Polychore Conway notation

[begvend]

Hi all,

Is there a polychore Conway notation?

.

[quickfur]

Currently, we don't know if Conway generalized his notation to higher dimensions. Certainly, some of the Conway operators have trivial generalizations: the d operator (dual) works for all polytopes, the a operator (ambo, or rectify) also works for all polytopes. Likewise the t, k, and e operators, although with e, there are two kinds of expansions possible: expansion of cells, and expansion of (2-)faces. The j operator may not easily generalize above 3D, although j, t, e, b can be replaced with an analogous series of operations in 4D and higher (truncate, runcinate, cantellate, stericate, etc.).

However, some operators have no trivial generalizations: the s (snub) operator can't be easily generalized; we'd have to introduce an alternation operator (which unfortunately only works for polytopes with even elements), and define s as the alternation of an omnitruncate (bevel).

Other aspects of Conway's notation may require some care in higher dimensions: e.g., the t and k operators may take an integer argument in 3D, representing the degree of the polygon being truncated/pyramidized. In 4D and higher, an integer argument is too ambiguous to be useful; for example, if we take the integer as the number of faces, then k12 could mean either kis the pentagonal prisms (12 faces) or the dodecahedra (12 faces also) - and both types of cells can occur in the same polytope. If we take the integer as the number of vertices, we also have a problem: both an octahedron and a triangular prism have 6 vertices, and both can easily occur in the same polytope.

So we will probably need other ways of designating what type of face a particular operator is to be applied to.

[wendy]

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