Now you guys are going too fast for me
!!those things you've found are extremely awesome (especially the J91/J92-thing).
If I'm right, these things look like J92-thing: xofxF(Vo)Fxfox3xFxoo(xo)ooxFx5xoxFf(oV)fFxox&#xt, where the parentheses mean those things are in the same layer.
and xofxfox3xFxoxFx5xoxFxox&#xt for the J91/J92-thing.
Is the Bilbro-cut also possible on a general o5x3o3o?
quickfur wrote:I'm also thinking we need to revisit the "castellated prism" name as well. It was appropriate when there was only a single castellated prism, but now that these things are turning out to be rather common, especially here where they are based of various strata of the o5x3o3o, I think we need a more generally-applicable naming scheme, as I'm expecting more CRFs to turn up with similar constructions.
I think so too!. Maybe an ad-hoc new naming scheme is usefull, we could make a wiki-page about polystratic CRF's, and index them according to that. What about P for polystratic, then a number for the order of polystratic-ness, so P3. for a tristratic CRF, and then a number, so P3.1 would be my thawrochoron.
Below an attempt is made to explain an idea for a more constructive naming scheme of mine.
The things with such a construction basically take a lace-tower of a Johnson solid, and then place another node to the CD-graph. This node can get different "values" (x, f, F etc.). Most of the time these new values get either x or o. (e.g. castellated prism: xFoFx3ooooo5xofox&#xt, it has ooooo, meaning it closes after the bilbro is completed., another example: oxFx3xfox5xoxx&#xt, my thawro-thing. it has xoxx on the new node, meaning it has oxFx(2)xoxx&#xt and xfox5xoxx&#xt, both can be closed after the thing).
basically what you do when you make such a polytope, you take a Johnson solid %%%%N%%%%&#xt, add a node: %%%%N%%%%P????&#xt. This means the nodes of the N are fixed, and the other node can be changed. This means, if N=3, that you place triangles atop each other, that make the shape %%%%3%%%%&#xt. These triangles are symmetrically placed on the .N.P.-thing, so it's like placing %%%%N%%%%&#xt's symmetrically around a .N.P.
Let's take the ursachora as an example: xfo3oox4ooo&#xt. The triangle gives a trid. ike: xfo3oox&#xt. these triangles are placed on the vertices of an o3o4x, so it's somewhat related to the dual of o3o4x. In the same way the xFoFx3ooooo5xofox&#xt is somewhat related to the dual of o3x5o, because the bilbro's are placed along the verticeds of a o3x5o.
All layers of the trid. ike in xfo3oox4ooo&#xt should be in the same 3-space (i.e. the dichoral angle between xf3oo4??&#xt and fo3ox4??&3xt should be 180 at the f3o). This means that if you choose two ??'s, all other ??'s are set as well. If the other stacks (%%%%P????&#xt and %%%%(2)????&#xt ) give nice things as wel, we have a new polytope. if ???? consists of x's and o's, these stacks are very likely to work out. I think, because of this, we should call things witth only x's and o's on the extra node "simple." if there are other things that x's and o's, it is then considered "complex", and a naming scheme will be difficult to make.
If you have a valid thing %%%%N%%%%P????&#xt, you can also have %%%%N%%%%P(?+l)(?+l)(?+l)(?+l)&#xt, i.e., you can add a constant "l" to all ?'s. Most of the time this only works for ????=oooo=>xxxx. for not-simple polytopes this might also work, but for simple polytopes with ???? not being oooo, it would make it complex.
Clearly for every symmetry there are at most two possible simple polytopes based on a build of a Johnson-solid, and only two if one has only o's, the other one will have only x's. This means we could call the tetahedral ursachoron the simple teddi-based tetahedral polychoron, maybe a shortened form is desired. The castelated prism then is the simple bilbro-based icosahedral polychoron. Things like bilbro's might give problems, as they have multiple lace-tower buildups. Expanded can still be called expanded. This explanation wasn't clear.
Now something I am able to explain: the thawro-ing of the o5x3o3o. To me it seems like a little more complex cut of a o5x3o3o. Compare it with a cut along a dodecahedron of a 600-cell: things get rearranged a bit (tetahedra get replaced by pentagonal pyramids), and if you place the two halves back on each other, the thing isn't CRF anymore. Furthermore it doesn't affect the rest of the polytope. Therefore I am pretty sure the mega-wedge is possible.
The bilbro-ing of the o5x3o3o has some similar properties, and therefore it may be seen as a special cut as well (maybe it's comparable with making a square orthobicupola out of an rhombicuboctahedron). this means both of these operations can be combined with other diminishings, yielding combinatorial explosion
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
-Stern/Multatuli/Eduard Douwes Dekker