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 student91 » Mon May 26, 2014 7:14 pm

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'll agree on that.
[...]
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.
I agree on that as well. up till now, I've seen partial expansion as a 3-step process, with 2 restrictions:
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:
Consider e.g. the here used partial expansion of ike to bilbiro: Image
[...]

--- rk
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.
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.
Last edited by student91 on Sat May 31, 2014 11:08 am, edited 2 times in total.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby quickfur » Mon May 26, 2014 7:34 pm

I wonder if this modified Stott expansion (i.e., faceting + "regular" Stott expansion) would itself be sufficient material for a journal paper. I'm almost certain that the icosahedron -> bilbiro/thawro/pentagonal_orthocupolarotunda transformation is unprecedented, and may in itself already be journal-worthy. Add to that the 4D analogues (even just the simplest cases!) and we already have a solid submission, IMO.

The rest of the CRF stuff can be postponed to a subsequent paper.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Mon May 26, 2014 8:02 pm

I know, awesome stuff right?
however, we do have to write a paper about the node-devices we've been using here all the time, as that hasn't been published yet. (It should also include the negative nodes and the rules for transition from positive nodes to negative ones). I don't like this, as that is complicated matter that I already intuitively understand, and It might discourage people if they have to read two articles instead of 1. Furthermore I would really like it if the article could be published before the 20th of October, as then I would've published my first scientific paper under 18 years old :D. (though I am quite busy with my school (I still have my school project btw) and thus I don't think it to be that important, it would just be a funny achievement if I could do that next to my schoolwork)
Apart from that, in my view this topic has always been about such a paper, as all CRF's we've found so far don't really have something that encompasses them all, so it would be a weird article if it included all of them. If the article would therefore be about partial expansions, it would still include some of the CRF's we've found, and it would be nicely closed.
furthermore note we're already a bit further:
goal: investigate all possible partial Stott-expansions, determine when they are convex, write a paper, be awesome and eat a cookie (long term goal)
We now are able to determine when they are convex, so we don't have to investigate all possible partial Stotts anymore, and now the only things we have to do are writing the article, being awesome and eating a cookie
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby quickfur » Mon May 26, 2014 8:16 pm

Well, the paper could just have two contributions: Wendy's node diagram device (i.e., treating CD diagrams as oblique coordinate systems instead of mere mirror-edge constructions), and partial Stott expansions built on top of it, followed by ample illustrations of 3D and 4D cases. I think that makes a very solid paper!

Now I don't know about what kind of media restrictions to journal submissions there are, but it would be awesome if we could include some nice full-color povray renders of the examples. ;) Even just selected animation frames from the icosahedron -> bilbiro / thawro / J32 transformations would be eyebrow-raising enough to convince the editors to publish the paper, I think. :P
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Tue May 27, 2014 7:30 am

I don't think these two subjects are suited to be placed in one paper. The node-subject is a general subject about notation, that might get a lot of references because it is a very useful and intuitive device, so that's a very general subject with wide applications.
The partial expansions on the other hand have some kind of closure, therefore won't be used that much, and thus it would be weird if someone that wants to refer to the node-device also has to refer to the partial expansions. On the other hand, the article about partial expansions will become quite long if it had to include a comprehensive explanation about the node devices. I think they both make very solid papers on their own. the only problem is that I don't want to spend much time on writing a comprhensive overview about the nodes, as I would like it much more if I could spent this time on investigating the partial expansions.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Wed May 28, 2014 9:11 pm

Something terrible is going on: now that I am able to check an expansion to work, I have been forced to conclude that all expansions we know of that aren't applied to the 600-cell, don't give CRF-polytopes in their "pure" forms. That means, they do give patches we can see in things as D4.5 and D4.6 etc, but when you apply the complete expansion, there are some parts, though they're not the most interesting parts of the polytope, that aren't CRF. Examples:

rox with .3.5x-expansion:
step 1: x3o5o || o3x5o || x3o5x || F3o5o || o3f5o || f3o5x || o3x5x || f3x5o || (V3o5o+x3o5f) ||etc.
step 2: x3o5\o || o3x5\o ||x3o5\x=>x3f5\(-x) ||F3o5\o ||o3f5\o || f3o5\x=> f3f5\(-x) || o3x5\x => o3F5\(-x) || f3x5\o || (V3o5\o+x3o5\f) ||etc.
you can see at the step x3o5\x=>x3f5\(-x), needed for restriction 2, that restriction 1 is violated, as a x3o is deleted at the "front" side of the mirror, leaving a non-convex limit. Furthermore at step o3x5\x => o3F5\(-x), also needed for restriction 2, restriction 1 is violated as well, because a o3x is taken away.
(superfluous) step 3: x3o5x || o3x5x || x3f5o || F3o5x || o3f5x || f3f5o ||o3F5o || o3F5o || f3x5x || (V3o5x+x3o5F) ||etc.
As explained at step 2, this isn't CRF. Nevertheless it has some interesting patches, e.g. the last things o3F5o||f3x5x||x3o5F(+V3o5x)||f3x5x||o3F5o can be spotted in D4.5.4.

tex with .3.5x-expansion:
step 1: x3o5o||u3o5o||x3x5o||u3o5x||F3o5o||x3o5u||o3f5x||A3o5o||f3o5u||B3o5x||o3x5u||B3x5o||o3u5x||f3u5o||C3o5o||x3x5f||etc.
step 2: It already fails at restriction 2: the 6th thing from the middle has a u that is part of a tut's hexagon, together with the 4th and 7th's x. this hexagon can't be expanded in a CRF-way, as x||u||x can't be expanded as such. therefore, this one is impossible as well (Although the final expansion gives the middle slice of D4.6, but the rest isn't CRF everywhere).

rox with demitess-expansion (x3.3.*b3.):
step 1: A3x3o*b3o+o3x3A*b3o+o3x3o*b3A + F3o3f*b3x+x3o3F*b3f+f3o3x*b3F + o3f3x*b3f+f3f3o*b3x+x3f3f*b3o
step 2: \A3x3o*b3o+\o3x3A*b3o+\o3x3o*b3A + \F3o3f*b3x+\x3o3F*b3f=>\(-x)3x3F*b3f+\f3o3x*b3F + \o3f3x*b3f+\f3f3o*b3x+\x3f3f*b3o=>\(-x)3F3f*b3o
Both steps \x3o3F*b3f=>\(-x)3x3F*b3f and \x3f3f*b3o=>\(-x)3F3f*b3o were necessary for restriction 2. Furthermore, both of them could be deleted individually, retaining convex limits. Unfortunately, when placed together, they connect with an x-edge that makes the deletion non-CRF. Hence this one also isn't CRF. However, the finished expansion, though not CRF, gives some patches of the D4.8-family.

This means the statement I made at my first post in this article is wrong: I am not able to describe every polytope we've made so far using partial expansions, though I can give constructions that give patches of these, which can be made CRF using patches of other polytopes. (though the x3.3.-, and the .3x3.-expansion of rox seem to be promising to give full CRF-polytopes)

Furthermore, the .3.3.-representation of rox and ex have shown me that my second assumption made here (the one about the new notation and inversions and stuff) is wrong as well, as these aren't symmetrical (i.e. they don't have the same on the "north" side as on the "south" side) This means there are a bit less expansions we have to check, but also that there won't be a general and convenient writing method for the article. (althogh we could use the \ to denotate these with the new notation as well, but that feels too artificial to me).
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student5 » Thu Jun 05, 2014 7:35 am

quickfur wrote:I wonder if this modified Stott expansion (i.e., faceting + "regular" Stott expansion) would itself be sufficient material for a journal paper. I'm almost certain that the icosahedron -> bilbiro/thawro/pentagonal_orthocupolarotunda transformation is unprecedented, and may in itself already be journal-worthy. Add to that the 4D analogues (even just the simplest cases!) and we already have a solid submission, IMO.

