Site Map Polytopes Dynkin Diagrams Vertex Figures, etc. Incidence Matrices Index

Grünbaum-Coxeter Polytopes

Grünbaum in 1977 found some special abstract polytope which since then is known as 11-cell. Coxeter in 1982 came up with a related figure, the 57-cell.

The essential of the construction of these abstract polytopes needs some side step first. This one leads from spherical space to elliptical space. Spherical spaces are those of the surface of a ball of any dimension. Elliptical spaces are locally the same ones, even adopting the corresponding metrics, but do identify every pair of opposite points.

The polytopes of elliptical space   (up)

  1. Any convex polytope of any dimension can be projected from its center onto some circumscribing sphere of according dimension. This derives some spherical tesselation from the polytope. 

     ©
    		 → 
     ©

  2. If the polytope did have point symmetry with respect to its center, the tesselation too will have. In these cases those spherical tesselations would lead to corresponding elliptical tesselations, right by identification of opposite points. All polytopes with inversion symmetry therefore will lead to some elliptical polytopes, here-below denoted by a leading lower case letter e as operant for the elliptical space counterpart (e(...)).

As these again are abstract polytopes, one could thus speak of elliptical polytopes as well. Elliptical polytopes can alternatively be considered as polytopes of projective space.

The following listing is not ment to be complete. They only show the general procedures, and will be given explicitely whenever they will be used for the Grünbaum-Coxeter polytopes below.

2D
e(x-2n-o)   (n>1)
hemi-2n-gon 
hemi(. .) | n | 2
----------+---+--
hemi(x .) | 2 | n
  
3D
e(o3x3o)
elloct
hemi-octahedron 
hemi(. . .) | 3 | 4 | 2 2
------------+---+---+----
hemi(. x .) | 2 | 6 | 1 1
------------+---+---+----
hemi(o3x .) | 3 | 3 | 2 *
hemi(. x3o) | 3 | 3 | * 2
  
e(x3o3x)
elco
hemi-cuboctahedron 
hemi(. . .) | 6 | 2 2 | 1 2 1
------------+---+-----+------
hemi(x . .) | 2 | 6 * | 1 1 0
hemi(. . x) | 2 | * 6 | 0 1 1
------------+---+-----+------
hemi(x3o .) | 3 | 3 0 | 2 * *
hemi(x . x) | 4 | 2 2 | * 3 *
hemi(. o3x) | 3 | 0 3 | * * 2
e(x3x3x)
eltoe
truncated hemi-octahedron 
hemi(. . .) | 12 | 1 1 1 | 1 1 1
------------+----+-------+------
hemi(x . .) |  2 | 6 * * | 1 1 0
hemi(. x .) |  2 | * 6 * | 1 0 1
hemi(. . x) |  2 | * * 6 | 0 1 1
------------+----+-------+------
hemi(x3x .) |  6 | 3 3 0 | 2 * *
hemi(x . x) |  4 | 2 0 2 | * 3 *
hemi(. x3x) |  6 | 0 3 3 | * * 2
e(x3o4o)
elloct
hemi-octahedron 
hemi(. . .) | 3 | 4 | 4
------------+---+---+--
hemi(x . .) | 2 | 6 | 2
------------+---+---+--
hemi(x3o .) | 3 | 3 | 4
e(x4o3o)
elcube
hemi-cube
wikipedia 
hemi(. . .) | 4 | 3 | 3
------------+---+---+--
hemi(x . .) | 2 | 6 | 2
------------+---+---+--
hemi(x4o .) | 4 | 4 | 3
e(o3x4o)
elco
hemi-cuboctahedron 
hemi(. . .) | 6 |  4 | 2 2
------------+---+----+----
hemi(. x .) | 2 | 12 | 1 1
------------+---+----+----
hemi(o3x .) | 3 |  3 | 4 *
hemi(. x4o) | 4 |  4 | * 3
  
e(x3x4o)
eltoe
truncated hemi-octahedron 
hemi(. . .) | 12 | 1  2 | 2 1
------------+----+------+----
hemi(x . .) |  2 | 6  * | 2 0
hemi(. x .) |  2 | * 12 | 1 1
------------+----+------+----
hemi(x3x .) |  6 | 3  3 | 4 *
hemi(. x4o) |  4 | 0  4 | * 3
e(o3x4x)
eltic
truncated hemi-cube 
hemi(. . .) | 12 |  2 1 | 1 2
------------+----+------+----
hemi(. x .) |  2 | 12 * | 1 1
hemi(. . x) |  2 |  * 6 | 0 2
------------+----+------+----
hemi(o3x .) |  3 |  3 0 | 4 *
hemi(. x4x) |  8 |  4 4 | * 3
e(x3o4x)
elsirco
small rhombated hemi-cuboctahedron

