Those are defined solely by their vertices and their edges (sides). But the spanned membran (their area or body) simply is neglected. Then regular polygons no longer need to be flat. They can be finite or infinite zig-zags, can be helical (with a flat regular polygonal projection image) or can be simply the alraidy well-known linear apeirogon. In his days he used them to introduce as what then became the Petrie polygon of a regular polyhedron or even of higher dimensional regular polytopes.

It even is possible to define a Petrie dual for regular polytopes by re-using the same edge skeleton, but interchanging the former faces by the Petrie polygons. This procedure then happens to be involutoric, that is, being applied twice, it does return to the starting figure.

Coxeter then used such skeletal polygons for vertex figures of "new" regular polyhedra. Those were {4, 6 | 4}, {6, 4 | 4}, and {6, 6 | 3}. I.e. 6 squares per vertex in the first case. But those are not arranged flat (as for hyperbolic tilings), rather the vertex figure is a skew zig-zag-Hexagon. Thereby induced one gets further recurrencies. This is what the number behind the pipe are for: in the first 2 cases there are square loops (holes). In fact the first one is a substructure of the squares of the cubic honeycomb x4o3o4o = x4o3o4x, the second is the skew polyhedral manifold of the hexagons in the bitruncatedcubic honeycomb o4x3x4o, and the latter one is the skew polyhedral manifold of the hexagons in the quarter cubic honeycomb x3x3o3o3*a.

Recently now researchers around Branko Grünbaum and Egon Schulte are considering those skew skeletal polygons not only for vertex figures, but also for true faces of according regular polyhedra. This then brings back the symmetry of {p, q}. Also potential Wythoffians of all those are considered.

So far on the current state of the art.

Here now my question: could those Wythoffians likewise be described by (somehow extended) Dynkin symbols?

Schulte already kind of distinguishes the polygons, writing e.g. {4

_{c}, 6

_{s}| 4

_{c}}. "c" here means a (flat) convex regular polygon, "s" here means a skew regular polygon. So we probably have to distinguish these by according Indices. But then (provided the "correct" symmetry (undecorated diagram) is being found, a decoration could be applied as for "normal" Wythoffians. Ain't it?

--- rk