User:Tomruen/uniform polychoron

Operation	Johnson Notation	Coxeter-Stott notation	Cell (3)	Cell (2)	Cell (1)	Cell (0)
Parent	{p,q,r}	t₀{p,q,r}	${\begin{Bmatrix}p,q\end{Bmatrix}}$	--	--	--
Truncated	t{p,q,r}	t_0,1{p,q,r}	$t{\begin{Bmatrix}p,q\end{Bmatrix}}$	--	--	${\begin{Bmatrix}q,r\end{Bmatrix}}$
Rectified	r{p,q,r}	t₁{p,q,r}	${\begin{Bmatrix}p\\q\end{Bmatrix}}$	--	--	${\begin{Bmatrix}q,r\end{Bmatrix}}$
Cantellated	c{p,q,r}	t_0,2{p,q,r}	$r{\begin{Bmatrix}p\\q\end{Bmatrix}}$	--	$t{\begin{Bmatrix}2,r\end{Bmatrix}}$	${\begin{Bmatrix}q\\r\end{Bmatrix}}$
Cantitruncated	ct{p,q,r}	t_0,1,2{p,q,r}	$t{\begin{Bmatrix}p\\q\end{Bmatrix}}$	--	$t{\begin{Bmatrix}2,r\end{Bmatrix}}$	$t{\begin{Bmatrix}q,r\end{Bmatrix}}$
Bitruncated	bt{p,q,r}	t_1,2{p,q,r}	$t{\begin{Bmatrix}q,p\end{Bmatrix}}$	--	--	$t{\begin{Bmatrix}q,r\end{Bmatrix}}$
Cantitruncated dual	ct{r,q,p}	t_1,2,3{p,q,r}	$t{\begin{Bmatrix}q,p\end{Bmatrix}}$	$t{\begin{Bmatrix}2,p\end{Bmatrix}}$	--	$t{\begin{Bmatrix}q\\r\end{Bmatrix}}$
Cantellated dual	c{r,q,p}	t_1,3{p,q,r}	${\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}2,p\end{Bmatrix}}$	--	$r{\begin{Bmatrix}q\\r\end{Bmatrix}}$
Rectified dual	r{r,q,p}	t₂{p,q,r}	${\begin{Bmatrix}q,p\end{Bmatrix}}$	--	--	${\begin{Bmatrix}q\\r\end{Bmatrix}}$
Truncated dual	t{r,q,p}	t_2,3{p,q,r}	${\begin{Bmatrix}q,p\end{Bmatrix}}$	--	--	$t{\begin{Bmatrix}r,q\end{Bmatrix}}$
Dual	{r,q,p}	t₃{p,q,r}	--	--	--	${\begin{Bmatrix}r,q\end{Bmatrix}}$

Runcinated	rr{p,q,r}	t_0,3{p,q,r}	${\begin{Bmatrix}p,q\end{Bmatrix}}$	$t{\begin{Bmatrix}2,p\end{Bmatrix}}$	$t{\begin{Bmatrix}2,r\end{Bmatrix}}$	${\begin{Bmatrix}r,q\end{Bmatrix}}$
Runcitruncated	rt{p,q,r}	t_0,1,3{p,q,r}	$t{\begin{Bmatrix}p,q\end{Bmatrix}}$	$t{\begin{Bmatrix}2,p\end{Bmatrix}}$	$t{\begin{Bmatrix}2,r\end{Bmatrix}}$	$r{\begin{Bmatrix}q\\r\end{Bmatrix}}$
Runcitruncated dual	rt{r,q,p}	t_0,2,3{p,q,r}	$r{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}2,p\end{Bmatrix}}$	$t{\begin{Bmatrix}2,r\end{Bmatrix}}$	$t{\begin{Bmatrix}r,q\end{Bmatrix}}$
Omnitruncated	ot{p,q,r}	t_0,1,2,3{p,q,r}	$t{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}2,p\end{Bmatrix}}$	$t{\begin{Bmatrix}2,r\end{Bmatrix}}$	$t{\begin{Bmatrix}q\\r\end{Bmatrix}}$

External link for Coxeter-Stott notation:
- [1]