(more general:
e(y3o4x) with "y" = x,q,f,...) 
hemi(. . .) | 12 |  2  2 | 1 2 1
------------+----+-------+------
hemi(x . .) |  2 | 12  * | 1 1 0
hemi(. . x) |  2 |  * 12 | 0 1 1
------------+----+-------+------
hemi(x3o .) |  3 |  3  0 | 4 * *
hemi(x . x) |  4 |  2  2 | * 6 *
hemi(. o4x) |  4 |  0  4 | * * 3
e(x3x4x)
elgirco
great rhombated hemi-cuboctahedron 
hemi(. . .) | 24 |  1  1  1 | 1 1 1
------------+----+----------+------
hemi(x . .) |  2 | 12  *  * | 1 1 0
hemi(. x .) |  2 |  * 12  * | 1 0 1
hemi(. . x) |  2 |  *  * 12 | 0 1 1
------------+----+----------+------
hemi(x3x .) |  6 |  3  3  0 | 4 * *
hemi(x . x) |  4 |  2  0  2 | * 6 *
hemi(. x4x) |  8 |  0  4  4 | * * 3
e(x3o5o)
ellike
hemi-icosahedron
wikipedia 
hemi(. . .) | 6 |  5 |  5
------------+---+----+---
hemi(x . .) | 2 | 15 |  2
------------+---+----+---
hemi(x3o .) | 3 |  3 | 10
e(x5o3o)
eldoe
hemi-dodecahedron
wikipedia 
hemi(. . .) | 10 |  3 | 3
------------+----+----+--
hemi(x . .) |  2 | 15 | 2
------------+----+----+--
hemi(x5o .) |  5 |  5 | 6
e(o3x5o)
ellid
hemi-icosidodecahedron 
hemi(. . .) | 15 |  4 |  2 2
------------+----+----+-----
hemi(. x .) |  2 | 30 |  1 1
------------+----+----+-----
hemi(o3x .) |  3 |  3 | 10 *
hemi(. x5o) |  5 |  5 |  * 6
  
e(x3x5o)
elti
truncated hemi-icosahedron 
hemi(. . .) | 30 |  1  2 |  2 1
------------+----+-------+-----
hemi(x . .) |  2 | 15  * |  2 0
hemi(. x .) |  2 |  * 30 |  1 1
------------+----+-------+-----
hemi(x3x .) |  6 |  3  3 | 10 *
hemi(. x5o) |  5 |  0  5 |  * 6
e(o3x5x)
eltid
truncated hemi-dodecahedron 
hemi(. . .) | 30 |  2  1 |  1 2
------------+----+-------+-----
hemi(. x .) |  2 | 30  * |  1 1
hemi(. . x) |  2 |  * 15 |  0 2
------------+----+-------+-----
hemi(o3x .) |  3 |  3  0 | 10 *
hemi(. x5x) | 10 |  5  5 |  * 6
e(x3o5x)
elsrid
small rhombated hemi-icosidodecahedron

(more general:
e(y3o5x) with "y" = x,q,f,...) 
hemi(. . .) | 30 |  2  2 |  1  2 1
------------+----+-------+--------
hemi(x . .) |  2 | 30  * |  1  1 0
hemi(. . x) |  2 |  * 30 |  0  1 1
------------+----+-------+--------
hemi(x3o .) |  3 |  3  0 | 10  * *
hemi(x . x) |  4 |  2  2 |  * 15 *
hemi(. o5x) |  5 |  0  5 |  *  * 6
e(x3x5x)
elgrid
great rhombated hemi-icosidodecahedron 
hemi(. . .) | 60 |  1  1  1 |  1  1 1
------------+----+----------+--------
hemi(x . .) |  2 | 30  *  * |  1  1 0
hemi(. x .) |  2 |  * 30  * |  1  0 1
hemi(. . x) |  2 |  *  * 30 |  0  1 1
------------+----+----------+--------
hemi(x3x .) |  6 |  3  3  0 | 10  * *
hemi(x . x) |  4 |  2  0  2 |  * 15 *
hemi(. x5x) | 10 |  0  5  5 |  *  * 6
4D
e(x3o3o4o)
elhex
hemi-hexadecachorron 
hemi(. . . .) | 4 ♦  6 | 12 | 8
--------------+---+----+----+--
hemi(x . . .) | 2 | 12 |  4 | 4
--------------+---+----+----+--
hemi(x3o . .) | 3 |  3 | 16 | 2
--------------+---+----+----+--
hemi(x3o3o .) ♦ 4 |  6 |  4 | 8
e(x4o3o3o)
eltes
hemi-tesseract 
hemi(. . . .) | 8 ♦  4 |  6 | 4
--------------+---+----+----+--
hemi(x . . .) | 2 | 16 |  3 | 3
--------------+---+----+----+--
hemi(x4o . .) | 4 |  4 | 12 | 2
--------------+---+----+----+--
hemi(x4o3o .) ♦ 8 | 12 |  6 | 4
e(o3x3o4o)
ellico
hemi-icositetrachorron 
hemi(. . . .) | 12 ♦  8 |  4  8 | 4 2
--------------+----+----+-------+----
hemi(. x . .) |  2 | 48 |  1  2 | 2 1
--------------+----+----+-------+----
hemi(o3x . .) |  3 |  3 | 16  * | 2 0
hemi(. x3o .) |  3 |  3 |  * 32 | 1 1
--------------+----+----+-------+----
hemi(o3x3o .) ♦  6 | 12 |  4  4 | 8 *
hemi(. x3o4o) ♦  6 | 12 |  0  8 | * 4
  
e(x3o4o3o)
ellico
hemi-icositetrachorron 
hemi(. . . .) | 12 ♦  8 | 12 |  6
--------------+----+----+----+---
hemi(x . . .) |  2 | 48 |  3 |  3
--------------+----+----+----+---
hemi(x3o . .) |  3 |  3 | 48 |  2
--------------+----+----+----+---
hemi(x3o4o .) ♦  6 | 12 |  8 | 12
  
