (3.4.4.4) |
(3.4.4) |
(3.4.3.4) |
144 | 720 | 864 | 288 | |
| cantitruncated 24-cell | t0,1,2{3,4,3} | (4.6.8) |
(3.4.4) |
(3.8.8) |
144 | 720 | 1152 | 576 | |
| runcitruncated 24-cell | t0,1,3{3,4,3} | (4.6.6) |
(4.4.6) |
(3.4.4) |
(3.4.4.4) |
240 | 1104 | 1440 | 576 |
| *bitruncated 24-cell | t1,2{3,4,3} | (3.8.8) |
(3.8.8) |
48 | 336 | 576 | 288 | ||
| *runcinated 24-cell | t0,3{3,4,3} | (3.3.3.3) |
(3.4.4) |
(3.4.4) |
(3.3.3.3) |
240 | 672 | 576 | 144 |
| *omnitruncated 24-cell | t0,1,2,3{3,4,3} | (4.6.8) |
(4.4.6) |
(4.4.6) |
(4.6.8) |
240 | 1392 | 2304 | 1152 |
| †snub 24-cell | s{3,4,3} |  (3.3.3.3.3) |
(3.3.3) (oblique) |
(3.3.3) |
144 | 480 | 432 | 96 | |
This family has diploid icositetrachoric symmetry, of order 24*48=1152: the 48 symmetries of the octahedron for each of the 24 cells.
*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 (the extended icositetrachoric group).
†The snub 24-cell, despite its common name, is not analogous to the snub cube; rather, it is derived by asymmetric rectification: each of its 96 vertices cuts an edge of the parent 24-cell in the golden ratio. Because of this skew, its symmetry number is only 576 (the ionic diminished icositetrachoric group). Of all regular polychora only the 24-cell can be treated in this way while preserving uniformity, because only it has a vertex figure in which edges can alternate.
[edit] The {3,3,5} (120-cell/600-cell) family
| Name | Extended Schläfli symbol |
Cell counts by location | Element counts | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Cells (120) |
Cells (720) |
Cells (1200) |
Cells (600) |
Cells | Faces | Edges | Vertices | ||
| 120-cell | {5,3,3} |  (5.5.5) |
120 | 720 | 1200 | 600 | |||
| 600-cell | {3,3,5} | (3.3.3) |
600 | 1200 | 720 | 120 | |||
| truncated 120-cell | t0,1{5,3,3} |  (3.10.10) |
(3.3.3) |
720 | 3120 | 4800 | 2400 | ||
| truncated 600-cell | t0,1{3,3,5} | (3.3.3.3.3) |
(3.6.6) |
720 | 3600 | 4320 | 1440 | ||
| rectified 120-cell | t1{5,3,3} |  (3.5.3.5) |
(3.3.3) |
720 | 3120 | 3600 | 1200 | ||
| rectified 600-cell | t1{3,3,5} | (3.3.3.3.3) |
(3.3.3.3) |
720 | 3600 | 3600 | 720 | ||
| cantellated 120-cell | t0,2{5,3,3} |  (3.4.5.4) |
(3.4.4) |
(3.3.3.3) |
1920 | 9120 | 10800 | 3600 | |
| cantellated 600-cell | t0,2{3,3,5} | (3.5.3.5) |
 (4.4.5) |
(3.4.3.4) |
1440 | 8640 | 10800 | 3600 | |
| cantitruncated 120-cell | t0,1,2{5,3,3} |  (4.6.10) |
(3.4.4) |
(3.6.6) |
1920 | 9102 | 14400 | 720 | |
| cantitruncated 600-cell | t0,1,2{3,3,5} |  (5.6.6) |
(4.4.5) |
(4.6.6) |
1440 | 8640 | 14400 | 7200 | |
| runcitruncated 120-cell | t0,1,3{5,3,3} | (3.10.10) |
 (4.4.10) |
(3.4.4) |
(3.4.3.4) |
2640 | 13440 | 18000 | 7200 |
| runcitruncated 600-cell | t0,1,3{3,3,5} | (3.4.5.4) |
(4.4.5) |
(4.4.6) |
(3.6.6) |
2640 | 13440 | 18000 | 7200 |
| bitruncated 120-cell also bitruncated 600-cell |
t1,2{5,3,3} | (5.6.6) |
(3.6.6) |
720 | 4320 | 7200 | 3600 | ||
| runcinated 120-cell also runcinated 600-cell |
t0,3{5,3,3} | (5.5.5) |
(4.4.5) |
(3.4.4) |
(3.3.3) |
2640 | 7440 | 7200 | 2400 |
| omnitruncated 120-cell also omnitruncated 600-cell |
t0,1,2,3{5,3,3} | (4.6.10) |
(4.4.10) |
(4.4.6) |
(4.6.6) |
2640 | 17040 | 28800 | 14400 |
This family has diploid hexacosichoric symmetry, of order 120*120=24*600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra.
