Construction of BT-polytopes via partial Stott-expansion

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Tue Apr 28, 2015 4:31 pm

wendy wrote:Is this what i am supposed to condense for the paper. I've read it somewhat.

You feel like you ar supposed to? I'd rather say there were several contributers to these contained topics and so ought be wellcome to contribute to the papaer as well, no?

As to the topic. Yes this one surely is one of them. Others are
and propably some other mails scattered in further threads, by reason of breakup of that Johnsonians thread...
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby wendy » Wed Apr 29, 2015 12:47 pm

Reading this, i might have to heavily revamp the lace-prism stuff into a handbook-like construct. It's long overdue.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Wed Apr 29, 2015 4:34 pm

The basics part of our paper ought be a good place. ;)

And you should add an (perhaps appendical) outline on how those occur as the vertex figure of the Wythoffians. - I've done that already within the verf-page of my incmats-website. But a published version still is missing, I fear. (We might re-use that for a start?)

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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Tue May 19, 2015 10:26 pm

Okay, back to the topic of this thread!

Commenced to consider now sadi wrt. o2o2o2o subsymmetry.
sadi itself then can be written as:
ooo|xxx|fff|FFF 2 Fxf|oFf|xFo|fxo 2 xfF|Ffo|Fox|xof 2 fFx|foF|oxF|ofx &#zx

Layers and potential flips each here are:
Code: Select all
A: oFxf > A3: oF(-x)f
B: oxfF > B2: o(-x)fF
C: ofFx > C4: ofF(-x)

D: xoFf > D1: (-x)oFf
E: xFfo > E1: (-x)Ffo
F: xfoF > F1: (-x)foF

G: fxFo > G2: f(-x)Fo
H: fFox > H4: fFo(-x)
I: foxF > I3: fo(-x)F

J: Ffxo > J3: Ff(-x)o
K: Fxof > K2: F(-x)of
L: Fofx > L4: Fof(-x)


Omitting from all combinations those which would result in u-edges, i.e. combies with
Code: Select all
A  + I3,J3
A3 + I,J
B  + G2,K2
B2 + G,K
C  + H4,L4
C4 + H,L
D  + E1,F1
D1 + E,F
E  + F1
E1 + F
G  + K2
G2 + K
H  + L4
H4 + L
I  + J3
I3 + J
as well as symbol intrinsic symmetry then results in a quite manageable amount of to be considered cases:
Code: Select all
ABCDEFGHIJKL             oooxxxfffFFF 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fFxfoFoxFofx &#zx
ABCD1E1F1GHIJKL          ooo(-x)(-x)(-x)fffFFF 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fFxfoFoxFofx &#zx
ABC4DEFGH4IJKL4          oooxxxfffFFF 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fF(-x)foFo(-x)Fof(-x) &#zx
ABC4D1E1F1GH4IJKL4       ooo(-x)(-x)(-x)fffFFF 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fF(-x)foFo(-x)Fof(-x) &#zx
AB2CDEFG2HIJK2L          oooxxxfffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 xfFFfoFoxxof 2 fFxfoFoxFofx &#zx
AB2CD1E1F1G2HIJK2L       ooo(-x)(-x)(-x)fffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 xfFFfoFoxxof 2 fFxfoFoxFofx &#zx
AB2C4DEFG2H4IJK2L4       oooxxxfffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 xfFFfoFoxxof 2 fF(-x)foFo(-x)Fof(-x) &#zx
AB2C4D1E1F1G2H4IJK2L4    ooo(-x)(-x)(-x)fffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 xfFFfoFoxxof 2 fF(-x)foFo(-x)Fof(-x) &#zx
A3BCDEFGHI3J3KL          oooxxxfffFFF 2 FxfoFfxFofxo 2 (-x)fFFfoFo(-x)(-x)of 2 fFxfoFoxFofx &#zx
A3BCD1E1F1GHI3J3KL       ooo(-x)(-x)(-x)fffFFF 2 FxfoFfxFofxo 2 (-x)fFFfoFo(-x)(-x)of 2 fFxfoFoxFofx &#zx
A3BC4DEFGH4I3J3KL4       oooxxxfffFFF 2 FxfoFfxFofxo 2 (-x)fFFfoFo(-x)(-x)of 2 fF(-x)foFo(-x)Fof(-x) &#zx
A3BC4D1E1F1GH4I3J3KL4    ooo(-x)(-x)(-x)fffFFF 2 FxfoFfxFofxo 2 (-x)fFFfoFo(-x)(-x)of 2 fF(-x)foFo(-x)Fof(-x) &#zx
A3B2CDEFG2HI3J3K2L       oooxxxfffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 (-x)fFFfoFo(-x)(-x)of 2 fFxfoFoxFofx &#zx
A3B2CD1E1F1G2HI3J3K2L    ooo(-x)(-x)(-x)fffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 (-x)fFFfoFo(-x)(-x)of 2 fFxfoFoxFofx &#zx
A3B2C4DEFG2H4I3J3K2L4    oooxxxfffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 (-x)fFFfoFo(-x)(-x)of 2 fF(-x)foFo(-x)Fof(-x) &#zx
A3B2C4D1E1F1G2H4I3J3K2L4 ooo(-x)(-x)(-x)fffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 (-x)fFFfoFo(-x)(-x)of 2 fF(-x)foFo(-x)Fof(-x) &#zx