e(x3o3o5o)
ellex
hemi-hexacosachorron 
hemi(. . . .) | 60 ♦  12 |  30 |  20
--------------+----+-----+-----+----
hemi(x . . .) |  2 | 360 |   5 |   5
--------------+----+-----+-----+----
hemi(x3o . .) |  3 |   3 | 600 |   2
--------------+----+-----+-----+----
hemi(x3o3o .) ♦  4 |   6 |   4 | 300
e(x5o3o3o)
elhi
hemi-hecatonicosachorron
Davis manifold 
hemi(. . . .) | 300 ♦   4 |   6 |  4
--------------+-----+-----+-----+---
hemi(x . . .) |   2 | 600 |   3 |  3
--------------+-----+-----+-----+---
hemi(x5o . .) |   5 |   5 | 360 |  2
--------------+-----+-----+-----+---
hemi(x5o3o .) ♦  20 |  30 |  12 | 60
e(x  x3o4o)
ellope
hemi - octahedron-prism 
hemi(. . . .) | 6 | 1  4 | 4 4 | 4 1
--------------+---+------+-----+----
hemi(x . . .) | 2 | 3  * | 4 0 | 4 0
hemi(. x . .) | 2 | * 12 | 1 2 | 2 1
--------------+---+------+-----+----
hemi(x x . .) | 4 | 2  2 | 6 * | 2 0
hemi(. x3o .) | 3 | 0  3 | * 8 | 1 1
--------------+---+------+-----+----
hemi(x x3o .) ♦ 6 | 3  6 | 3 2 | 4 *
hemi(. x3o4o) ♦ 6 | 0 12 | 0 8 | * 1
e(x  x4o3o)
eltes
hemi-tesseract 
hemi(. . . .) | 8 ♦ 1  3 | 3 3 | 3 1
--------------+---+------+-----+----
hemi(x . . .) | 2 | 4  * | 3 0 | 3 0
hemi(. x . .) | 2 | * 12 | 1 2 | 2 1
--------------+---+------+-----+----
hemi(x x . .) | 4 | 2  2 | 6 * | 2 0
hemi(. x4o .) | 4 | 0  4 | * 6 | 1 1
--------------+---+------+-----+----
hemi(x x4o .) ♦ 8 | 4  8 | 4 2 | 3 *
hemi(. x4o3o) ♦ 8 | 0 12 | 0 6 | * 1
e(x4o  x4o)
eltes
hemi-tesseract 
hemi(. . . .) | 8 ♦ 2 2 | 1 4 1 | 2 2
--------------+---+-----+-------+----
hemi(x . . .) | 2 | 8 * | 1 2 0 | 2 1
hemi(. . x .) | 2 | * 8 | 0 2 1 | 1 2
--------------+---+-----+-------+----
hemi(x4o . .) | 4 | 4 0 | 2 * * | 2 0
hemi(x . x .) | 4 | 2 2 | * 8 * | 1 1
hemi(. . x4o) | 4 | 0 4 | * * 2 | 0 2
--------------+---+-----+-------+----
hemi(x4o x .) ♦ 8 | 8 4 | 2 4 0 | 2 *
hemi(x . x4o) ♦ 8 | 4 8 | 0 4 2 | * 2
  