The rest of the CRF stuff can be postponed to a subsequent paper.

Please note that there are other possible partial stott expansions in 3D aswell, I listed them somewhere on this forum, and they consist of all partial expansions of other symetry groups.
E.g tet -> tricup or tet->oct
o3o||x3o->x3o||o3x
o3o||x3o ->o3x||x3x
I have listed them somewhere in the forum, I'm just wondering where that is, but they have no precedent either as far as my (limited) knowledge goes
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student5 » Thu Jun 05, 2014 8:18 am

Found the post, it was in partial stott expansions of nonconvex figures.
student5 wrote:[...]
so now for the octahedron:
Code: Select all
#no x's in the left tower, so it remains the same
o4o    o4o       x4o o4x #orthobicup and cuboc (gyrobicup)
o4x -> q4(-x) -> x4x q4o
o4o    o4o       x4o o4x
#symetric, so only one modification view
x3o    x3o       x3x #tricup
o3x -> x3(-x) -> x3o
#left tower again
o2x    o2(-x) x2x o2o #elongated square pyramid and a non-CRF thingy
q2o -> q2o -> Q2o q2x
o2x    o2(-x) x2x o2o

and the tetrahedron:
Code: Select all
x3o    x3o      x3x #tricup
o3x -> x3(-x) -> x3o

x2o    x2o        x2x #squippy
o2x -> o2(-x) -> o2o

I think these expansions can be fun too, and I'm reasonably confident they have their 4D analogies.

I think expansions in ex's symetry group other tgan ex will not prove very worthwile, except fore some patching which will be extremely difficult and tedious because there are so many different patches. The true stottexpansion of newtonian bodies is interesting, even in 3D but it doesn't work there either, as I was trying to expand o3x5o trigonally and pentagonally, both without success. I think it is more useful to contuniue expanding simplexes, since they work in 3D and must surely work in 4D :)
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Thu Jun 05, 2014 12:02 pm

Hmm,

Code: Select all
o4o    o4o       o4x
o4x -> q4(-x) -> q4o
o4o    o4o       o4x

is - in effect - nothing different from x3o4o -> x3x4o -> o3x4o, i.e. a mixture of classical Stott expansion and contraction. Thus not too surprising here. So being obtained here differently in your new quirky mode...

Code: Select all
o4o    x4o
o4x -> x4x
o4o    x4o

is an already known partial Stott expansion, just as being described here. It truely is a partial one, because of just planely applying the usual (non-quirky) expansion, but only with respect to a subsymmetry.

Same holds here:
Code: Select all
o2x    x2x
q2o -> Q2o
o2x    x2x

already described there, as being a true (non-quirky) partial Stott expansion.

That one then
Code: Select all
x3o    x3o       x3x
o3x -> x3(-x) -> x3o

though being quirky again, could be decomposed into a multistep transformation, an usual Stott expansion, such an usual contraction, and than a partial Stott expansion: o3x3o -> x3x3o -> x3o3o = ox3oo&#x -> ox3xx&#x.

Thus, the somehow most surprising one is the last in your list, the transformation from a 3-fold pyramid into a 4-fold one:
Code: Select all
x2o    x2o       x2x
o2x -> o2(-x) -> o2o


--- rk

PS: wrt. the outstanding paper I get the feeling that we should differentiate a bit into
a) classical Stott expansions / contractions (finally published at 1913*),
b) my recent partial ones (found 2013 - you spott the coincidence?) and
c) those new quirky ones (i.e. found in 2014).

(* Submission was already in 1910.)
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby quickfur » Thu Jun 05, 2014 2:43 pm

Along the lines of partial Stott expansion, we could also include these series:

1) triangular bipyramid (J12) -> elongated triangular bipyramid (expand along axis of symmetry)
J12 -> triangular orthobicupola (expand perpendicularly to axis of symmetry)
J12 -> elongated triangular orthobicupola (expand both ways)

2) octahedron (regarded as square bipyramid) -> elongated square bipyramid (expand along axis of symmetry)
square bipyramid -> square orthobicupola (expand perpendicularly to axis of symmetry)
square bipyramid -> elongated square orthobicupola (i.e. rhombicuboctahedron)(expand both ways)

3) octahedron (regarded as triangular antiprism) -> {faceting} -> triangular cupola (this is similar to ico -> {faceting} -> J92)
triangular antiprism -> {faceting} -> triangular gyrobicupola (coincides with cuboctahedron)

4) pentagonal bipyramid (J13) -> elongated pentagonal bipyramid (expand along axis of symmetry)
J13 -> pentagonal orthobicupola (expand perpendicularly to axis of symmetry)
J13 -> elongated pentagonal orthobicupola (expand both ways)
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Thu Jun 05, 2014 5:37 pm

student5 wrote:[...]
I think expansions in ex's symetry group other tgan ex will not prove very worthwile, except fore some patching which will be extremely difficult and tedious because there are so many different patches. The true stottexpansion of newtonian bodies is interesting, even in 3D but it doesn't work there either, as I was trying to expand o3x5o trigonally and pentagonally, both without success. I think it is more useful to contuniue expanding simplexes, since they work in 3D and must surely work in 4D :)

I think we have to nuance this a bit. (we could have a similar apprach as in the Segmetochora-article). trinagles and squares both have expansions in 1D-subsymmetry (o||x "=" o||(-x)->x||o and x||x "=" (-x)||(-x) -> o||o. pentagons (and bigger polygons) don't have such a expansion from CRF to CRF. This means that if there is a big polygon normally intersection the mirror that is subject of expansion, this will yield non-CRF parts. This means things with big polygons are very unlikely to give valid expansions. (in fact this is only possible if we can choose a subsymmetry that has mirrors with no big polygons normally intersecting this, e.g. id=>el. pent. gyrobirotunda, with 1D-subsymmetry. such subsymmetries are either trivial, or hard to find (I doubt their existance) but not trivial)
This is the reason tex, and a bunch of others I've not even looked at, won't easilly allow such expansions. The reason rox seemingly doesn't allow expansions is because all expansions we've tried so far fail on "restricion 1". This does not mean rox doesn't have such expansions, I'm very hopeful it has some of them, though much less than ex. (this because ex can have loads of vertices deleted while staying CRF, in contrary to rox) .3.3.-expansions on it seem to be hopeful.

Klitzing wrote:[...]
PS: wrt. the outstanding paper I get the feeling that we should differentiate a bit into
a) classical Stott expansions / contractions (finally published at 1913*),
b) my recent partial ones (found 2013 - you spott the coincidence?) and
c) those new quirky ones (i.e. found in 2014).

(* Submission was already in 1910.)


Sounds good to me. We could have a) as an introduction, b) as some interesting things, and then c) as things you can only understand after having read b). seems a very good setup to me
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby quickfur » Thu Jun 05, 2014 5:56 pm

student91 wrote:[...]
Klitzing wrote:[...]
PS: wrt. the outstanding paper I get the feeling that we should differentiate a bit into
a) classical Stott expansions / contractions (finally published at 1913*),
b) my recent partial ones (found 2013 - you spott the coincidence?) and
c) those new quirky ones (i.e. found in 2014).

(* Submission was already in 1910.)


Sounds good to me. We could have a) as an introduction, b) as some interesting things, and then c) as things you can only understand after having read b). seems a very good setup to me

I agree. Sounds like (b) can include things like elongated pyramids, orthobicupolae, and 4D things like the bridge-augmented tesseract that I discovered, and the other partial expansions Klitzing found. These are relatively easy to understand.