Well, the first is sadi itself.

Then considered the next. That kaleido-faceting then allows for a partial Stott expansion wrt. to the first node position. The outcome then is xxxoooFFFAAA 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fFxfoFoxFofx &#zx, which clearly can be given easily as a lace city too. In the one direction it will look like
Code: Select all
        x x   F o     o f     F o   x x       
                      (F)     (I)   (B)       
                                              
                                              
o o           x F     A x     x F           o o
                      (K)     (A)           (D)
                                              
x f     A o           F F           A o     x f
                      (H)           (L)     (C)
                                              
                                              
F x     o F   A f             A f   o F     F x
                              (J)   (E)     (G)
                                              
                                              
x f     A o           F F           A o     x f
                                              
                                              
o o           x F     A x     x F           o o
                                              
                                              
                                              
        x x   F o     o f     F o   x x       
while within the orthogonal direction it will look like
Code: Select all
        o x           x f     f o     x f           o x       
                              (E)     (A)           (H)       
                                                              
                                                              
x o                   F x     o F     F x                   x o
                              (F)     (C)                   (J)
                                                              
o f     F o           f F             f F           F o     o f
                                      (B)           (G)     (K)
                                                              
                                                              
f x     x F                   F f                   x F     f x
                              (D)                   (I)     (L)
                                                              
                                                              
o f     F o           f F             f F           F o     o f
                                                              
                                                              
x o                   F x     o F     F x                   x o
                                                              
                                                              
                                                              
        o x           x f     f o     x f           o x       


And indeed, this EKF fellow happens to work out as a true new CRF!
Its total cell count is
  6 bilbiroes
  2 ikes
  36 squippies
  40 teddies
  36 tets
  8 trips

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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Fri May 22, 2015 8:16 am

... and the remaining above listed cases either are dublicates - in fact, there is the following inner symmetry of the symbol:
Code: Select all
(ABCDEFGHIJKL)(1234) → (KFHAJIELBGDC)(3124) → (ILDGBKCFJAHE)(2431) → (BCAEFDHIGKLJ)(1423) → (JEGCLHDKAIFB)(4132) → (GJEHCLADKBIF)(4321) → (ABCDEFGHIJKL)(4321)
- and/or are impossible as solely unit-edged convex figures (after according expansions).

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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Wed Jun 17, 2015 6:26 pm

Coming back onto EKF (expanded kaleido faceted - as those are called meanwhile and no longer do restrict onto BT or BPT (or some commutation thereof), i.e. ones which include bilbiroes (J91), thawroes (J92) resp. pocuroes (J32)) polytopes on a more meta level.

The very EKF notion itself asks to consider some (possibly subsymmetrical) faceting (of any chosen) starting figure, and then to apply some (accordingly subsymmetrical) Stott expansion in order to get the figure again convex. Further it aims to restrict then to CRF (convex regular faced) results.

Our so far considered realm was driven by student91's quirks (edge flips), i.e. applications of the type x -> (-x) to the starting figure within being considered subsymmetrical Dynkin graph display. That surely results in afore mentioned kaleido facetings.

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.

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.) :sweatdrop:

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?

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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Wed Jun 17, 2015 11:26 pm

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.) :sweatdrop:

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.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Thu Jun 18, 2015 12:32 pm

student91 wrote:I am not completely sure what you are up to.

Not too hard to explain. Way back, with publication in 2002, I was doing a brute force computer aided research for all "edge facetings" for any given uniform polyhedron, which show up at least an true axial symmetry. (Cf. the corresponding paper or the corresponding website, where all the results not only are listed, but all findings are represented by 3-view pic and VRML.) "Edge faceting" here implies getting regular faced polyhedra only.

That research then was just considering all possible faces and true facets (up to oriented axial symmetry) and progressively putting faces together until all edges are hit an even number of times (dyadicity, but several edges might coincide).