5D
e(x3o3o3o4o)
eltac
hemi-triacontiditeron 
hemi(. . . . .) | 5 ♦  8 | 24 | 32 | 16
----------------+---+----+----+----+---
hemi(x . . . .) | 2 | 20 ♦  6 | 12 |  8
----------------+---+----+----+----+---
hemi(x3o . . .) | 3 |  3 | 40 |  4 |  4
----------------+---+----+----+----+---
hemi(x3o3o . .) ♦ 4 |  6 |  4 | 40 |  2
----------------+---+----+----+----+---
hemi(x3o3o3o .) ♦ 5 | 10 | 10 |  5 | 16
e(x4o3o3o3o)
elpent
hemi-penteract 
hemi(. . . . .) | 16 ♦  5 | 10 | 10 | 5
----------------+----+----+----+----+--
hemi(x . . . .) |  2 | 40 ♦  4 |  6 | 4
----------------+----+----+----+----+--
hemi(x4o . . .) |  4 |  4 | 40 |  3 | 3
----------------+----+----+----+----+--
hemi(x4o3o . .) ♦  8 | 12 |  6 | 20 | 2
----------------+----+----+----+----+--
hemi(x4o3o3o .) ♦ 16 | 32 | 24 |  8 | 5
e(o3x3o3o4o)
elrat
rectified hemi-triacontiditeron 
hemi(. . . . .) | 20 ♦  12 |  6  24 | 12 16 |  8 2
----------------+----+-----+--------+-------+-----
hemi(. x . . .) |  2 | 120 |  1   4 |  4  4 |  4 1
----------------+----+-----+--------+-------+-----
hemi(o3x . . .) |  3 |   3 | 40   * |  4  0 |  4 0
hemi(. x3o . .) |  3 |   3 |  * 160 |  1  2 |  2 1
----------------+----+-----+--------+-------+-----
hemi(o3x3o . .) ♦  6 |  12 |  4   4 | 40  * |  2 0
hemi(. x3o3o .) ♦  4 |   6 |  0   4 |  * 80 |  1 1
----------------+----+-----+--------+-------+-----
hemi(o3x3o3o .) ♦ 10 |  30 | 10  20 |  5  5 | 16 *
hemi(. x3o3o4o) ♦  8 |  24 |  0  32 |  0 16 |  * 5
e(o3o3x3o4o)
elnit
hemi-penteractitriacontiditeron 
hemi(. . . . .) | 40 ♦  12 |  12  12 |  4 12  3 |  4 3
----------------+----+-----+---------+----------+-----
hemi(. . x . .) |  2 | 240 |   2   2 |  1  4  1 |  2 2
----------------+----+-----+---------+----------+-----
hemi(. o3x . .) |  3 |   3 | 160   * |  1  2  0 |  2 1
hemi(. . x3o .) |  3 |   3 |   * 160 |  0  2  1 |  1 2
----------------+----+-----+---------+----------+-----
hemi(o3o3x . .) ♦  4 |   6 |   4   0 | 40  *  * |  2 0
hemi(. o3x3o .) ♦  6 |  12 |   4   4 |  * 80  * |  1 1
hemi(. . x3o4o) ♦  6 |  12 |   0   8 |  *  * 20 |  0 2
----------------+----+-----+---------+----------+-----
hemi(o3o3x3o .) ♦ 10 |  30 |  20  10 |  5  5  0 | 16 *
hemi(. o3x3o4o) ♦ 24 |  96 |  32  64 |  0 16  8 |  * 5
e(x4o  x3o4o)
elsquoct
hemi - square-octahedron-duoprism 
hemi(. . . . .) | 12 |  2  4 | 1  8  4 | 4  8 1 | 4 2
----------------+----+-------+---------+--------+----
hemi(x . . . .) |  2 | 12  * | 1  4  0 | 4  4 0 | 4 1
hemi(. . x . .) |  2 |  * 24 | 0  2  2 | 1  4 1 | 2 2
----------------+----+-------+---------+--------+----
hemi(x4o . . .) |  4 |  4  0 | 3  *  * | 4  0 0 | 4 0
hemi(x . x . .) |  4 |  2  2 | * 24  * | 1  2 0 | 2 1
hemi(. . x3o .) |  3 |  0  3 | *  * 16 | 0  2 1 | 1 2
----------------+----+-------+---------+--------+----
hemi(x4o x . .) ♦  8 |  8  4 | 2  4  0 | 6  * * | 2 0
hemi(x . x3o .) ♦  6 |  3  6 | 0  3  2 | * 16 * | 1 1
hemi(. . x3o4o) ♦  6 |  0 12 | 0  0  8 | *  * 2 | 0 2
----------------+----+-------+---------+--------+----
hemi(x4o x3o .) ♦ 12 | 12 12 | 3 12  4 | 3  4 0 | 4 *
hemi(x . x3o4o) ♦ 12 |  6 24 | 0 12 16 | 0  8 2 | * 2
  

The construction of Grünbaum-Coxeter polytopes   (up)

Essentially Grünbaum-Coxeter polytopes are set up like usual polytopes. But instead of usual spherical space polytopes for facets, they use elliptical space ones. And additionally instead of spherical space polytopes for vertex figures, they use elliptical space ones.

Instead of this bottom-up construction by doing elliptical identifications of elements first and then adding small things to larger structures in whatever spaces these thingies might live in, also a top-down construction can be applied. One could well start with usual (regular) tesselations {p, q, ..., r}, and apply kind of modulo-wrappings to them in order to derive GC{p, q, ..., r} by identifications of elements. Kind of like this has been done according to the cellular and vertex figural identifications of opposite incident elements.

GC(x4o4o) as mod-wrapp of x4o4o GC(x6o6o) as mod-wrapp of x6o6o GC(x4o4o6*a) as mod-wrapp of x4o4o6*a

Surprisingly it comes out in 3D that there is some general isomorphism between GC(x-2N-o-2M-o) and x-N-o-M-o, as can be seen from the incidence matrices shown below. – This can be understood as follows. The face polygons of GC(x-2N-o-2M-o) are e(x-2N-o   .), which virtually function like x-N-o. On the other hand the vertex figure of GC(x-2N-o-2M-o) is (up to scaling) e(.   x-2M-o), which in turn virtually functions like x-M-o. Those two objects interlink mutually and will produce (bottom-up) finally the logical structure of x-N-o-M-o.

The same holds true for the closed loop Dynkin symbols of that dimension, i.e. for GC(x-2P-o-2Q-o-2R-*a). Here the faces are e(x-2P-o   .   ) and e(x   .   o-2R-*a), which virtually function like x-P-o resp. x-R-o. The vertex figure of GC(x-2P-o-2Q-o-2R-*a) is e(.   a-2Q-b   ), where the edge sizes a and b depend on 2P resp. 2R and additionally on the elliptical identification of these faces. The final functionality of that vertex figure is like x(P)-Q-x(R). Therefore GC(x-2P-o-2Q-o-2R-*a) comes out to be isomorphic to x-P-o-Q-o-R-*a. (It should be noted, that here either of P, Q, R might be 2 as well, but that the reduction of these digons is not allowed, cf. digons within abstract polytopes.)