Then (c) can include the really wild things like the BT polychora, pretasto, etc.. I think that will make for a solid paper!

As a way to lead from (b) to (c), we could use the J91/J92 pseudopyramids as intermediate examples. The J91 pseudopyramid can be constructed from the well-known icosahedral pyramid via the faceting/Stott expansion, and is simple enough for most people to grasp (the line segment pseudo-apex arises directly from the Stott expansion of the ico pyramid's point apex). Ditto for ico pyramid -> J92 pseudo-pyramid. Then having given a taste of J91 and J92 appearing in simple 4D polychora, we can move on to give wilder examples like pretasto, the various bilbiro'd and thawro'd polychora, etc..

I'd like to have some way of including some of the more obscure BT polychora we found, like D4.8.x, but that may be a bit out of the scope of the paper (you cannot construct it via simple Stott expansion, but it seems to be a mix-and-match mosaic of different surface patches from Stott-expanded uniform polychora). But maybe there's a pattern to it (a method to the madness :P)?

Also, I don't know what the restrictions on diagrams might be for publishing in a journal, but if at all possible, I'd love to include some nice renders of these shapes in full color. ;)
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Marek14 » Thu Jun 05, 2014 8:19 pm

quickfur wrote:
student91 wrote:[...]
Klitzing wrote:[...]
PS: wrt. the outstanding paper I get the feeling that we should differentiate a bit into
a) classical Stott expansions / contractions (finally published at 1913*),
b) my recent partial ones (found 2013 - you spott the coincidence?) and
c) those new quirky ones (i.e. found in 2014).

(* Submission was already in 1910.)


Sounds good to me. We could have a) as an introduction, b) as some interesting things, and then c) as things you can only understand after having read b). seems a very good setup to me

I agree. Sounds like (b) can include things like elongated pyramids, orthobicupolae, and 4D things like the bridge-augmented tesseract that I discovered, and the other partial expansions Klitzing found. These are relatively easy to understand.

Then (c) can include the really wild things like the BT polychora, pretasto, etc.. I think that will make for a solid paper!

As a way to lead from (b) to (c), we could use the J91/J92 pseudopyramids as intermediate examples. The J91 pseudopyramid can be constructed from the well-known icosahedral pyramid via the faceting/Stott expansion, and is simple enough for most people to grasp (the line segment pseudo-apex arises directly from the Stott expansion of the ico pyramid's point apex). Ditto for ico pyramid -> J92 pseudo-pyramid. Then having given a taste of J91 and J92 appearing in simple 4D polychora, we can move on to give wilder examples like pretasto, the various bilbiro'd and thawro'd polychora, etc..

I'd like to have some way of including some of the more obscure BT polychora we found, like D4.8.x, but that may be a bit out of the scope of the paper (you cannot construct it via simple Stott expansion, but it seems to be a mix-and-match mosaic of different surface patches from Stott-expanded uniform polychora). But maybe there's a pattern to it (a method to the madness :P)?

Also, I don't know what the restrictions on diagrams might be for publishing in a journal, but if at all possible, I'd love to include some nice renders of these shapes in full color. ;)


Not sure if colors are accepted in general, but of course you could publish a link to the full-color images.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Thu Jun 05, 2014 8:57 pm

quickfur wrote:... As a way to lead from (b) to (c), we could use the J91/J92 pseudopyramids as intermediate examples.

Don't forget the J32 (pocuro) !

The J91 pseudopyramid can be constructed from the well-known icosahedral pyramid via the faceting/Stott expansion, and is simple enough for most people to grasp (the line segment pseudo-apex arises directly from the Stott expansion of the ico pyramid's point apex).

Sorry, "ico" = icositetrachoron = 24-cell, while icosahedron = "ike" ! (The latter is what you are refering here.)

Ditto for ico pyramid -> J92 pseudo-pyramid. Then having given a taste of J91 and J92 appearing in simple 4D polychora, we can move on to give wilder examples like pretasto, the various bilbiro'd and thawro'd polychora, etc..
...


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

Postby student91 » Thu Jun 05, 2014 9:59 pm

quickfur wrote:[...]
Also, I don't know what the restrictions on diagrams might be for publishing in a journal, but if at all possible, I'd love to include some nice renders of these shapes in full color. ;)

Klitzings article has full-colour images as well, so I don't think it to be impossible, though I only know it for the online version. I guess it thus depends entirely on the publisher.
Klitzing wrote:
quickfur wrote:... As a way to lead from (b) to (c), we could use the J91/J92 pseudopyramids as intermediate examples.

Don't forget the J32 (pocuro) !

Don't forget those are the only possible ones for the icosahedral uniforms. Outside of these uniforms, loads of other things are possible (like oct=>3-cup, tet=>squippy etc., as I am listing these, I'm not sure anymore if there are gthat many of them) We should not forget on these non-icosahedral ones. (in 4D you also have (o4o)(x4o)+(x4o)(o4o) "=" (o4o)(x4o)+((-x)4q)(o4o) => (x4o)(x4o)+(o4q)(o4o) =tetradiminished ico. 3D equivalent of this is oct=>cuboct, i.e. not special, but having (oct||cube)=> (cuboct||8-prism) as a consequence)
This means we have to analyse a load of possibilities. It would be very helpfull if this could be done systematically, like Polyhedron Dude seems to be doing with the uniforms. I'm not very good at making such structures in talks. as far as I see it, we have the following subjects:
less than 3D:
fundamental, though not interesting stuff

3D:
non-complex expansions:
these mostly occur in demicubic/cubic/tetrahedral symmetry and dihedral stuff (prism/antiprism/pyramid symmetries). Only exception known so far is id=>elongated pentagonal gyrobirotunda.
complex expansions:
icosahedral symmetry: pocuro, bilbiro and thawro
other symmetry: few are known, more to be explored

4D:
non-complex expansions:
there must be loads of these, how many are not on Klitzings site, is your investigation done exhaustively?

complex expansions:
in ex-symmetry:
expansions of ex: presumably for every subsymmetry, though not sure
expansion of rox: maybe in tetrahedral subsymmetry, others to be investigated.
in other symmetry: loads unknown, how to be investigated?


the complex expansions in other symmetries are the ones that are bothering me: there are probably quite some of them, and checking them all must be horrible if done in one thread. I don't know where to split this in order to find a balance between amounds of threads and length of threads. in general I mean I don't know where to start with these. But first of all, I think it to be best to do exhaustive research on the ex-things.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Fri Jun 06, 2014 10:16 am

student91 wrote:non-complex expansions:
there must be loads of these, how many are not on Klitzings site, is your investigation done exhaustively?

There is obviously some systematics in that, and I was running along that line. But as already stated in the theorem on that page, it is still not known how to set up an encompassing exhaustive set of preconditions. Several cases exceeding the actual set are already contained on that page. But no complete theory. - This btw. is why I did shy away from publication so far.

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

Postby student91 » Sun Jun 08, 2014 9:28 pm

Klitzing wrote:[...] But as already stated in the theorem on that page, it is still not known how to set up an encompassing exhaustive set of preconditions. Several cases exceeding the actual set are already contained on that page.[...]