A to be expanded KF now would in fact be nothing else, just going up one dimension (for our current research for CRFs), and furthermore omitting that restriction of having at least a true axial symmetry. I.e. choosing any uniform polychoron, considering all its faces (cells), all its edge-respecting facets, but also any of their intersections (i.e. diminishings) as well, which on their own are orbiform Johnson solids (subdimensional CRFs). Then one once again would have to consider all their combinatorical adjoins to full, subsymmetrically edge-faceted polychora. And finally, for getting the final EKF, those derived polychora then have to be expanded to get convex again.

On a second view it is clear that this process at least would produce EKFs. But then those cannot be all EKFs. - Why? - In fact the internal intersections - before expansion - not even need to be convex themselves. Rather they might be reflex, showing up retrograde parts (negative density regions) on their own. They just have to get convex after an according expansion. Else those new figures would not be obtained, i.e. bilbiro, thawro and pocure could not become potential facets, when only orbiform unit-edged polyhedra would have been used for faces...

Therefore my main issue here would be, whether or not any such combinatorically obtained polychoron (as a facet adjoin of such faces and/or facets) could be described by Dynkin diagrams. Or at least classifying which ones can. - Provided such an Dynkin diagram described KF would be given, then the application of partial Stott expansions would be rather easy.


And a second point of request, so not completely Independent, was whether your quirks (layer-wise independently applied edge flips) would be enough to describe all those.

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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Fri Jun 26, 2015 12:36 pm

Klitzing wrote:[...]A to be expanded KF now would in fact be nothing else, just going up one dimension (for our current research for CRFs), and furthermore omitting that restriction of having at least a true axial symmetry. I.e. choosing any uniform polychoron, considering all its faces (cells), all its edge-respecting facets, but also any of their intersections (i.e. diminishings) as well, which on their own are orbiform Johnson solids (subdimensional CRFs). Then one once again would have to consider all their combinatorical adjoins to full, subsymmetrically edge-faceted polychora. And finally, for getting the final EKF, those derived polychora then have to be expanded to get convex again.
I'm sorry, but a to be expaded KF would be a bit more restricted than what you just posted. This is because a to be expanded KF must become CRF after some number of (+x)-expansions. (It is not that hard to see that a different expansion will not give CRF's)
This means that a to be expanded KF must have only (-x)-negative nodes. (or maybe also (-u)'s, but I doubt we will ever enconter such tings). Furthermore the Stott-expansion must be done according to some subsymmetric coordinate system. This means that, when one restricts himself to easily found to be expanded KF's, those have to have the same symmetry as the symmetry of the expansion.
Although you seem to hope to be able to bypass these restrictions, I think it is very valuable to look only for KF's that do also allow these restrictions.
On a second view it is clear that this process at least would produce EKFs. But then those cannot be all EKFs. - Why? - In fact the internal intersections - before expansion - not even need to be convex themselves. Rather they might be reflex, showing up retrograde parts (negative density regions) on their own. They just have to get convex after an according expansion. Else those new figures would not be obtained, i.e. bilbiro, thawro and pocure could not become potential facets, when only orbiform unit-edged polyhedra would have been used for faces...

Therefore my main issue here would be, whether or not any such combinatorically obtained polychoron (as a facet adjoin of such faces and/or facets) could be described by Dynkin diagrams. Or at least classifying which ones can. - Provided such an Dynkin diagram described KF would be given, then the application of partial Stott expansions would be rather easy.

And a second point of request, so not completely Independent, was whether your quirks (layer-wise independently applied edge flips) would be enough to describe all those.
I've been digging into that research of yours about facetings, and I think I have been able to desribe some of the facetings in a dynkin-way, and say of some others that they are impossible to descirbe as such with the used algorithm for obtaining a polyhedron out of a dynkin-coordinates set.
Take e.g. id: only id-10-6-3-b is describeable in dynkin-notation, as xFFVFx3(-x)(-f)o(-f)(-x)o&#xt. When interpreted as \xFFVFx3\(-x)(-f)o(-f)(-x)o&#xt, this gives half of a decagon, with a little triangle erected on one of its edges. Now xFFVFx3\(-x)(-f)o(-f)(-x)o&#xt gives the decagon (if one is kind enough to ignore the triangle in this reflection), and \xFFVFx3(-x)(-f)o(-f)(-x)o&#xt gives the patch of pentagons and triangles. This thus means that xFFVFx3(-x)(-f)o(-f)(-x)o&#xt gives the desired faceting. When one tries to describe other facetings of id, e.g. 10-6-3-a, this way, one gets a problem: one could describe the vertices as such that they give a decagon on the one expansion, thus one gets oxFVFx3\xfo(-f)(-F)(-F)&#xt. This does give decagons in the right orientation, but unfortunately \oxFVFx3xfo(-f)(-F)(-F)&#xt does not give the other desired patches :( .

