Uniform polychoron

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.

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** Ongoing: Thousands of nonconvex uniform polychora are known, but mostly unpublished.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.

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:
** 6 convex regular 4-polytopes;
** 10 nonconvex regular 4-polytopes.

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

Geometric derivations for 46 Wythoffian polychora

Summary chart of truncation operations

Example locations of kaleidoscopic generator point on fundamental domain.

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-Dynkins 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- Dynkins diagram	Description
Truncation	t0,1{p,q,r}	Dynkins-1100.png	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.
Rectification	t1{p,q,r}	Dynkins-0100.png	Truncation operation applied until the original edges are degenerated into points.
Bitruncation	t1,2{p,q,r}	Dynkins-0110.png	A truncation between a rectified form and the dual rectified form
Cantellation	t0,2{p,q,r}	Dynkins-1010.png	A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
Runcination (or expanded)	t0,3{p,q,r}	Dynkins-1001.png	A truncation applied to the cells, faces, and edges and defines a progression between a regular form and the dual.
Cantitruncated	t0,1,2{p,q,r}	Dynkins-1101.png	Both the cantellation and truncation operations applied together.
Runcitruncated	t0,1,3{p,q,r}	Dynkins-1011.png	both the runcination and truncation operations applied together.
omnitruncated (or more specifically runcicantitruncated)	t0,1,2,3{p,q,r}	Dynkins-1111.png	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.

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	t1,2,3{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.

The {3,3,4} (tesseract/16-cell) family

The hypercube - 8 cubes wrapped into a four dimensional box

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 also 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 also 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 also 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.

The {3,4,3} (self-dual 24-cell) family

{3,4,3}

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.

The {3,3,5} (120-cell/600-cell) family

{5,3,3}

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.

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).

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.

Polyhedral hyperprisms

Pentagonal-prism-first orthogonal projection into 3-dimensional space of a dodecahedral hyperprism with 2 dodecahedra connected perpendicularly by 12 pentagonal prisms

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.

Duoprisms

The second is the infinite family of duoprisms, products of two 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.

See also

*Polychoron
*Semiregular 4-polytopes
*Duoprism

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 0387004246.
Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.

External links

* Convex uniform polychora
**Polytope in R⁴, Marco Möller
*** 2004 Dissertation (German): VierdimensionaleArhimedishe Polytope
**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