I guess I'm missing some things here. First of all, what cases are you referring to that are exceeding the actual set?
(note that Euclidean and hyperbolic geometry both allow for infinite amounts of subsymmetries, and therefore have infinitely many partial expansions. Therefore, I think it's best to not include them in the already quite big paper)

Secondly, I did not understand what you were saying here on your site:
"Esp. with respect to the following section it should be pointed out, that full Stott expansion may be considered w.r.t. individual directions. But any such direction then implies that simultanuous expansions will have to be applied in any further direction as well, which is an image of the chosen one under the orbit of the refered to symmetry. – Note that this original idea of Mrs. Stott applies not only to Coxeter groups only, i.e. to reflection derived groups which thus can be given by Dynkin diagrams. This applies to any other symmetry as well. (Esp. as Mrs. Stott derived her ideas already years before the set up of Dynkin diagrams!)"
I do not understand the part " which is an image of the chosen one under the orbit of the refered to symmetry", it's a bit too cryptic to me. Furthermore I do not understand completely what symmetries you are referring to by "This applies to any other symmetry as well". Do you mean symmetries like the ones on which id=>elongated pentagonal gyrobirotunda is based (antiprismatic), or do you mean something else, and could you maybe give more examples?

Thirdly, do you think the following is an exhaustive set of preconditions, though a bit hard to exhaustively investigate:
determine equal limits, these limits must be similar in the sense that they are congruent.
represent the limits as points. these points must be placed as if they were part of a reflective symmetry.
move these points apart according to this symmetry. Of course you should move the limits in the same manner.

the example id=>long name should then be
limits: cupola + cupola, in gyro orientation
points representation: this gives two points with the (obvious) reflective symmetry of a mirror exactly in between.
moving apart: the points now are moved away from the mirror in between them. This makes the elongated pentagonal gyrobirotunda
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Thu Jun 12, 2014 8:21 pm

student91 wrote:
Klitzing wrote:[...] But as already stated in the theorem on that page, it is still not known how to set up an encompassing exhaustive set of preconditions. Several cases exceeding the actual set are already contained on that page.[...]


I guess I'm missing some things here. First of all, what cases are you referring to that are exceeding the actual set? ...

I was refering there to the preconditions of the there provided theorem, i.e. of:
  • Let S1 be any pre-fix Dynkin subsymbol which is a linear sequence from o3 and x3.
  • Let S2 be any post-fix Dynkin subsymbol which is a linear sequence from 3o and 3x.
  • Then there will be a partial Stott expansion series starting at S1-o-4-o-S2 up to S1-o-4-x-S2.
  • Also there will be a partial Stott expansion series starting at o-4-S1-o-4-o up to o-4-S1-o-4-x
  • Also there will be a partial Stott expansion series starting at x-4-S1-o-4-o up to x-4-S1-o-4-x
  • Also there will be a partial Stott expansion series starting at o-4-o-S2-4-o up to o-4-x-S2-4-o
  • Also there will be a partial Stott expansion series starting at o-4-o-S2-4-x up to o-4-x-S2-4-x
  • Let |S1| be further the number of -3- links of that subsequence (possibly 0).
  • Then the number of transition steps of any such series can be given uniformly as: |S1| + 2.

Note that this theorem was set up just by investigation of the provided examples (and their obvious extensions to longer and longer Dynkin symbols). The proof provided always runs the same way: the easier cases (with few nodes ringed) do allow for a different Dynkin representation with respect to a subsymmetry. Wrt that subsymmetry the usual Stott expansion can be applied in multiple steps, whareas in the larger symmetry all subsequent expansions would have to be applied at once, if they would have been described by usual Stott expansion. But then, those higher symmetry Dynkin symbols allow for further decorations (having more nodes ringed), which then cannot been transfered to that lower symmetry any more. But the corresponding series of (then truely) partial Stott expansions can easily be shown (by example) to exist then still!

Outside of the preconditions of this theorem obviously is e.g. the tiling x3o6o, provided there, few lines below. As shown, it allows also for a 3-step partial Stott expansion (shown there), running through 2 intermediate, non-uniform stages.

... (note that Euclidean and hyperbolic geometry both allow for infinite amounts of subsymmetries, and therefore have infinitely many partial expansions. Therefore, I think it's best to not include them in the already quite big paper)

Secondly, I did not understand what you were saying here on your site:
"Esp. with respect to the following section it should be pointed out, that full Stott expansion may be considered w.r.t. individual directions. But any such direction then implies that simultanuous expansions will have to be applied in any further direction as well, which is an image of the chosen one under the orbit of the refered to symmetry. – Note that this original idea of Mrs. Stott applies not only to Coxeter groups only, i.e. to reflection derived groups which thus can be given by Dynkin diagrams. This applies to any other symmetry as well. (Esp. as Mrs. Stott derived her ideas already years before the set up of Dynkin diagrams!)"
I do not understand the part " which is an image of the chosen one under the orbit of the refered to symmetry", it's a bit too cryptic to me. ...

You choose one direction. Then "the orbit of" that "under the action of the symmetry" simply is the set of all symmetry equivalent directions. - but be aware that this highly depends on the refered to symmetry! E.g. a cube, a square prism, and a brique all might have the same shape, but just are considered by very different symmetries. And so, accordingly, the 6 faces either do belong or do not belong to the same orbit resp. equivalence class.

... Furthermore I do not understand completely what symmetries you are referring to by "This applies to any other symmetry as well". Do you mean symmetries like the ones on which id=>elongated pentagonal gyrobirotunda is based (antiprismatic), or do you mean something else, and could you maybe give more examples?

Applications of usual Stott expansions usually are given with respect to Dynkin symbols of Wythoffian polytopes. That is wrt. polytopes, which can be described by reflectional symmetry groups (also called Coxeter groups). But there are also rotational symmetries, glide reflections etc., which might belong to some polytope, but there is no purely reflectional division of those elements possible, which still would be a symmetry of the considered object.

Thirdly, do you think the following is an exhaustive set of preconditions, though a bit hard to exhaustively investigate:
determine equal limits, these limits must be similar in the sense that they are congruent.
represent the limits as points. these points must be placed as if they were part of a reflective symmetry.
move these points apart according to this symmetry. Of course you should move the limits in the same manner.

the example id=>long name should then be
limits: cupola + cupola, in gyro orientation
points representation: this gives two points with the (obvious) reflective symmetry of a mirror exactly in between.
moving apart: the points now are moved away from the mirror in between them. This makes the elongated pentagonal gyrobirotunda

No, you are taking the opposite way here, not providing any necessary condition for the existance, but just requirements instead, which individually have to be investigated in any explicite case. But on the other hand it is a versatile concept on how to proceed. Even so it allows not for classes of to be found cases which haven't to be investigated individually in order to know of there existance. My examples are only investigated thereafter, in order to look for their detailed structure. Not in order to know whether they would exist or not. The existance as such already is provided by the preconditions of the theorem.

My question therefore where pointing into a different direction: whether there would be different or more general cases, which likewise allow for a general conclusion of their existance.

Singular further examples beyond my theorem e.g. were already provided by Wendy: xNo4o and x5o3o4o. - These Point obviously in a slightly different direction and so cannot too easily described by an unifying theory (in the short run).

While x3o6o and the lateron described derivatives like x3o6o3o should be unifiable much more easily with the existing Theorem, I suppose. It also should be extendable then to x-3-o-2N-o in general...

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

Postby student91 » Sun Jun 15, 2014 11:40 am

Just on a sidenote, I found a funny Euclidean partial expansion:
Take the x3o6o-tiling. This can be represented as (A=3; B=4)
x3x3x3*a + A3o3o3*a + o3A3o3*a + o3o3A3*a
There is a 3-step transition series from this to o3o6x:
u3x3x3*a + B3o3o3*a + x3A3o3*a + x3o3A3*a
u3u3x3*a + B3x3o3*a + x3B3o3*a + x3x3A3*a
u3u3u3*a + B3x3x3*a + x3B3x3*a + x3x3B3*a = o3o6x
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Sun Jun 15, 2014 11:29 pm

Oh yeah, I see.