Furthermore the ike-facetings 13-3, 6-6-b, 10-6 and 5-7 are describeable (as the well-known (-x)ofo3xfox&#xt, or as xfo(-x)3(-x)ofx&#xt, or o(-x)oo5ofxo&#xt and o(-x)o5ofx&#xt resp.)

A lot of the srid and sirco-facetings are not describeable using dynkin-notation. (those that are are sir-8-6-4, sir-4-9-3-a and sir-4-9-3-b; and quite some srid-facetings)

Now what do the facetings that are writeable have in common?
Well, they can all be made convex by applying some expansions to the mirrors of the posessed symmetry. This means, that a to be expanded KF can only become CRF, if it is writeable in dynkin notation.
This is an intrinsic property of the used definition of making polyhedra from the dynkin-coordinates.
Therefore, it is best to look for "quirked" facetings instead of "faceted" facetings, because the quirks give only those facetings that do allow CRFs to be created.

P.S. You might have noticed that some of the (e.g. id-10-6-3-a) facetings do have a open downside, and thus do not become convex after expansion. This is, in the id case, because the downsie actually has a x3(-x) triangle that connects the two patches, but has been ommitted.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Sat Jun 27, 2015 3:15 pm

Already a first reply to your very first answering paragraph:
student91 wrote:I'm sorry, but a to be expaded KF would be a bit more restricted than what you just posted. This is because a to be expanded KF must become CRF after some number of (+x)-expansions. (It is not that hard to see that a different expansion will not give CRF's)
This means that a to be expanded KF must have only (-x)-negative nodes. (or maybe also (-u)'s, but I doubt we will ever enconter such tings). Furthermore the Stott-expansion must be done according to some subsymmetric coordinate system. This means that, when one restricts himself to easily found to be expanded KF's, those have to have the same symmetry as the symmetry of the expansion.


It is a striking fact, that we have to do an afterward expansion, and the expansion has to apply with that subsymmetry of consideration. Thus you are right, we need only consider "edge"-flips.

But(!!) esp. when considering previous decompositions into layers, i.e. "&#xt" or even "&#zx" thingies, then I don't see, why we should have to restrict to x-edges only, where the flips apply. Why not f-"edges", q-"edges" etc. as well?

In fact, I will provide an easy counter example:

Consider the regular octagon with x-edges. If k=2sin(67.5°)=1.847759 is the size of its shortchord, then that very octagon could be provided as ko-4-ok-&#zx, right? (hull of compound of the 2 inscribed dual squares). Now consider the edge-flip of one of these squares, i.e. k(qk)-4-o(-k)-&#zx - so far still describing (as a hull) the very same octagon. Now apply a k-size partial Stott expansion to that second node position, resulting in k(qk)-4-ko-&#zx.

Yes, you'll be right, that one is not a CRF, even so convex: it describes a non-regular dodecagon with edge cycle (x-x-k-x-x-k-x-x-k-x-x-k). In fact, you'll have still 4 vertices with 135° (the ones between 2 x-edges), and the remaining ones (between x and k) are of size 157.5°. (135° is the vertex angle of the regular octagon, 157.5° is the vertex angle of the regular hexadecagon.)

But in the same run we could apply a further partial Stott expansion onto the (other) first node each, but this time not of size +k, but rather of size -(k-x). Thus we'd get (k-k+x)(qk-k+x)-4-ko-&#zx or just x(qk-k+x)-4-ko-&#zx. And this then is indeed a non-regular dodecagon with all unit edges!! (Having the same vertex angles.)

Thus, I think, we have to look into that a bit more intense. Don't you agree?

--- rk
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Thu Jul 09, 2015 2:59 pm

Klitzing wrote:[...]
It is a striking fact, that we have to do an afterward expansion, and the expansion has to apply with that subsymmetry of consideration. Thus you are right, we need only consider "edge"-flips.

But(!!) esp. when considering previous decompositions into layers, i.e. "&#xt" or even "&#zx" thingies, then I don't see, why we should have to restrict to x-edges only, where the flips apply. Why not f-"edges", q-"edges" etc. as well?

In fact, I will provide an easy counter example:
[...]
Thus, I think, we have to look into that a bit more intense. Don't you agree?