summary

	Parent	Truncated	Rectified	Bitruncated	Runcinated	Omnitruncated	Runcitruncated	Cantellated	Cantitruncated
Schläfli symbol	t_{0_{p,q,r}}	t_{0,1_{p,q,r}}	t_{1_{p,q,r}}	t_{1,2_{p,q,r}}	t_{0,3_{p,q,r}}	t_{0,1,2,3_{p,q,r}}	t_{0,1,3_{p,q,r}}	t_{0,2_{p,q,r}}	t_{0,1,2_{p,q,r}}
{3,3,3}	5-cell	Truncated 5-cell	Rectified 5-cell	Bitruncated 5-cell	Runcinated 5-cell	Omnitruncated 5-cell	Runcitruncated 5-cell	Cantellated 5-cell	Omnitruncated 5-cell
{4,3,3}	8-cell	Truncated 8-cell	Rectified 8-cell	Bitruncated 8-cell	Runcinated 8-cell	Omnitruncated 8-cell	Runcitruncated 8-cell	Cantitruncated 8-cell	Omnitruncated 8-cell
{3,3,4}	16-cell	Truncated 16-cell	Rectified 16-cell	Bitruncated 16-cell	Runcinated 16-cell	Omnitruncated 16-cell	Runcitruncated 16-cell	Cantitruncated 16-cell	Omnitruncated 16-cell
{3,4,3}	24-cell	Truncated 24-cell	Rectified 24-cell	Bitruncated 24-cell	Runcinated 24-cell	Omnitruncated 24-cell	Runcitruncated 24-cell	Cantitruncated 24-cell	Omnitruncated 24-cell
{5,3,3}	120-cell	Truncated 120-cell	Rectified 120-cell	Bitruncated 120-cell	Runcinated 120-cell	Omnitruncated 120-cell	Runcitruncated 120-cell	Cantitruncated 120-cell	Omnitruncated 120-cell
{3,3,5}	600-cell	Truncated 600-cell	Rectified 600-cell	Bitruncated 600-cell	Runcinated 600-cell	Omnitruncated 600-cell	Runcitruncated 600-cell	Cantitruncated 600-cell	Omnitruncated 600-cell
{4,3,4}	Cubic honeycomb	Truncated cubic honeycomb	Rectified cubic honeycomb	Bitruncated cubic honeycomb	Runcinated cubic honeycomb	Omnitruncated cubic honeycomb	Runcitruncated cubic honeycomb	Cantitruncated cubic honeycomb	Omnitruncated cubic honeycomb

Geometric derivations

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
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
faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below. (Note: the following definitions use the word facet, which must be distinguished from face. A facet of a polytope is an element of the next lower dimension; a face is two-dimensional. A facet of a polyhedron (3D polytope) is a face; a facet of a polychoron (4D polytope) is a cell.)

A truncated polytope has each original vertex cut off so that the middle of each original edge remains. Where the vertex was, there appears a new facet, the parent's vertex figure. Each original facet is likewise truncated.
A rectified polytope is similar, but the truncation is carried further so that the new facets meet at the center of each original edge. The cuboctahedron, which may be called rectified cube, illustrates the process in three dimensions.
A bitruncated polychoron, as the name suggests, has the truncation carried further still: the parent's cell is reduced to its truncated dual. The growing vertex cell now collides with its siblings, so it too is truncated at the plane which bisected the parent's edge.
In a runcinated polytope the parent's facets are reduced to make room for those of its dual, without changing the shape
orientation of either. This leaves gaps corresponding to the parent's faces and edges, which are filled with prisms. The process is illustrated in three dimensions by the small rhombicosidodecahedron, whose squares can be considered two-dimensional prisms.
A cantitruncated polytope is rectified and then truncated.
A cantellated polytope is twice rectified.
Omnitruncation is similar to runcination, except that the facets of the parents are first cantitruncated.
TO DO: explain runcitruncation

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

If two polychora are duals of each other (such as the 16-cell and 16-cell,
the 120-cell and 600-cell), then bitruncating, runcinating
omnitruncating either produces the same figure as the same operation to the other.

The 5-cell family {3,3,3}

#	Name	5 (C)	10 (F)	10 (E)	5 (V)
1	5-cell	(3.3.3)	triangles	(threefold edges)	(vertices)
3	truncated 5-cell	(3.6.6)			(3.3.3)
2	rectified 5-cell	(3.3.3.3)		(vertices)	(3.3.3)
6*	bitruncated 5-cell	(3.6.6)			(3.6.6)
4	cantellated 5-cell	(3.4.3.4)		3.4.4	(3.3.3.3)
7	cantitruncated 5-cell	(4.6.6)		3.4.4	(3.6.6)
5*	runcinated 5-cell	(3.3.3)	3.4.4	3.4.4	(3.3.3)
8	runcitruncated 5-cell	(3.6.6)	(4.4.6)	3.4.4	(3.4.3.4)
9*	omnitruncated 5-cell	(4.6.6)	(4.4.6)	(4.4.6)	(4.6.6)

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 8-cell/16-cell family {4,3,3} and {3,3,4}

{3,3,4} 16-cell

#	Name	8 (V)	24 (E)	32 (F)	16 (C)
12	16-cell	(vertices)	(fourfold edges)	triangles	(3.3.3)
17	truncated 16-cell	(3.3.3.3)			(3.6.6)
**	rectified 16-cell	(3.3.3.3)	(vertices)		(3.3.3.3)
16	bitruncated 16-cell	(4.6.6)			(3.6.6)
**	cantellated 16-cell	(3.4.3.4)	(4.4.4)		(3.4.3.4)
**	cantitruncated 16-cell	(4.6.6)	(4.4.4)		(4.6.6)
15	runcinated 16-cell	(4.4.4)	(4.4.4)	3.4.4	(3.3.3)
20	runcitruncated 16-cell	(3.4.4.4)	(4.4.4)	(4.4.6)	(3.6.6)
21	omnitruncated 16-cell	(4.6.8)	(4.4.8)	(4.4.6)	(4.6.6)

