I'll agree on that.Klitzing wrote:student91 wrote:[...]Klitzing wrote:Ehh, not too difficult, after all: isn't this just iddip (oo3xx5oo&#x) augmented by 12 pippies (ox2ox5oo&#x)?
Well, it is true that it will occur just as a part (segment) within a larger structure, which deserves lots of interest on its own, but I think kind of minimalistic here: it is a smaller CRF, valid on itself. So, at least, worthwhile to mention!
I agree on that as well. up till now, I've seen partial expansion as a 3-step process, with 2 restrictions:[...]
What I am after here all the time, is that the mere mechanics of your Dynkin symbol transformations is rather clear to me. But its implications onto esp. the face structures of the to be shifted elements is not described thereby and will have to be digged out separately.
step 1: take a subsymmetry
step 2: determine the limits according to this subsymmetry
step 3: move the limits apart.
restriction 1: the part that is moving outwards must be CRF
restriction 2: the new things that are made where things are pulled apart have to be CRF.
step 2 is the most interesting step, and although I've implicitly used it all the time, I've never really explained it well. This step can be seen as coloring the vertices that should be moved together with the same colour. (the vertices that do o->x get 2 colours) these limits should meet restriction 1. therefore, it may be seen as useful to be able to see the limits as isolated things (in fact I've seen these expansions all the time as limits moving away from each other, and afterwards the part in between the limits gets filled up). Now ho do we locate the limits? The way I used to think of locating the limits is that you construct your polytope using the wythoff-constructions of the subsymmetry, but you disable the mirrors that are subject of the expansion. Every time you mirror this limit in that mirror, you get a new limit, and thus we can determine all limits this way. example: ike => bilbiro
step 1: ike=>f2o2x+x2f2o+o2x2f
step 2: Here I need a symbol to show I disable a mirror. I'll use \, as I've been programming lately, and the backslash disables things. The limit then is \f2o2x+\x2f2o+\o2x2f. This limit looks like 8 triangles placed around an edge. The limit nicely passes restriction 1. However, restriction 2 causes problems, as the x of \x will become u, and that's not CRF. If we change this x in -x, the limit becomes \f2o2x+\(-x)2f2o+\o2x2f. That is, two pentagons and two triangles placed around an edge. These are the green pentagons and the blue triangles in your image:
Now we have two limits that are CRF, and that will pass restriction 2, so we can proceed to the next step, and finally make the bilbiro.Consider e.g. the here used partial expansion of ike to bilbiro:
[...]
--- rk
Most of the time, restriction 2 is the one that forces you to change x in (-x), and restriction 1 is the restriction that might be violated by this. I often fix the restriction 2 problem by first assuming that there's only x's and o's at the boundary of the limit (risky assumption here) and then I can solve this by changing all x's in (-x)'s, as then the boundaries will still be made of x's and o's, which is always CRF, and thus restriction 2 is met. In the mean time, I'm hoping restriction 1 won't be violated by this (overlooking this sometimes, just as in the demitesseractic expansion of rox).
so in summary, I see expansion as
take subsymmetry -> determine limits -> expand
and you seem to see expansion as
take subsymmetry -> take faceting -> expand
I like my way a bit more, as then you can check for restriction 1 and 2 to be met in-process. The difference with your way is that I neglect the yellow, purple and light blue triangles, and just move the limits themselves. I have to say that your drawings helped me make up this process of expansion. Also note the similarity between the limits and your facetings.