--- rk


Up till now, I have been discarting such expansions, because of the following argument:
All 2-dimensional 'CRF' EKFs I know of are:
x3(-x) => x3o
(-x)||o => o||x
o||o => x||x
x2o => x2x
(-x)2x => o2x
These are all (+x)-expansions of regular polygons to regular polygons I know of.
All 3-dimensional CRF EKF's must, at the mirror where the expansion happens, make one of these transitions happen to the faces there.
Of course this is not restricted to the things just listed, but in fact any 2-dimensional EKF CRF could be used here, I just don't know any more.
A similar argument holds for 4-dimensional CRF EKF's, but here you can restrict to all possible 3-dimensional EKF's as well.

It is very well possible that some not x-expanded 2D 'CRF' EKF's exist, but you will need more than just 1 such expansion to be able to derive a 3D CRF EKF of this (unless the symmetry of expansion is the symmetry the polytope can be constructed with by an ordinary Wythoff-construction).
Esp. when the expansion-symmetry is quite a bit smaller than the construction-symmetry, you get a big chance that a part will be expanded only once by subsequent expansions, thereby creating faces with non-x edge lengths.
Now in 4D, this problem has become huge because of the lack of numerous such expansions in 3D.

But still, finding such EKF's in 2D is indeed very interesting. We could search for an expansion that gives, by the described expansion, an x-edged convex polygon. I have not found such things yet, but everything is still possible.
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.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Thu Jul 09, 2015 4:31 pm

Well, in retrospective I've to relativate my own recent 2D find a bit.

You are most probably right in that any higher-D EKF has to raise apon the set of found lower-D ones.
But, and this is my redrawel, even so my find is well a valide EKF, it results NOT in a regular polygon (by purpose of construction), and therefore any higher-D EKF, which hypothetically would use it, right for that reason would not become a CRF.

--- rk
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Thu Jul 09, 2015 4:44 pm

Just to add to your list, student91:
I recently stumbled upon that one: a 60° / 120° rhomb can be expanded along its long diagonal and thereby becoming a regular hexagon.
In fact, this is also being used, when a stack of a tet apon an oct (mono-augmented trigonal antiprism) gets partially Stott expanded normal to its axis, and thus becoming a tut (truncated tetrahedron). In that un-expanded figure the lateral triangles happen to be co-realmic, i.e. they actually join into that described rhomb.

--- rk
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Sun Aug 09, 2015 9:40 pm

In last july of last year student91 came up within this thread with a neat new CRF / EKF. Then he only provided the construction device, supposing that it ought exist:
Code: Select all
                    o2x         o2x                   
                                N                     
          o2o       x2F   f2x   x2F       o2o         
                          E     A         H           
                                                      
    x2x   f2F       F2o   o2A   F2o       f2F   x2x   
                          F     C         Q     J     
    o2F   F2x       f2A         f2A       F2x   o2F   
                                B         G     K     
                                                      
o2x f2o   x2A      Vo2xB  F2F  Vo2xB      x2A   f2o o2x
                          D     OP        I     L   M 
                                                      
    o2F   F2x       f2A         f2A       F2x   o2F   
                                                      
    x2x   f2F       F2o   o2A   F2o       f2F   x2x   
                                                      
                                                      
          o2o       x2F   f2x   x2F       o2o         
                                                      
                    o2x         o2x                   

as being derived from ex, when that one is described in o2o2o2o subsymmetry, applying then 2 quirks and then the accordingly required partial Stott expansions.

Then, in october, he himself elaborated on that figure, proving its existance and providing its cell counts. For that purpose he did not take refuge to incidence matrices, as I would usually do, but restricted his exposal more to applied combinatoric elaborations.

Halas, it took me until now to find the spare time, to return to that find and elaborate on that independently. Having now done so, I clearly can support his counts. I.e. xxxoooFFFAAABxxxF FxfoFfxFofxooVoof xfFFfoFoxxofooVof FAoFxAxoAxFoxxxBF&#zx (where F = f+x = ff, V = 2f, A = F+x = f+2x, B = V+x = 2f+x = fff) has indeed as total cell count:
    16 bilbiroes (J91)
    16 gyepips (J11)
    64 squippies (J1)
    16 teddies (J63)
    24 tets
    24 trips

The above mentioned tegum sum symbol here is quite long and nasty. But it has a further intrinsic symmetry. Within this ordering the A-th and G-th layers are equivalent in 4D, so are accordingly the B-th and J-th, the C-th and E-th, the D-th and H-th, the F-th and L-th, I-th and K-th, M-th and P-th, N-th and O-th, and the Q-th layer finally is automorph. (These letters also appear in one quadrant of the above lace city display, always below the corresponding position.) This additional intrinsic symmetry then allows for a more reduced incidence matrix description (still huge enough, but then manageable - to be elaborated by hand). - (Stay tuned for my next website update, where it will be included then.)

--- rk
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