*)
In cases denoted by this asterix, the pre-images of the facets under the identifications (top-down method) are cells of the respective tiling, which are monostratic only, i.e. have vertices in exactly 2 parallel layers. Therefore the identification will pass on the vertex numbers of the cellular top-facet to the bottom-facet, and all cellular latteral facets too do have the same numbered vertices. Accodingly any neighbouring cell will pass on this very number set to its opposite vertices, etc. Those Grünbaum-Coxeter polytopes, marked by this asterix, are, like dihedra, somehow degenerate: for there, the vertex count of GC(x-P-o-...-o-Q-o-R-o-S-o) equals the vertex count of its cells e(x-P-o-...-o-Q-o-R-o), and that vertex count in turn equals moreover the vertex count of the ridges x-P-o-...-o-Q-o.
3D
GC(x¯4¯o¯2n¯o)
(n>1)   *) 
mod(e(. . .)) | 2 | n | n
--------------+---+---+--
mod(  x . . ) | 2 | n | 2
--------------+---+---+--
mod(e(x4o .)) | 2 | 2 | n
GC(x¯2n¯o¯4¯o)
(n>1) 
mod(e(.  . .)) | n | 2 | 2
---------------+---+---+--
mod(  x  . . ) | 2 | n | 2
---------------+---+---+--
mod(e(x2no .)) | n | n | 2
GC(o4x4o) 
mod(e(. . .)) | 2 | 2 | 1 1
--------------+---+---+----
mod(  . x . ) | 2 | 2 | 1 1
--------------+---+---+----
mod(e(o4x .)) | 2 | 2 | 1 *
mod(e(. x4o)) | 2 | 2 | * 1
  
GC(x6o6o) 
mod(e(. . .)) | 4 | 3 | 3
--------------+---+---+--
mod(  x . . ) | 2 | 6 | 2
--------------+---+---+--
mod(e(x6o .)) | 3 | 3 | 4
  
GC(o6x6o) 
mod(e(. . .)) | 3 | 2 | 1 1
--------------+---+---+----
mod(  . x . ) | 2 | 3 | 1 1
--------------+---+---+----
mod(e(o6x .)) | 3 | 3 | 1 *
mod(e(. x6o)) | 3 | 3 | * 1
  
GC(x6o8o) 
mod(e(. . .)) | 6 |  4 | 4
--------------+---+----+--
mod(  x . . ) | 2 | 12 | 2
--------------+---+----+--
mod(e(x6o .)) | 3 |  3 | 8
GC(x8o6o) 
mod(e(. . .)) | 8 |  3 | 3
--------------+---+----+--
mod(  x . . ) | 2 | 12 | 2
--------------+---+----+--
mod(e(x8o .)) | 4 |  4 | 6
GC(x6o10o) 
mod(e(. .  .)) | 12 |  5 |  5
---------------+----+----+---
mod(  x .  . ) |  2 | 30 |  2
---------------+----+----+---
mod(e(x6o  .)) |  3 |  3 | 20
GC(x10o6o) 
mod(e(.  . .)) | 20 |  3 |  3
---------------+----+----+---
mod(  x  . . ) |  2 | 30 |  2
---------------+----+----+---
mod(e(x10o .)) |  5 |  5 | 12
GC(x¯4¯o¯4¯o¯2n¯*a)
(n>1) 
mod(e(. . .    )) | n |  4 | 2 2
------------------+---+----+----
mod(  x . .     ) | 2 | 2n | 1 1
------------------+---+----+----
mod(e(x4o .    )) | 2 |  2 | n *
mod(e(x . o2n*a)) | n |  n | * 2
GC(o¯4¯x¯4¯o¯2n¯*a)
(n>1) 
mod(e(. . .    )) | 2 | 2 | 1 1
------------------+---+---+----
mod(  . x .     ) | 2 | 2 | 1 1
------------------+---+---+----
mod(e(o4x .    )) | 2 | 2 | 1 *
mod(e(. x4o    )) | 2 | 2 | * 1
GC(x6o4o6*a) 
mod(e(. . .   )) | 6 |  4 | 2 2
-----------------+---+----+----
mod(  x . .    ) | 2 | 12 | 1 1
-----------------+---+----+----
mod(e(x6o .   )) | 3 |  3 | 4 *
mod(e(x . o6*a)) | 3 |  3 | * 4
  
GC(x6o4o8*a) 
mod(e(. . .   )) | 12 |  4 | 2 2
-----------------+----+----+----
mod(  x . .    ) |  2 | 24 | 1 1
-----------------+----+----+----
mod(e(x6o .   )) |  3 |  3 | 8 *
mod(e(x . o8*a)) |  4 |  4 | * 6
  
GC(x6o4o10*a) 
mod(e(. . .    )) | 30 |  4 |  2  2
------------------+----+----+------
mod(  x . .     ) |  2 | 60 |  1  1
------------------+----+----+------
mod(e(x6o .    )) |  3 |  3 | 20  *
mod(e(x . o10*a)) |  5 |  5 |  * 12
  
4D
GC(o3x3o4o)
*) 
mod(e(. . . .)) | 3 ♦ 4 | 2 4 | 2 1
----------------+---+---+-----+----
mod(  . x . . ) | 2 | 6 | 1 2 | 2 1
----------------+---+---+-----+----
mod(  o3x . . ) | 3 | 3 | 2 * | 2 0
mod(  . x3o . ) | 3 | 3 | * 4 | 1 1
----------------+---+---+-----+----
mod(e(o3x3o .)) ♦ 3 | 6 | 2 2 | 2 *
mod(e(. x3o4o)) ♦ 3 | 6 | 0 4 | * 1
  