While edges A not only have thrice the unit size, they indeed always are used as three consecutive unit edges. Whereas edges B are to be meant as having four times the size of unit edges, but they only are used as a drawn unit edge, followed by two transparent ones, and then a to be drawn one again! (Else you would have to deal with lots of halved hexagons, i.e. x-x-x-u trapezia. But those then would not be any longer unit edged solely, cf. "CRF".))

  • Then your xAoo3xoAo3xooA3*a&#zxt clearly is nothing but the uniform trat (triangular tiling), i.e. having vertex figure [3^6] only.
  • Your next uBxx3xoAo3xooA3*a&#zxt will have still some vertices of type [3^6], but you would have [3^4,6] and [(3,6)^2] as well.
  • Your further uBxx3uxBx3xooA3*a&#zxt will have still some [(3,6)^2], but additionally vertices of types [3^2,6^2] and [6^3].
  • And finally your uBxx3uxBx3uxxB3*a&#zxt again is the uniform hexat (hexagonal tiling) with just the single vertex figure [6^3].
(Always provided that the B edges would have the mentioned transparent medial part...)

In fact, edges here work always as being kind of
Code: Select all
x = +---+

u = +---+---+

A = +---+---+---+

          +---+
         /     \
B = +---+       +---+
         \     /
          +---+


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

Postby Klitzing » Sat Jun 21, 2014 10:38 am

Klitzing wrote:...
  • Then your xAoo3xoAo3xooA3*a&#zxt clearly is nothing but the uniform trat (triangular tiling), i.e. having vertex figure [3^6] only.
  • Your next uBxx3xoAo3xooA3*a&#zxt will have still some vertices of type [3^6], but you would have [3^4,6] and [(3,6)^2] as well.
  • Your further uBxx3uxBx3xooA3*a&#zxt will have still some [(3,6)^2], but additionally vertices of types [3^2,6^2] and [6^3].
  • And finally your uBxx3uxBx3uxxB3*a&#zxt again is the uniform hexat (hexagonal tiling) with just the single vertex figure [6^3].


Rethought your new "partial Stott expansion sequence" with these 2 intermediate planar "CRF tilings". Here is a pic how it would most probably look like:
xAoo3xoAo3xooA3a_zx-sequence.jpg
(90.06 KiB) Not downloaded yet


Even so that sequence of tilings obviously always remains within CRF (only providing regular triangles and regular hexagons), it is NOT a true partial Stott expansion sequence. :o - Why?

Well, in the left-most triangle tiling (trat) you have 6 types of triangles: green and lime ones, blue and aquamarine ones, resp. red and rose ones. Then you'll note that in the transition into the next tiling, some vertices get blown up to gold resp. purple triangles. - This is complete valid. - But the blue and aquamarine triangles get combined into sky-blue hexagons! And this is not how Stott expansions do work (partial or not). :cry:

At the next transition we get new teal triangles out of former vertices. And gold triangles get blown up to hexagons. - Again both allowed transformations. - But also the red and rose triangles combine into salmon hexagons. - Thus again not valid. :cry:

And finally at the last transition purple and teal triangles get blown up to hexagons. - Again allowed. - But the green and lime triangles get combined into neon hexagons. - Thus again not valid. :cry:

That is, after all, this is not a true (partial) Stott expansion sequence! :oops:


But it could be saved, for sure 8) : by just combining those pairs of triangles into rhombs, eg. cf. the following pic:
rhombic_xoo3xoAo3xooA3a_zx-sequence.jpg
rhombic_xoo3xoAo3xooA3a_zx-sequence.jpg (84.17 KiB) Viewed 73588 times


In that sequence now there is always a one-to-one correspondance: vertices to triangles, rhombs to hexagons, or triangles to hexagons. Thus a completely valid partial Stott expansion sequence.

Just a pitty that on the other hand 3 of those tilings (and esp. these intermediate ones) no longer are CRF, right because of the existance of rhombs! :cry:

--- rk


PS @keiji : don't understand why the one pic comes out to become a downloadable file only, while the other one gets readily displayed... :?:
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby student91 » Sun Jun 22, 2014 6:57 pm

Hmm, yes, that's a funny thing that happens in this funny expansion. I guess this expansion then is an expansion that doesn't belong among normal partial expansions. But why can't it belong to quirky partial expansions (those that need a faceting before rightful application).
The quirky stuff is happening where the limits are connected (quirky stuff is always happening in that region). Here you have the expansion o||x||o (=two triangles, or a rhomb) => x||u||x (=hexagon). Now if you change the triangles that will undergo this expansion in rhombs before doing the expanion, everything is valid. Thus you would get
x3o6o-tiling => facteing thereof => 1-st step => faceting thereof => 2-nd step => faceting thereof => hexagonal tiling
xAoo3xoAo3xooA3a_zx-sequence.png
(350.87 KiB) Not downloaded yet


Now apart from that sidestep: we have to determine the basic structure our paper will get.
It should include 3 kinds of expansion: a)Stotts partial expansions b)Klitzings partial expansions c)quirky expansions (those that need a faceting beforehand)
I would like it if the article had some kind of closure. This means we have to do the following:
1. define what a partial expansion is
2. provide all partial expansions
(3. prove these are all)

1. is where we have to work on: Klitzing clearly has a different definition than I have, and this makes that quite some times, I call something a partial expansion, and he thinks it is a true expansion or an invalid one.
I mostly think of a partial expansion as something that makes a polytope b from a polytope a by moving the vertices of a apart. When you define a as the set of vertices of the polytope (of course, only possible with convex polytopes), then the partial expansion is a function a->b. Furthermore the expansion must be an expansion in the sense that the vertices are moved in some kind of symmetry, and the expansion musn't be writeable in one diagram (i.e. it must have multiple classes of vertices that are moving).

2. can only be done after 1., and 3 is optional, and depends mostly on the time we want to wait for publishing

So I would really like to start discussing what a good definition is for a partial expansion. Furthermore I would like to know what defenition Klitzing is using. As far as I got it, he's using something like: It must be a Stott-expansion, and it must be done according to a lower symmetry than the polytope you begin with has.
I always try to bypass facetings. (though do you think the following to be true: if the vertices of polytope a can be moved to a polytope b according to some symmetry, there exists a faceting of a that can be used to make b from a by expansion)
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Sun Jun 22, 2014 10:54 pm

Dear student91,
I'm based on the definition of Mrs Stott herself (1913: Geometrical deduction of semiregular from regular polytopes and space fillings, p5: Definition of expansion).
Mrs. Stott wrote:Let O be the centre of a regular polytope and M1, M2, M3.. the centre of its limits l1, l2, l3... The operation of expansion consists in moving the limits lk to equal distances away from O each in the direction of the line O Mk which joins O to its centre, the limits lk remaining parallel to their original positions, retaining their original size, and being moved over such a distance that the two new positions of any vertex, which was common to two adjacent edges in the original polytope, shall be separated by a length equal to an edge.

The examples following that definition then make clear that she is speaking of the "elements" in the parlance of the abstract polytopes when refering to "limits", and that the index is meant to describe the dimension of these elements. (In other context these are also known as "k-faces".) - Sure, she is working in the context of regular starting figures. Thus all elements of the same dimension are equivalent. But little later, at least as soon as she is dealing with concatenation of these operations, she gets into the realm where there might occur multiple inequivalent elements of the same dimension. Then she is assuming to restrict to just one of those being her "subject".

