- we would need some basics on Coxeter Dynkin symbols and Wythoff's kaleidoscopical construction to be reviewed for the ease of the reader and also for some selfcontainedness of the article.
- we need to outline Wendy's skew coordinates (mainly the ability to scale the different edge types obtained in the Wythoff construction freely and independently - in its effect to the represented polytopes this already is known since the days of Max Brückner as "Varietäten" (varieties), but as notational concept applied to (ASCII linearizations of) Coxeter Dynkin diagrams it still remains unpublished.
- we need some review on Alicia Boole-Stott's definition on what is now called Stott expansion and its translation to Coxeter Dynkin diagrams. - That part in retrospective kind of seems trivial, but in fact her paper was written way before the notational concepts of Dynkin and Coxeter. It rather contributed to the setup of the latter.
- we then will have to go beyond the scope of her article and at least outline the different paths of generalizations of her concept, e.g. as already being done on my corresponding webpage (also no printed publication on that so far).
- we then need to outline the concept of edge flips introduced by student91. Here not only the notational techniques ought to be explained, but also its relation to the kaleidoscopical construction with seed points lying outside the fundamental domain. - That topic then surely has been investigated earlier, some references should be spottable, perhaps even one or two more general pics deduced therefrom. - But here we restrict to those special positions at the outside, which occur as constructed vertex set of some given (other) seed point within. A further genuinity of the ideas in consideration amount in not to translate that outside seed point into one within some larger fundamental domain with multiplicity (order, index) greater than 1, as usually being done in the past, but still to use the original (elementary) fundamental domain plus an extrapolation of Wendy's concept (cf. list item 2) towards negative edge lengths.
- the to be described BTP polytopes then result in some specific applications of the last 2 list items.

The idea behind that last list item then always runs as follows (as meanwhile being boiled down to):

- start with some specific polytope, e.g. the icosahedron.
- define some specific subsymmetry to be chosen, e.g. the brique symmetry o2o2o.
- decompose the vertex set of your polytope into subsets with the chosen subsymmetry

Here: x3o5o -> x2f2o + f2o2x + o2x2f - apply one or more edge flip(s) to one or more component(s)

E.g.: (-x)2f2o + f2o2x + o2x2f

(the thus described vertex set as a whole will have to be equivalent to that of the starting figure. In more detail, in the sense of a kaleidoscopical constructed polytope, it just defines an edge faceting of the starting polytope.) - now use a correspondingly subsymmetric Stott expansion uniformly to all the components, which then lead back to prograde edges only

Here: (x-x=o)2f2o + (x+f=F)2o2x + (x+o=x)2x2f - in order to result in a CRF polytope some restrictions have to be met here in general (the chosen example surely will pass)

The outcome in our chosen example then is the bilunabirotunda (J91).

Why do I refer to "BTP" here?

B represents bilbiro (bilunabirotunda, J91). T represents thawro (triangular hebesphenorotunda, J92). P represents pocuro (pentagonal orthocupolarotunda, J32). All three can be derived as just outlined from ike (icosahedron), when dealing with o2o2o, o2o3o, resp. o2o5o subsymmetry.

In fact, x3o5o = x3o || o3f || f3o || o3x -> (-x)3x || o3f || f3o || o3x -> (x-x=o)3x || (x+o=x)3f || (x+f=F)3o || (x+o=x)3x = thawro. Respectively x3o5o = o5o || x5o || o5x || o5o -> o5o || (-x)5f || o5x || o5o -> (x+o=x)5o || (x-x=o)5f || (x+o=x)5x || (x+o=x)5o = pocuro.

The most, if not all of the above mentioned thread on BT polytopes then is about similar applications to polychora, i.e. within 4D. Mainly ex = hexacosachoron = 600-cell = x3o3o5o is being used for starting figure.

--- rk