GC(x3o4o3o)
*) 
mod(e(. . . .)) | 3 ♦ 4 | 6 | 3
----------------+---+---+---+--
mod(  x . . . ) | 2 | 6 | 3 | 3
----------------+---+---+---+--
mod(  x3o . . ) | 3 | 3 | 6 | 2
----------------+---+---+---+--
mod(e(x3o4o .)) ♦ 3 | 6 | 4 | 3
  
GC(x4o3o4o)
*) 
mod(e(. . . .)) | 4 ♦ 3 | 6 | 4
----------------+---+---+---+--
mod(  x . . . ) | 2 | 6 | 4 | 4
----------------+---+---+---+--
mod(  x4o . . ) | 4 | 4 | 6 | 2
----------------+---+---+---+--
mod(e(x4o3o .)) ♦ 4 | 6 | 3 | 4
  
GC(x4o3o5o)
*) 
mod(e(. . . .)) | 4 ♦  6 | 15 | 10
----------------+---+----+----+---
mod(  x . . . ) | 2 | 12 |  5 |  5
----------------+---+----+----+---
mod(  x4o . . ) | 4 |  4 | 15 |  2
----------------+---+----+----+---
mod(e(x4o3o .)) ♦ 4 |  6 |  3 | 10
GC(x5o3o4o) 
mod(e(. . . .)) | 10 ♦  3 |  6 | 4
----------------+----+----+----+--
mod(  x . . . ) |  2 | 15 |  4 | 4
----------------+----+----+----+--
mod(  x5o . . ) |  5 |  5 | 12 | 2
----------------+----+----+----+--
mod(e(x5o3o .)) ♦ 10 | 15 |  6 | 4
GC(x3o5o3o)
"11-cell"
wikipedia 
mod(e(. . . .)) | 11 ♦ 10 | 15 |  6
----------------+----+----+----+---
mod(  x . . . ) |  2 | 55 |  3 |  3
----------------+----+----+----+---
mod(  x3o . . ) |  3 |  3 | 55 |  2
----------------+----+----+----+---
mod(e(x3o5o .)) ♦  6 | 15 | 10 | 11
  
GC(x5o3o5o)
"57-cell"
wikipedia 
mod(e(. . . .)) | 57 ♦   6 |  15 | 10
----------------+----+-----+-----+---
mod(  x . . . ) |  2 | 171 |   5 |  5
----------------+----+-----+-----+---
mod(  x5o . . ) |  5 |   5 | 171 |  2
----------------+----+-----+-----+---
mod(e(x5o3o .)) ♦ 10 |  15 |   6 | 57
  