We should note then that she is assuming to use the full symmetry of the object which divide the elements of some chosen dimension into subsets. - Here I went a step beyond, when I was speaking of "partial Stott expansions": I just use a subsymmetry of the whole figure. And I use only the radial vectors of that subsymmetry for expansion directions. That same subsymmetry also usually maps larger complexes or patches only transiently around, i.e. patches with more than just a single element. Thus the expansion then is assumed to act on these larger patches as a whole, i.e. as larger moving units. Moreover such a patch itself does not need to have a specific facet which is orthogonal to the expansion direction. E.g. when expanding the octahedron towards the elongated square bipyramid, it is a patch of 4 triangles which is moved, and none of the triangles is orthogonal to the moving direction.

Further we should note that she is NOT mentioning l0 in her definition. In fact, whenever we are considering subdimensional elements only, we get armys, regiments, etc. of polytopes which all have the same defining struts. And so we get into some wild speculation onto which member to choose.

We finally should have a look too on the effects of Mrs. Stotts expansion. When she moves some elements out, those do not change, neither in shape nor in count - only in position. At the former ridges, peaks, etc. further elements may arise. But always we have kind a one-to-one relation: all former facets remain facets, some ridges become further facets, some peaks might become facets too, etc. In fact, any facet of the final outcome is derived by exactly one specific element (possibly of some subdimension) of the starting figure. (Esp. there is no such thing that 2 triangles have to be joined first into a rhomb in order to get expanded into an hexagon.)

This also is how I was looking e.g. onto these recent ike to bilbiro transformations too. We are not allowed to resort to the mere vertex set of the icosahedron and move that apart into just 2 opposing directions. We well have to consider the full patches that move. And, when doing that from the opposite direction, i.e. by contracting the bilunabirotunda correspondingly, we can see quite nicely which faceting of the icosahedron originally would have to be used here.

All that "quirks" stuff with using negative edges in Dynkin symbols just results in choosing an appropriate faceting first. I think that this is well allowed, but is a independent step to be done first, and has nothing to do with the Stott expansion itself. - Whether a rhomb, being derived from 2 neighbouring triangles, can truely be called a faceting, depends. Euclidean space is kind a singularity here. Everywhere else a faceting face cannot use 2 consecutive sides of a specific face without getting forced to reproduce that very face. But such a rhomb would just do that. - This very singularity of vanishing curvature also lead e.g. to the redefinition of "scaliformity", now requiring a third axiom which asks that any to be used element from any dimension has to be orbiform itself. Thus explicitely excluding rhombs, shields, and other weird stuff.

But when considering the tri-hexagonal rhombic tiling (i.e. the dual of o3x6o), the 3-step partial Stott expansion thereon is clearly valid (as shown in my last post). No weird stuff is needed to interfere. Esp. no intermediate facetings (whether being allowable or not).

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

Postby Klitzing » Wed Jun 25, 2014 1:05 pm

Just an note related to student91's (-x) edges.

You could any part of the Dynkin diagram containing ...---n/d---(-x)---m/b---...
simply replace by ...---n/(n-d)---x---m/(m-b)---...
And esp. m=2, b=1 surely shows that the trailing link here would be allowed to be missing as well.
As invers of the former also further links could be added in the same way, e.g. when considering a bifurcation node of the graph.

E.g.
  • x3(-x) = x3/2x = x6/2o = Grünbaumian doubly wound hexagon = double covered triangle
  • x3(-x)3o = (-x)3x3o = x3/2x3o = x3/2x3/2o = Grünbaumian triple covered tetrahedron (each face plane consists of a triangle and a coincident doubly wound hexagon)
  • x3(-x)4o = (-x)3x4o = x3/2x4o = x3/2x4/3o = Grünbaumian double covered octahedron with all faceting squares, also double covered (faces are 8 doubly wound hexagons plus 4 squares)
  • x3(-x)5o = (-x)3x5o = x3/2x5o = x3/2x5/4o = Grünbaumian double covered icosahedron with inscribed (single covered) great dodecahedron (faces are 20 doubly wound hexagons plus 12 pentagons)
  • etc.

Further it shall be mentioned that the incidence matrices, i.e. the abstract polytopes would not differ at all, when the denominator of these link marks would just be dropped. It is only the geometric relative placement of elements which differs (and in cases of Grünbaumian polytopes just happens to become completely coincident with some of their elements).

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

Postby student91 » Mon Jun 30, 2014 11:15 pm

Klitzing wrote:Dear student91,
I'm based on the definition of Mrs Stott herself (1913: Geometrical deduction of semiregular from regular polytopes and space fillings, p5: Definition of expansion). [...]
Of course we all base our definitions on this definition. What I was trying to discuss is what a partial expansion is, and esp. what makes it partial (the expansion part was indeed already provided by mrs Stott). Following this you already recognize two different definitions of partial:
We should note then that she is assuming to use the full symmetry of the object which divide the elements of some chosen dimension into subsets. - Here I went a step beyond, when I was speaking of "partial Stott expansions": I just use a subsymmetry of the whole figure. And I use only the radial vectors of that subsymmetry for expansion directions. That same subsymmetry also usually maps larger complexes or patches only transiently around, i.e. patches with more than just a single element. Thus the expansion then is assumed to act on these larger patches as a whole, i.e. as larger moving units. Moreover such a patch itself does not need to have a specific facet which is orthogonal to the expansion direction. E.g. when expanding the octahedron towards the elongated square bipyramid, it is a patch of 4 triangles which is moved, and none of the triangles is orthogonal to the moving direction.

That is, an expansion may be called partial, or at least not a normal expansion, if either:
1] You are expanding according to a subsymmetry of the whole figure, or
2] You use only the radial vectors of that subsymmetry for expansion directions. This means you use whole complexes (or patches) as limits. (if you use patches, you have to use radial vectors, and you only need to use radial vectors if the direction of expansion is ambiguous. Therefore these two parts have to be together).
sidequestion is if we are going to restrict ourselves to reflective subsymmetry only, or include more symmetries as well

These two give rise to different sets: When you are using 1], you have to discard things with facetings or middle-steps of expansion-chains (like esquidpy=>squobcu etc.) When you are using 2] you include these, but you exclude things like o3x3o->o3x3x etc. Therefore 2] does not correspond to the way Stott uses the partial expansions. I think both definitions are valuable, but we have to be able to distinguish between them, to prevent more ambiguities. I suggest we keep calling 1] a partial expansion, (as it is being used like this in Stotts paper), and the call 2] an extended partial expansion, or EPE, or something like that. (I don't like that name, but discussions about names are so horribly useless, much more useless than discussions about definitions. when you have a better proposition I'll stick to that one). Note that it is incredibly difficult to make a comprehensive set of definitions to include both of the sets 1] and 2]. I guess in hindsight that is one of the reasons I was using vertex configuration only, as then more of both sets are included. However, in hindsight this approach indeed seems less perfect.

When we have a definition, I guess we can start investigating the subject. I hope you agree 2] is best to make subject of the article. If you do so, first thing we have to do is investigate what subsymmetries the uniforms have (though our definition as we have it doesn't exclude expansions of non-uniforms. if we choose to restrict ourselves to uniforms, it would make the field of investigation much less unknown, but it would also exclude ike-pyramid=>pseudopyramids. I guess here we have to find a compromise)
I though this to be done excessively, though the existence of the demicubic symmetry of ex made me doubt this, as I could find nothing about that on the free sources I use. If we choose to use other symmetries than reflective this investigation might very well come out to be quite big. However, Wendy seemed to know about the demicubic subsymmetry of ex, so I hope she can refer me to somewhere where I can see this is already done.

