by wendy » Sat Apr 03, 2010 8:29 am
There are actually something like 22 operators in three dimensions, but fewer are needed to produce these.
In any case, one should note that Conway's operators assumes any given figure is regular, and that one can go from any regular figure to its dual, or derived archemedian or catalan. So, eg in 3d, the prototype is xoo, the dual is variously oox or moo, and there are then 6 derived archemedian and 6 catalans from these. Add the gyrated and snub, and then the propellor etc to this.
In four dimensions, one has operators with 4 symbols, eg xooo = ooom, and 14 derived elements, and snubs/dual thereof. In five dimensions, the base is xoooo or oooom, each with 30 dependents.
Equities in the singular are of the form d = dual = mooo... = ...ooox . In the compound, the only form is the Conway-Kepler rule, which gives axoo...!..oxo... as .oxaxo.. where a is either x (cantetruncate) or o (cantellate). The other equity is 'd' (x) = m and d(m) = x. That is the dual of xxoo is mmoo.
Geo. Hart used the equities to create things like xox from oxo!oxo, and xxx from xxo!oxo. However, one should not consider something like xxx or xox thus derived. In practice, all of the base operators simply represent vectors in space, implemented from the origin, in the form (1,0,0) + (0,1,0) etc. In order to acheive this, one turns to 'pennant theory'.
A pennant is simply a simplex (triangle, tetrahedron, etc), that is cell of a tiling of such, such that the resulting vertices can be numbered 0,1,2,... consistantly. One way to get this is from the Coxeter-Dynkin graph, another is to use 'flags' or simplices formed by the notional cemtres of the vertex, edge, hedron, choron, ... of a polytope. Of course, the dimension number provides the vertex-order.
Since each triangle is given correct coordinates, it makes sense for eg (1,1,0) ie halfway between v0 and v1, etc. The coordinate is moved by flip (in Wythoff construction, reflection), to every pennant, and edges etc are drawn in as per wythoff construction. The corresponding catalan, comes by removing the walls except where an 'm' retains them.
The cantellate operator, says that if you apply first a single vector vz, and then add v0,v1, then you get v(z-1),v0*vz,v(z+1), which is read either as a vertex (x) or a catalan's face-tangent (m).
Alternation of vertices x, m gives s or g. With Klitzing's rule, you can have any set as alternating, eg ssx (v+e = even, h all), or in a later form even halving by different levels of s/g. An example of such a figure is ssox{3,4,3} = s3s4o3x. This has an assortment of faces as; 24 truncated tetrahedra, 96 triangular cupola, and 24 icosahedra. So there it goes.