GC(x4o3o4o3*a) 
mod(e(. . . .   )) | 7 ♦ 12 | 12 12 | 4 6 3
-------------------+---+----+-------+------
mod(  x . . .    ) | 2 | 42 |  2  2 | 1 2 1
-------------------+---+----+-------+------
mod(  x4o . .    ) | 4 |  4 | 21  * | 1 1 0
mod(  x . . o3*a ) | 3 |  3 |  * 28 | 0 1 1
-------------------+---+----+-------+------
mod(e(x4o3o .   )) ♦ 4 |  6 |  3  0 | 7 * *
mod(e(x4o . o3*a)) ♦ 6 | 12 |  3  4 | * 7 *
mod(e(x . o4o3*a)) ♦ 3 |  6 |  0  4 | * * 7
GC(x5o3o5o3*a)
(N so far unknown) 
mod(e(. . . .   )) |  N ♦  30 | 30  30 | 10 15 6
-------------------+----+-----+--------+--------
mod(  x . . .    ) |  2 | 15N |  2   2 |  1  2 1
-------------------+----+-----+--------+--------
mod(  x5o . .    ) |  5 |   5 | 6N   * |  1  1 0
mod(  x . . o3*a ) |  3 |   3 |  * 10N |  0  1 1
-------------------+----+-----+--------+--------
mod(e(x5o3o .   )) ♦ 10 |  15 |  6   0 |  N  * *
mod(e(x5o . o3*a)) ♦ 15 |  30 |  6  10 |  *  N *
mod(e(x . o5o3*a)) ♦  6 |  15 |  0  10 |  *  * N
GC(x4o3o5o3*a)
(N so far unknown) 
mod(e(. . . .   )) | 2N ♦  30 |  30  30 | 10 15  6
-------------------+----+-----+---------+---------
mod(  x . . .    ) |  2 | 30N |   2   2 |  1  2  1
-------------------+----+-----+---------+---------
mod(  x4o . .    ) |  4 |   4 | 15N   * |  1  1  0
mod(  x . . o3*a ) |  3 |   3 |   * 20N |  0  1  1
-------------------+----+-----+---------+---------
mod(e(x4o3o .   )) ♦  4 |   6 |   3   0 | 5N  *  *
mod(e(x4o . o3*a)) ♦  6 |  12 |   3   4 |  * 5N  *
mod(e(x . o5o3*a)) ♦  6 |  15 |   0  10 |  *  * 2N
GC(x5o3o4o3*a)
(N so far unknown) 
mod(e(. . . .   )) | 5N ♦  12 |  12  12 |  4  6  3
-------------------+----+-----+---------+---------
mod(  x . . .    ) |  2 | 30N |   2   2 |  1  2  1
-------------------+----+-----+---------+---------
mod(  x5o . .    ) |  5 |   5 | 12N   * |  1  1  0
mod(  x . . o3*a ) |  3 |   3 |   * 20N |  0  1  1
-------------------+----+-----+---------+---------
mod(e(x5o3o .   )) ♦ 10 |  15 |   6   0 | 2N  *  *
mod(e(x5o . o3*a)) ♦ 15 |  30 |   6  10 |  * 2N  *
mod(e(x . o4o3*a)) ♦  3 |   6 |   0   4 |  *  * 5N
5D
GC(x3o3o4o3o)
*) 
mod(e(. . . . .)) | 4 ♦ 12 | 48 | 48 | 12
------------------+---+----+----+----+---
mod(  x . . . . ) | 2 | 24 ♦  8 | 12 |  6
------------------+---+----+----+----+---
mod(  x3o . . . ) | 3 |  3 | 64 |  3 |  3
------------------+---+----+----+----+---
mod(  x3o3o . . ) ♦ 4 |  6 |  4 | 48 |  2
------------------+---+----+----+----+---
mod(e(x3o3o4o .)) ♦ 4 | 12 | 16 |  8 | 12
GC(x3o4o3o3o) 
mod(e(. . . . .)) | 12 ♦  8 | 16 | 12 | 4
------------------+----+----+----+----+--
mod(  x . . . . ) |  2 | 48 ♦  4 |  6 | 4
------------------+----+----+----+----+--
mod(  x3o . . . ) |  3 |  3 | 64 |  3 | 3
------------------+----+----+----+----+--
mod(  x3o4o . . ) ♦  6 | 12 |  8 | 24 | 2
------------------+----+----+----+----+--
mod(e(x3o4o3o .)) ♦ 12 | 48 | 48 | 12 | 4
GC(o3x3o4o3o) 
mod(e(. . . . .)) | 12 ♦  8 |  4 12 |  6  6 | 3 1
------------------+----+----+-------+-------+----
mod(  . x . . . ) |  2 | 48 ♦  1  3 |  3  3 | 3 1
------------------+----+----+-------+-------+----
mod(  o3x . . . ) |  3 |  3 | 16  * |  3  0 | 3 0
mod(  . x3o . . ) |  3 |  3 |  * 48 |  1  2 | 2 1
------------------+----+----+-------+-------+----
mod(  o3x3o . . ) ♦  6 | 12 |  4  4 | 12  * | 2 0
mod(  . x3o4o . ) ♦  6 | 12 |  0  8 |  * 12 | 1 1
------------------+----+----+-------+-------+----
mod(e(o3x3o4o .)) ♦ 12 | 48 | 16 32 |  8  4 | 3 *
mod(e(. x3o4o3o)) ♦ 12 | 48 |  0 48 |  0 12 | * 1
  
GC(x4o3o3o4o)
*) 
mod(e(. . . . .)) | 8 ♦  4 | 12 | 16 | 8
------------------+---+----+----+----+--
mod(  x . . . . ) | 2 | 16 ♦  6 | 12 | 8
------------------+---+----+----+----+--
mod(  x4o . . . ) | 4 |  4 | 24 |  4 | 4
------------------+---+----+----+----+--
mod(  x4o3o . . ) ♦ 8 | 12 |  6 | 16 | 2
------------------+---+----+----+----+--
mod(e(x4o3o3o .)) ♦ 8 | 16 | 12 |  4 | 8
  
GC(o4o3x3o4o) 
mod(e(. . . . .)) | 12 ♦  8 |  8  8 | 2  8 2 | 2 2
------------------+----+----+-------+--------+----
mod(  . . x . . ) |  2 | 48 ♦  2  2 | 1  4 1 | 2 2
------------------+----+----+-------+--------+----
mod(  . o3x . . ) |  3 |  3 | 32  * | 1  2 0 | 2 1
mod(  . . x3o . ) |  3 |  3 |  * 32 | 0  2 1 | 1 2
------------------+----+----+-------+--------+----
mod(  o4o3x . . ) ♦  6 | 12 |  8  0 | 4  * * | 2 0
mod(  . o3x3o . ) ♦  6 | 12 |  4  4 | * 16 * | 1 1
mod(  . . x3o4o ) ♦  6 | 12 |  0  8 | *  * 4 | 0 2
------------------+----+----+-------+--------+----
mod(e(o4o3x3o .)) ♦ 12 | 48 | 32 16 | 4  8 0 | 2 *
mod(e(. o3x3o4o)) ♦ 12 | 48 | 16 32 | 0  8 4 | * 2
  