{4,3,3}

#	Name	8 (C)	24 (F)	32 (E)	16 (V)
10	8-cell	(4.4.4)	squares	(threefold edges)	(vertices)
13	truncated 8-cell	(3.8.8)			(3.3.3)
11	rectified 8-cell	(3.4.3.4)		(vertices)	(3.3.3)
16	bitruncated 8-cell	(4.6.6)			(3.6.6)
14	cantellated 8-cell	(3.4.4.4)		3.4.4	(3.3.3.3)
18	cantitruncated 8-cell	(4.6.8)		3.4.4	(3.6.6)
15	runcinated 8-cell	(4.4.4)	(4.4.4)	3.4.4	(3.3.3)
19	runcitruncated 8-cell	(3.8.8)	(4.4.8)	3.4.4	(3.4.3.4)
21	omnitruncated 8-cell	(4.6.8)	(4.4.8)	(4.4.6)	(4.6.6)

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 24-cell family {3,4,3}

#	Name	24	96	96	24
22	24-cell	(3.3.3.3)	(triangles)	(fourfold edges)	(vertices)
24	truncated 24-cell	(4.6.6)			(4.4.4)
23	rectified 24-cell	(3.4.3.4)		(vertices)	(4.4.4)
27*	bitruncated 24-cell	(3.8.8)			(3.8.8)
25	cantellated 24-cell	(3.4.4.4)		3.4.4	(3.4.3.4)
28	cantitruncated 24-cell	(4.6.8)		3.4.4	(3.8.8)
26*	runcinated 24-cell	(3.3.3.3)	3.4.4	3.4.4	(3.3.3.3)
29	runcitruncated 24-cell	(4.6.6)	(4.4.6)	3.4.4	(3.4.4.4)
30*	omnitruncated 24-cell	(4.6.8)	(4.4.6)	(4.4.6)	(4.6.8)
31†	snub 24-cell	(3.3.3.3.3)		(3.3.3) (oblique)	(3.3.3)

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 120-cell/600-cell family {5,3,3} and {3,3,5}

{3,3,5}

#	Name	120 (V)	720 (E)	1200 (F)	600 (C)
35	600-cell	(vertices)	(fivefold edges)	triangles	(3.3.3)
41	truncated 600-cell	(3.3.3.3.3)			(3.6.6)
34	rectified 600-cell	(3.3.3.3.3)	(vertices)		(3.3.3.3)
39	bitruncated 600-cell	(5.6.6)			(3.6.6)
40	cantellated 600-cell	(3.5.3.5)	(4.4.5)		(3.4.3.4)
45	cantitruncated 600-cell	(5.6.6)	(4.4.5)		(4.6.6)
38	runcinated 600-cell	(5.5.5)	(4.4.5)	3.4.4	(3.3.3)
44	runcitruncated 600-cell	(3.4.5.4)	(4.4.5)	(4.4.6)	(3.6.6)
46	omnitruncated 600-cell	(4.6.10)	(4.4.10)	(4.4.6)	(4.6.6)

{5,3,3}

#	Name	120 (C)	720 (F)	1200 (E)	600 (V)
32	120-cell	(5.5.5)	pentagons	(threefold edges)	(vertices)
36	truncated 120-cell	(3.10.10)			(3.3.3)
33	rectified 120-cell	(3.5.3.5)		(vertices)	(3.3.3)
39	bitruncated 120-cell	(5.6.6)			(3.6.6)
37	cantellated 120-cell	(3.4.5.4)		3.4.4	(3.3.3.3)
42	cantitruncated 120-cell	(4.6.10)		3.4.4	(3.6.6)
38	runcinated 120-cell	(5.5.5)	(4.4.5)	3.4.4	(3.3.3)
43	runcitruncated 120-cell	(3.10.10)	(4.4.10)	3.4.4	(3.4.3.4)
46	omnitruncated 120-cell	(4.6.10)	(4.4.10)	(4.4.6)	(4.6.6)

This family has diploid hexacosichoric symmetry, of order 120*120=24*600=14400: 120 for each of the 120 dodecahedra,
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 (3.3.3). This is analogous to the band of triangles joining the two opposite faces in an antiprism. However, the grand antiprism is not a member of an infinite family of uniform polychora.

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.

The more obvious family of prismatic polychora is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a polychoron are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (one of which, the cube-prism, is listed above as the 16-cell), 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.

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 16-cell can be considered a 4,4-duoprism, though its symmetry is higher than that implies.)

There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms.

External links

Uniform polychora by Jonathan Bowers, who discovered many of them.
Uniform Polytopes in Four Dimensions by George Olshevsky, from which the data in the tables were taken
Uniform Polychora by Richard Klitzing