[...]
We finally should have a look too on the effects of Mrs. Stotts expansion. When she moves some elements out, those do not change, neither in shape nor in count - only in position. At the former ridges, peaks, etc. further elements may arise. But always we have kind a one-to-one relation: all former facets remain facets, some ridges become further facets, some peaks might become facets too, etc. In fact, any facet of the final outcome is derived by exactly one specific element (possibly of some subdimension) of the starting figure.
Note that in the more complex expansions (such as id=>el.pent.gyrobirotunda) you always have a one-to-one relation of limit-to-limit (rotunda remains rotunda), and the things that formerly connected the limits do crazy things (like o||o=>x||x). That's basically why I like the segmentogora-format for the article, as the crazy things are always subdimensional. therefore a listing of the subdimensional expansions might directly exclude loads of expansions, as there is no appropriate crazy thing.
(Esp. there is no such thing that 2 triangles have to be joined first into a rhomb in order to get expanded into an hexagon.)
This thus is indeed a very funny exceptional but comparable crazy thing. It would also not be listed in the subdimensional things. Indeed it is exceptional, let's not make this head subject of this topic.
[...]
All that "quirks" stuff with using negative edges in Dynkin symbols just results in choosing an appropriate faceting first. I think that this is well allowed, but is a independent step to be done first, and has nothing to do with the Stott expansion itself. - Whether a rhomb, being derived from 2 neighbouring triangles, can truely be called a faceting, depends.[(my opinion here is that it can, but it might not be a regular, true, scaliform or whatever-it-is-not faceting, but just an irregular appropriate faceting)] Euclidean space is kind a singularity here. Everywhere else a faceting face cannot use 2 consecutive sides of a specific face without getting forced to reproduce that very face. But such a rhomb would just do that. - This very singularity of vanishing curvature also lead e.g. to the redefinition of "scaliformity", now requiring a third axiom which asks that any to be used element from any dimension has to be orbiform itself. Thus explicitely excluding rhombs, shields, and other weird stuff.
[...]
--- rk

I've never really investigated Euclidean and hyperbolean geometry that well. (just as non-convex polytopes, that last post of you gives rise to interesting directions of investigation, but I'm currently too (pre)occupied with other things(on a sidenote, my school-project is making progress, I have a very comprehensible explanation about stott-expansions and Coxeter-Dynkin-diagrams etc. I guess I will upload a translation of it when I have time (though that will take long, I will have my exams next year, and I have to finish my school-project in december, so I might just not be very active here for some time). If someone is able to translate from dutch, It would be awesome if he did, you can find my school-project here, chapter 1.1.3 (without introduction of platonic solid, and exclusion of prisms and antiprisms)))) I have no idea how things like x5o4o3o etc. work.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby wendy » Tue Jul 01, 2014 7:33 am

Here's something. Has anyone considered what happens if ye take say s3s4o3x, and replace the icosahedra with a pyramid?

The balance of the faces are variously expanded tetrahedra (ie xo3xx&#t ) = triangular cupolae and Cuboctahedra x3o3x (for the tetrahedra).

But we know this beast is convex, because all of its faces are parallel to some {3,3,5}.

It might be something that student91 is looking for.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Tue Jul 01, 2014 8:46 am

Wendy suggests to consider prissi with ikes being augmented by ikepies.

prissi = s3s4o3x
i.e. it is derived from s3s4o3o (sadi) by expansion along .3.4.3x
the ikes remain unchanged in that expansion
but ike -> ikepy within sadi gets it back to ex

thus:
prissi with ike -> ikepy ought to be some ex expansion wrt. to some icoic subsym!

Here at least is the incmat of that suggested biform figure
Code: Select all
288  * |   2   1   4   1 |  1   2   3   4  2   1   4 |  2  1  3   2   3 prissi vertices
  * 24 |   0   0   0  12 |  0   0   0   0  0   6  24 |  0  0  0   8  12 ikepy tips ==> verf = ike (snit)
-------+-----------------+---------------------------+-----------------
  2  0 | 288   *   *   * |  1   0   0   2  1   0   0 |  1  1  2   0   0
  2  0 |   * 144   *   * |  0   0   2   0  2   1   0 |  0  1  2   0   2
  2  0 |   *   * 576   * |  0   1   1   1  0   0   1 |  1  0  1   1   1
  1  1 |   *   *   * 288 |  0   0   0   0  0   1   4 |  0  0  0   2   3
-------+-----------------+---------------------------+-----------------
  3  0 |   3   0   0   0 | 96   *   *   *  *   *   * |  0  1  1   0   0
  3  0 |   0   0   3   0 |  * 192   *   *  *   *   * |  1  0  0   1   0
  3  0 |   0   1   2   0 |  *   * 288   *  *   *   * |  0  0  1   0   1
  4  0 |   2   0   2   0 |  *   *   * 288  *   *   * |  1  0  1   0   0
  6  0 |   3   3   0   0 |  *   *   *   * 96   *   * |  0  1  1   0   0
  2  1 |   0   1   0   2 |  *   *   *   *  * 144   * |  0  0  0   0   2
  2  1 |   0   0   1   2 |  *   *   *   *  *   * 576 |  0  0  0   1   1
-------+-----------------+---------------------------+-----------------
  6  0 |   3   0   6   0 |  0   2   0   3  0   0   0 | 96  *  *   *   * trip
 12  0 |  12   6   0   0 |  4   0   0   0  4   0   0 |  * 24  *   *   * tut
  9  0 |   6   3   6   0 |  1   0   3   3  1   0   0 |  *  * 96   *   * tricu
  3  1 |   0   0   3   3 |  0   1   0   0  0   0   3 |  *  *  * 192   * tet
  3  1 |   0   1   2   3 |  0   0   1   0  0   1   2 |  *  *  *   * 288 tet

(Wendy is wrong wrt. coes. There are tuts instead.)
So some tets of ex (in fact an icoic subset) gets expanded into tuts.
All the 96 face incident tets then get expanded into tricues.
The trips thereby arise from former mere triangles: their lacing edges arise from the distortion from mere vertices into the hexagon-hexagon edges of the then tuts. Their lateral square faces obviously connect to the ones of the tricues.
And the remaining tets (600-24-96=480=192+288) herein remain unchanged.

I suppose that it might be harder to describe that transformation directly within full icoic symmetry, i.e. wrt .3.4.3. .
It might be easier to do a first trial within the common subsymmetry of both, icoic and hyic, i.e. wrt. .3.3. *b3.
In the latter ico is described as o3x3o *b3o.
Thus we would just need for some description of ex as some zero height lace prism (/tower?) decomposition in that subsymmetry, i.e. as (?)3(?)3(?) *3(?)&#zx(t), where the "(?)" have to be evaluated as a sequence of 2 or more letters each.

Sure, all three arms will look the same here, i.e. the symbol itself gets threefold symmetry. This then reflects the in fact being needed higher icoic symmetry. And this is how then potentially also a (?)3(?)4(?)3(?)&#zx(t) description might be found too.

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

Postby Klitzing » Tue Jul 01, 2014 1:04 pm

Hmmm,
the vertex set of ex = x3o3o5o is that of a compound of 5 vertex inscribed icoes = x3o4o3o, if scaled by f, for sure.
And sadi = s3s4o3o is that ex faceting, which arises when exactly 1 ico in that compound is missing.