[edit] The grand antiprism
The anomalous forty-seventh non-Wythoffian polychoron is known as the grand antiprism, and consists 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, the grand antiprism is not a member of an infinite family of uniform polytopes.
Its symmetry number is 400 (the ionic diminished Coxeter group).
[edit] Prismatic uniform polychora
There are two infinite families of uniform polychora that are considered prismatic, in that they generalize the properties of the 3-dimensional prisms. A prismatic polytope is a Cartesian product of two polytopes of lower dimension.
[edit] Polyhedral hyperprisms
The more obvious family of prismatic polychora is the polyhedral hyperprisms, 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), 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.
18 convex polyhedral hyperprisms (from 5 Platonic solid and 13 Archimedean solid)
- Tetrahedral hyperprism - 2 tetrahedra connected by 4 triangular prisms
- Cubic hyperprism (regular tesseract) - 2 cubes connected by 6 cubes.
- Octahedral hyperprism - 2 octahedra connected by 8 triangular prisms
- Dodecahedral hyperprism - 2 dodecahedra connected by 12 pentagonal prisms
- Icosahedral hyperprism - 2 icosahedra connected by 20 triangular prisms
- Truncated tetrahedral hyperprism - 2 truncated tetrahedra connected by 4 triangular prisms and 4 hexagonal prisms
- Truncated cubic hyperprism - 2 truncated cubes connected by 8 triangular prisms and 6 octagonal prisms.
- Truncated octahedral hyperprism - 2 truncated octahedra connected by 6 cubes and 8 hexagonal prisms.
- Truncated dodecahedral hyperprism - 2 truncated dodecahedra connected by 20 triangular prisms and 12 decagonal prisms.
- Truncated icosahedral hyperprism - 2 runcated icosahedra connected by 12 pentagonal prisms and 20 hexagonal prisms.
- Truncated cuboctahedral hyperprism - 2 truncated cuboctahedra connected by 12 cubes, 8 hexagonal prism, and 6 octagonal prisms.
- Truncated icosidodecahedral hyperprism - 2 truncated icosidodecahedra connected by 30 cubes, 20 hexagonal prisms, and 12 decagonal prisms
- Cuboctahedral hyperprism - 2 cuboctahedra connected by 8 triangular prisms and 6 cubes.
- Icosidodecahedral hyperprism - 2 icosidodecahedra connected by 20 triangular prisms and 12 pentagonal prisms.
- Rhombicuboctahedral hyperprism - 2 rhombicuboctahedra connected by 8 triangular prisms and 18 cubes.
- Rhombicosidodecahedral hyperprism - 2 rhombicosidodecahedra connected by 20 triangular prisms, 30 cubes, and 12 pentagonal prisms
- Snub cubic hyperprism - 2 snub cubes connected by 32 triangular prisms and 6 cubes
- Snub dodecahedral hyperprism - 2 snub dodecahedra connected by 80 triangular prisms, and 12 pentagonal prisms
Infinite sets of convex prismatic hyperprisms (from uniform prisms and antiprisms)
- p-gonal prismatic hyperprism (p≥3) - 2 p-gonal prisms, connected by 2 p-gonal prisms and p cubes.
- p-gonal antiprismatic hyperprism (p≥3) - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms.
[edit] Duoprisms
The second is the infinite family of uniform duoprisms, products of two regular polygons. Note that 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 p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can be considered a 4,4-duoprism, though its symmetry is higher than that implies.
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.
[edit] Geometric derivations for 46 Wythoffian polychora
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.
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. (180/n degrees) Circled nodes show which mirrors are active for each form. That is mirrors for which the generating point is located off the mirror.
| Operation | Schläfli symbol |
Coxeter- Dynkin diagram |
Description |
|---|---|---|---|
| Parent | t0{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 | 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 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 and 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 (or more specifically runcicantitruncated) |
t0,1,2,3{p,q,r} |  |
Has all three operators applied. |
See also 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.
[edit] See also
[edit] References
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- 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
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular and Semi Regular Polytopes, part I, mathematical magazine, Springer, Berlin, 1940
- 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
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- H.S.M. Coxeter, Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
- H.S.M. Coxeter, Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988
- 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
- M. Möller: Definitions and computations to the Platonic and Archimedean polyhedrons, thesis (diploma), University of Hamburg, 2001
- 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.
[edit] External links
- Convex uniform polychora
- Polytope in R4, Marco Möller
- Uniform Polytopes in Four Dimensions by George Olshevsky, from which the data in the tables were taken
- Regular and semi-regular convex polytopes a short historical overview
- Nonconvex uniform polychora
- Uniform polychora by Jonathan Bowers
