Klitzing wrote:[...]Even so those quirks can be applied on a simple technical level, the final seeving of the results for true CRFs is awkward longuish and in most to be considered cases would lead either to asked for longer edges, to non-regular polygons (broken parts thereof only), or even to non-CRFs of some higher subdimensional 2 < d < D.

So it might be worthwhile (in order to reduce the amount of all those dead ends) not to ignore all the so far simply transponed individual considerations onto the end, but to consider only such facetings right from the beginning, which would use true CRF facets in the start. (So far simply any combinations of all applicable quirks had been considered.) Depending on the chosen subsymmetry this then includes mutual intersections of those facets, and thus any number of corresponding diminishings. Thus this again could be considered in restricting the number of possible facetings.

I am not completely sure what you are up to. So far, if I'm right, we have only searched for EKF's of polytopes that do not have lower-dimensional surtopes that don't allow to create a CRF EKF. (in the [5,3,3]-family, we have only searched for things with o5o3x's or o5o3o's, because other cells don't allow to create EKF-cells). Unfortunately, this doesn't bring us much further, as there are a

lot of cells that do allow EKF's, in quite many subsymmetries.

In this regard, one can generally say that the following is true:

If a polytope 'A' has a surtope 'B' with full symmetry .p.q.r... (so if A is rox, then B can be an icosahedron, and then .p.q.r. is .5.3.),

One tries to make an EKF according to a subsymmetry that makes some of the B's be placed in a subsymmetry .k.l.m. (so for .5.3. this can be .2.2.)

if B doesn't have any EKF's in .k.l.m., then there is no EKF according to that subsymmetry of A.

This is why you can generally discard polytopes with pentagons under a symmetry that is not their full symmetry (or the pentagon part of a cartesian product) (at least one pentagon is then represented as x||f||o, without 'CRF' EKF's) .

I have been using some further deduction methods, though those were/are not comletely understood to be correct.

When one has a multilayered representation of a (uniform) polytope, one can determine the 'lacing distance' between the layers. If one now tries to 'quirk' a single node, three things can happen between two layers:

one layer is x=>(-x), the other one is o: the lacing distance stays the same

both layers x=>(-x): the lacing distance stays the same

one layer is x=>(-x), the other one is greater than x: the lacing distance increases

When multiple quirks are applied, I assumed the distance wouldn't increase, though I do not know any proof of this.

Assuming that for CRF-ity, every layer should have at least 1 lacing distance of x, one can hereby exclude quite some 'possible' to-be-EKF facetings.

Right from the view of the notion this ought be desirable. But I don't know how to transfer that onto that mere technical level, as those quirks were. (In fact this was the true genuine idea of student91, I fear.)

So, is there any further idea out there on how to describe

any diminishing or faceting of, say any Wythoffian polytope (or even uniform polytope) on a mere Dynkin diagrammal level?

--- rk

As of now, we are unable to describe every polytope in a dynkin-way. However, I have been thinking about ways to be able to describe

most polytopes in a dynkin-diagram-like-thing.

One could opt the possibility to add a '\' to the dynkin-diagram. This symbol would discard the mirroring properties of that node. this means that x2x2x means a full cube (or at least the vertices of a cube, depending on the construction you are using), and \x2\x2\x means a point in a standard coordinate-system, at a distance 'x' to every plane. (or a vertex surrounded by three quarter-squares, depending on your construction).

This way, one could write e.g. ike as xofo3ofox2\(f?)?(-?)(-f?), with ? some weird constant. When enough \'s are applied, one can thus represent any polytope in any dynkin-graph. (e.g. a q-edge tetrahedron could be \xooo2\oxoo2\ooxo2\ooox)

This device has some use, e.g. when one tries to define construction methods of 'intricate coordinates=>polytope'. One could define it as:

first, \ all nodes. (so xo2ox=>\xo2\ox)

then, connect the vertices in the most-convex way. (it is not that hard to define most-convex, but I'll do that another time) (now the example makes a line between the two vetrices)

then, take away the \'s that must be taken away, and mirror the constructed patch accordingly, filling up spaces in the 'most-convex' way. (now the line is mirrored 4 times, constructing 4 triangles, thus one has atetrahedron)

This defenition nicely makes fo(-x)2xfo2oxf&#zx look like the faceting Klitzing has drawn for the 'ike=>bilbiro'-process.

A final thing one should keep in mind is that the symmetries .3.4.3.; .4.3.3.; .p.2.p. and .p.2.q. can have multiple distinct representations in the same symmetry. This means that exploring all EKF's should be done with more care than when investigating .5.3.3. etc.

student91

P.S. The 'most convex' way needs some work, but now I'm playing with the idea to do it like "fill it up, such that when all vertices are connected to the origin, one gets a pyramid whose base is convex", as you see, it needs a little work.

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