Thus we get: ex = os3os4oo3fo&#zx:
Code: Select all
os3os4oo3fo&#zx

      o.3o.4o.3o.       | 24  * |  12   0   0 |   6  24  0   0  0 |   8  12  0  0
demi( .o3.o4.o3.o     ) |  * 96 |   3   3   6 |   3  12  3   9  3 |   6   9  1  4
------------------------+-------+-------------+-------------------+--------------
demi( oo3oo4oo3oo&#zx ) |  1  1 | 288   *   * |   1   4  0   0  0 |   2   3  0  0
      .. .s4.o ..       |  0  2 |   * 144   * |   1   0  0   2  2 |   0   2  1  2
sefa( .s3.s .. ..     ) |  0  2 |   *   * 288 |   0   2  1   2  0 |   2   2  0  1
------------------------+-------+-------------+-------------------+--------------
      .. os4oo ..&#x    |  1  2 |   2   1   0 | 144   *  *   *  * |   0   2  0  0
sefa( os3os .. ..&#x  ) |  1  2 |   2   0   1 |   * 576  *   *  * |   1   1  0  0
      .s3.s .. ..       |  0  3 |   0   0   3 |   *   * 96   *  * |   2   0  0  0
sefa( .s3.s4.o ..     ) |  0  3 |   0   1   2 |   *   *  * 288  * |   0   1  0  1
sefa( .. .s4.o3.o     ) |  0  3 |   0   3   0 |   *   *  *   * 96 |   0   0  1  1
------------------------+-------+-------------+-------------------+--------------
      os3os .. ..       |  1  3 |   3   0   3 |   0   3  1   0  0 | 192   *  *  *
sefa( os3os4oo ..&#x  ) |  1  3 |   3   1   2 |   1   2  0   1  0 |   * 288  *  *
      .. .s4.o3.o       |  0  4 |   0   6   0 |   0   0  0   0  4 |   *   * 24  *
sefa( .s3.s4.o3.o     ) |  0  4 |   0   3   3 |   0   0  0   3  1 |   *   *  * 96

(Note the usage of 'z' in that suffix &#zx : i.e. the height of that lace prism not only evaluates to zero, but the bases too are nothing but pseudo elements of the to be described polychoron.)

This display of ex might serve helpful, esp. when considering its representation within icoic subsymmetry.

But then the searched for description of that biform figure of Wendy would accordingly just read os3os4oo3Fx&#zx, which then is not truely that enlighting. Esp. as this description already might be read as "o3o4o3F || s3s4o3x", i.e. vertices of some huge ico atop a (starting) prissi, i.e. nothing but the symbolic direct translation of her construction device: the augmentation of the ikes. (Sure, in that other sense of reading partially snubbed lace prisms, as also the above matrix display of ex is outlined, this symbol well can also be read as "alternated faceting of ox3ox4ooFx&#zy with resp. to elements ..3..4.o3.x". That then is a bit more sofisticated in its statement.)

But perhaps we can derive a more enlighting different description too?

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

Postby student91 » Tue Jul 01, 2014 1:34 pm

Klitzing wrote:Wendy suggests to consider prissi with ikes being augmented by ikepies.

prissi = s3s4o3x
i.e. it is derived from s3s4o3o (sadi) by expansion along .3.4.3x
the ikes remain unchanged in that expansion
but ike -> ikepy within sadi gets it back to ex

thus:
prissi with ike -> ikepy ought to be some ex expansion wrt. to some icoic subsym!
[...]
So some tets of ex (in fact an icoic subset) gets expanded into tuts.
All the 96 face incident tets then get expanded into tricues.
The trips thereby arise from former mere triangles: their lacing edges arise from the distortion from mere vertices into the hexagon-hexagon edges of the then tuts. Their lateral square faces obviously connect to the ones of the tricues.
And the remaining tets (600-24-96=480=192+288) herein remain unchanged.

I suppose that it might be harder to describe that transformation directly within full icoic symmetry, i.e. wrt .3.4.3. .
It might be easier to do a first trial within the common subsymmetry of both, icoic and hyic, i.e. wrt. .3.3. *b3.
In the latter ico is described as o3x3o *b3o.
Thus we would just need for some description of ex as some zero height lace prism (/tower?) decomposition in that subsymmetry, i.e. as (?)3(?)3(?) *3(?)&#zx(t), where the "(?)" have to be evaluated as a sequence of 2 or more letters each.
Such a representation has been found in the D4.11-topic and onwards, though a bit cryptical. It is f3o3x*b3o + o3o3f*b3x + x3o3o*b3f + o3f3o*b3o, or foxo3ooof3xfoo*b3oxfo&#zx in shorter notation. The first three things give sadi (so fox3ooo3xfo*b3oxf&#zx). Prissi then would be this with an expansion on the middle o's, or fox3xxx3xfo*b3oxf&#zx. You can see the tuts here, as .3x3o*b3x + o3x3.*b3x + x3x3o*b3. The thing Wendy suggests then is foxo3xxxF3xfoo*b3oxfo&#zx. That is, a demicubic expansion of ex using the middle node. Using one of the other nodes gives the ico-augmented D4.11 (D4.11.1 or so?). I think it is weird we have never tried expanding according to this node.
Sure, all three arms will look the same here, i.e. the symbol itself gets threefold symmetry. This then reflects the in fact being needed higher icoic symmetry. And this is how then potentially also a (?)3(?)4(?)3(?)&#zx(t) description might be found too.
--- rk
As you see, the symbol doesn't have direct threefold symmetry. instead, it has three chiral pairs f3o3x*b3o + o3o3f*b3x + x3o3o*b3f, which can be morphed into each other by a rotation of the diagram. This shows ex has demicubic symmetry with a (chiral) trefoil turning symmetry. I like how the symmetries of the polytopes are reflected in the diagrams.
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Re: Construction of BT-polytopes via partial Stott-expansion

Postby Klitzing » Tue Jul 01, 2014 4:46 pm

student91 wrote:
Klitzing wrote:Wendy suggests to consider prissi with ikes being augmented by ikepies.
[...] I suppose that it might be harder to describe that transformation directly within full icoic symmetry, i.e. wrt .3.4.3. .
It might be easier to do a first trial within the common subsymmetry of both, icoic and hyic, i.e. wrt. .3.3. *b3.
In the latter ico is described as o3x3o *b3o.
Thus we would just need for some description of ex as some zero height lace prism (/tower?) decomposition in that subsymmetry, i.e. as (?)3(?)3(?) *3(?)&#zx(t), where the "(?)" have to be evaluated as a sequence of 2 or more letters each.
Such a representation has been found in the D4.11-topic and onwards, though a bit cryptical. It is f3o3x*b3o + o3o3f*b3x + x3o3o*b3f + o3f3o*b3o, or foxo3ooof3xfoo*b3oxfo&#zx in shorter notation. The first three things give sadi (so fox3ooo3xfo*b3oxf&#zx).

:D exactly what I was looking for. Didn't remember that you already described that one.
Prissi then would be this with an expansion on the middle o's, or fox3xxx3xfo*b3oxf&#zx. You can see the tuts here, as .3x3o*b3x + o3x3.*b3x + x3x3o*b3. The thing Wendy suggests then is foxo3xxxF3xfoo*b3oxfo&#zx. That is, a demicubic expansion of ex using the middle node. Using one of the other nodes gives the ico-augmented D4.11 (D4.11.1 or so?). I think it is weird we have never tried expanding according to this node.

so Wendy thus contributed to CRFebruary ...
Sure, all three arms will look the same here, i.e. the symbol itself gets threefold symmetry. This then reflects the in fact being needed higher icoic symmetry. And this is how then potentially also a (?)3(?)4(?)3(?)&#zx(t) description might be found too.
--- rk
As you see, the symbol doesn't have direct threefold symmetry. instead, it has three chiral pairs f3o3x*b3o + o3o3f*b3x + x3o3o*b3f, which can be morphed into each other by a rotation of the diagram. This shows ex has demicubic symmetry with a (chiral) trefoil turning symmetry. I like how the symmetries of the polytopes are reflected in the diagrams.

Yep, you are right in that. Ex does not have full icoic symmetry, for sure. Still those arms "look" rather equal. Kind a chirality as you state. So being a 3-value one. Even sadi can be described only as a snub of the icoic group: s3s4o3o.

--- rk
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