GC(x4o3o3o5o)
*) 
mod(e(. . . . .)) | 8 ♦  60 | 360 | 600 | 300
------------------+---+-----+-----+-----+----
mod(  x . . . . ) | 2 | 240 ♦  12 |  30 |  20
------------------+---+-----+-----+-----+----
mod(  x4o . . . ) | 4 |   4 | 720 |   5 |   5
------------------+---+-----+-----+-----+----
mod(  x4o3o . . ) ♦ 8 |  12 |   6 | 600 |   2
------------------+---+-----+-----+-----+----
mod(e(x4o3o3o .)) ♦ 8 |  16 |  12 |   4 | 300
GC(x5o3o3o4o) 
mod(e(. . . . .)) | 300 ♦   4 |  12 |  16 | 8
------------------+-----+-----+-----+-----+--
mod(  x . . . . ) |   2 | 600 ♦   6 |  12 | 8
------------------+-----+-----+-----+-----+--
mod(  x5o . . . ) |   5 |   5 | 720 |   4 | 4
------------------+-----+-----+-----+-----+--
mod(  x5o3o . . ) ♦  20 |  30 |  12 | 240 | 2
------------------+-----+-----+-----+-----+--
mod(e(x5o3o3o .)) ♦ 300 | 600 | 360 |  60 | 8
GC(x5o3o3o5o)
(N so far unknown) 
mod(e(. . . . .)) |   N ♦  60 | 360 | 600 | 300
------------------+-----+-----+-----+-----+----
mod(  x . . . . ) |   2 | 30N ♦  12 |  30 |  20
------------------+-----+-----+-----+-----+----
mod(  x5o . . . ) |   5 |   5 | 72N |   5 |   5
------------------+-----+-----+-----+-----+----
mod(  x5o3o . . ) ♦  20 |  30 |  12 | 30N |   2
------------------+-----+-----+-----+-----+----
mod(e(x5o3o3o .)) ♦ 300 | 600 | 360 |  60 |   N
  
6D
GC(x4o3o3o3o4o)
*) 
mod(e(. . . . . .)) | 16 ♦  5 | 20 | 40 | 40 | 16
--------------------+----+----+----+----+----+---
mod(  x . . . . . ) |  2 | 40 ♦  8 | 24 | 32 | 16
--------------------+----+----+----+----+----+---
mod(  x4o . . . . ) |  4 |  4 | 80 ♦  6 | 12 |  8
--------------------+----+----+----+----+----+---
mod(  x4o3o . . . ) ♦  8 | 12 |  6 | 80 |  4 |  4
--------------------+----+----+----+----+----+---
mod(  x4o3o3o . . ) ♦ 16 | 32 | 24 |  8 | 40 |  2
--------------------+----+----+----+----+----+---
mod(e(x4o3o3o3o .)) ♦ 16 | 40 | 40 | 20 |  5 | 16
  
GC(o4o3x3o3o4o) 
mod(e(. . . . . .)) | 40 ♦  12 |  12  24 |  3  24  16 |  6 16  2 | 4 2
--------------------+----+-----+---------+------------+----------+----
mod(  . . x . . . ) |  2 | 240 |   2   4 |  1   8   4 |  4  8  1 | 4 2
--------------------+----+-----+---------+------------+----------+----
mod(  . o3x . . . ) |  3 |   3 | 160   * |  1   4   0 |  4  4  0 | 4 1
mod(  . . x3o . . ) |  3 |   3 |   * 320 |  0   2   2 |  1  4  1 | 2 2
--------------------+----+-----+---------+------------+----------+----
mod(  o4o3x . . . ) ♦  6 |  12 |   8   0 | 20   *   * |  4  0  0 | 4 0
mod(  . o3x3o . . ) ♦  6 |  12 |   4   4 |  * 160   * |  1  2  0 | 2 1
mod(  . . x3o3o . ) ♦  4 |   6 |   0   4 |  *   * 160 |  0  2  1 | 1 2
--------------------+----+-----+---------+------------+----------+----
mod(  o4o3x3o . . ) ♦ 24 |  96 |  64  32 |  8  16   0 | 10  *  * | 2 0
mod(  . o3x3o3o . ) ♦ 10 |  30 |  10  20 |  0   5   5 |  * 64  * | 1 1
mod(  . . x3o3o4o ) ♦  8 |  24 |   0  32 |  0   0  16 |  *  * 10 | 0 2
--------------------+----+-----+---------+------------+----------+----
mod(e(o4o3x3o3o .)) ♦ 40 | 240 | 160 160 | 20  80  40 |  5 16  0 | 4 *
mod(e(. o3x3o3o4o)) ♦ 20 | 120 |  40 160 |  0  40  80 |  0 16  5 | * 4
  

For the regular ones the above list is complete, except that in dimensions above 6 additionaly only the analogue of the single regular one of 6D does exist. This is because that the pre-image structures within the parantheses might well be spherical, euclidean, or hyperbolic. But the latter two cases subject to the additional demand that all its subelements must be spherical again. Only then the vertex figures and the facets can be made elliptical by identifications across their centers (provided those all have inversion symmetry).

For GC(x5o3o3o5o) so far only the incidence structure is known to me. It well might be an infinite one none the same and therefore would then be omitted from this list again. If this will come out to be true, then the famous 11-cell and 57-cell are the only finite Coxeter-Grünbaum polytopes which neither are degenerate in the above mentioned sense (or dual to such) nor are isomorphic to usual regular polytopes of spherical space.

For the non-regular cases the above restrictions hold true again. (Especially the inversion symmetry of the vertex figure here is a severe limitation.) The here listed ones are complete as long one considers linear Dynkin diagrams for the pre-images only, and moreover the vertex figure polytope of those is uniform as well. (Cases having uniform cells, but inversion symmetric vertex figure polytopes with different edge sizes, for either pre-images, would exist in addition.) Examples for either addition are given in several cases, but in no way are meant to be complete.


© 2004-2013
top of page