Conway's operators exist in all dimensions, but the names Conway uses does not suggest this.
In essence, Conway's notation is based on flags, on the assumption that all polyhedra are regular. The various operators are designated PFE or PFM as they go to a uniform or catalan respectively. This table will show what goes on.
- Code: Select all
sub dual PFE PFM
C x4o3o 0 m4o3o - dual
O o4o3x 2 o4o3m dual -
tC x4x3o 0,1 m4m3o trunc kis
tO o4x3x 1,2 o4m3m td kd
CO o4x3o 1 o3m4o ambi join
rCO x4o3x 0,2 m3o4m expand ortho
tCO x4x3x 0,1,2 m3m4m bevel meta
sC s4s3s g3g4g snub gyro
Basically, you have two operators PFE (pennant-flip-edge) and PFM (pennant flip margin). When the flip is replaced by mirrors, you get the Uniforms and Catalans.
A pennant is a smplex with numbered vertices, tiling a space so that each vertex of the tiling has simplex-corners of the same number. An easy way to construct such a thing is to draw flags (triangles, with vertices at 0, 1, 2 (centres of the vertices, edges, faces). Groups represented by Dynkin graphs do as well.
For PFE or WME, you place a vertex somewhere in the triangle, including on the surface. If it lies on a given wall (opposite vertex m), then node m is marked 'o', If it lies off the wall, an edge forms between it and its reflection, giving an 'x'.
For PFM or WMM, you remove all walls marked 'o' and keep walls marked m. Because only the walls marked m are kept, the elevation on the o-corners must be set so the image matches the location. If you think of the pennant supported by poles at each corner, you can lift or drop corners so that the flag is flat across the opposite side, This is what a o requires.
You then can replace the Coxeter-Stott t_ style notation (t_0,2) by e_ or m_ to reflect the edge or margin flips.
The only rules that exist of several symbols, are
e_0 = m_n and e_n = t_0
x_1 , x_m = x_m-1,m+1 where x is t,m,e eg e=aa
x_0,1 x_m = x_m-1,m,m+1 eg b=ta