by wendy » Mon Oct 14, 2013 7:24 am
The actual space the orbifold does not really change things.
You can generally group the tetrahedron, the antiprisms 2 2 p, and the snubs 2 p q, under the tetrahedron 2 2 2 if you use the same colour for the opposite regions. Anything that is grouped under 'digon', can be replaced by 'polygon', so a red edge becomes variously, a red square, triangle, etc.
There is outside-wrap too. You can cut out a 'p' from two different forms, and glue the two halves together. This is what's essentially happening at the kinds of edges where i use ø-edges. You need to track these edges because they bound active regions, and so if there are three active points inside a block, you get triangles. The behaviour of this is generally understood.
This is the current heirachy of the dCT. It follows the bowers army thing, in that finer detail adds more constraints.
1. Army This us the undecorated CT, like *2 3 5 or 2 2 * 6 4, this is more a mathematical heading, rather than a real pen-on-paper group. None the less JHC has attributed to these a 'cost', which is some kind of area measurement.
2. Regiment, this is an ordered representation of the army, which does not traslate to other ordered representations. So 3 5 8 * and 8 5 3 * are the same regiment, but 5 3 8 * is a different regiment. The regiment is home to the passive elements, which work much the same way as CD things, but there are more possibilities. Where things get different, is that CT allows for 'active regions' and so forth.
3. Company, this is a partition of regiments into areas of common active regions. This is where i hope to find home for your null-set branch. ø, but i am not sure how many different null branches are allowed, ie are there null branches of the archiform [a,b], or are they all {a,b}? It is known that some null branches can be detected from the orbifold.
Paderborn is one of those towns that turns up often in the magazines i read. They're mostly railway magazines, because they have pretty pictures, and one can get involved at varying depths - useful when one is on alert. Apparently there was a railway workshop there. But i am happy that ye had much fun up there.