student91 wrote:So what we can do next is investigate all possible partial Stott-expansions, determine when they are convex, write a paper, be awesome and eat a cookie (long term goal)
quickfur wrote:This is exciting! So we should write a joint paper together, describing our findings?
I'm thinking the subject of the paper can be partial (or modified) Stott expansion itself, and the CRFs we discovered would serve as examples of the process, so we are not obligated to enumerate all possibilities (which I think will take a lot more work -- and I mean a lot more, given how many possible combinations there are). This way, we can just use our current discoveries (maybe plus some others we might find on the way) as the source data to draw from, which should reduce the total amount of work that needs to be done. Plus, it should keep the paper within reasonable length so that people will actually read it. The idea of partial expansion or modified expansion (e.g., ike -> bilbiro) seems like a good unifying concept for the paper to focus on.
I think the number of combinations is much less than we think. First of all, we "only" have to check all subsymmetries for all polytopes with ikes. (that's not completely true, you also have the case of id being treated differently.) furthermore I think investigations will show we can ignore even more expansions.quickfur wrote:I'm thinking the subject of the paper can be partial (or modified) Stott expansion itself, and the CRFs we discovered would serve as examples of the process, so we are not obligated to enumerate all possibilities (which I think will take a lot more work -- and I mean a lot more, given how many possible combinations there are)
This way, we can just use our current discoveries (maybe plus some others we might find on the way) as the source data to draw from, which should reduce the total amount of work that needs to be done. Plus, it should keep the paper within reasonable length so that people will actually read it. The idea of partial expansion or modified expansion (e.g., ike -> bilbiro) seems like a good unifying concept for the paper to focus on.
Just forgot to tell you, very few of the BT-polytopes are true partial expansions, most of them are built up from parts of expansions and parts of normal uniforms, or had some cutting after the expansion. So I wanted to say: those "pure" forms are probably low in number, with an awful lot of cutting possible.quickfur wrote:[...]They are all partial (or modified) Stott expansions of the 120-cell family uniforms. In other words, the 4D analogues of J91 and J92.
So then this leads to the next question: are there any crown jewels outside this class?
quickfur wrote:Anyway, back to the prospective paper on partial expansions: I think it's worthwhile to also include the 3D cases of modified Stott expansion we found: ike -> J91, ike -> J92, ike -> J32. AFAIK, these derivations are entirely new, and have never been considered before, especially the fact that J91 and J92 have direct derivations from ike: they are more than just some slice of the icosidodecahedron with random CRF bits closing it up; these "random bits" turn out to have direct correspondence with the faceted ike from which the rest of the shape is derived.
(But I could be wrong about this being new -- I'm not as well-read as I'd like to be. Klitzing should know, though. )
wendy wrote:Thanks for finding the paper! I've downloaded it, it seems pretty much what i imagined to be in it.
bilbiro
oxFxo 2 ofxfo &#xt
-> : (-x) 2 (-o)
<- : (+x) 2 (+o)
ike faceting
(-x)ofo(-x) 2 ofxfo &#xt
thawro
oxFx 3 xfox &#xt
-> : (-x) 3 (-o)
<- : (+x) 3 (+o)
ike faceting
(-x)ofo 3 xfox &#xt
pocuro
xoxx 5 ofxo &#xt
-> : (-x) 5 (-o)
<- : (+x) 5 (+o)
ike faceting
o(-x)oo 5 ofxo &#xt
As far as I got it, in that part she was describing contractions of octahedral polytopes according to tetrahedral subsymmetry, resp. contractions of tesseractic polytopes according to demitesseractic symmetry. She describes:Klitzing wrote:[...]
While re-reading it now once more, I even found that Alicia herself used the term "partial operations" (page 15 of the pdf) when refering to the application of expansion to the triangular "limits" (faces) only in the transition from cuboctahedron towards truncated cube. Thus my term of "partial Stott expansion / contraction / operation" already got a precursor. But still, my idea would have been new. As she there is still using full symmetry again, "partial" there was just refering to a subset of "limits" (facets).
In this context it should be noted that the transitions of the yesterday provided pics of "true" transitions from faceted ike to bilbiro resp. from (differently) faceted ike to thawro (as well as the there not yet provided further one from even different faceted ike to pocuro) all are partial in Alicia's sense only: they use a subset of faces, which have to be expanded, not a subsymmetry! It only surpasses Alicia's setup, that expansions would not be normal to the corresponding "limits" but here rather orthogonal to the axial symmetry of the objects.
In contrast in my partial expansions, eg. from oct to esquidpy (J15) even uses still all faces, but breaks the former symmetry instead! (Whereas in the following steps of that very sequence, i.e. from esquidpy to squobcu (J28), resp. from squobcu to sirco it then would be "partial" in the sense of student91 again only (subset of faces moving non-necessarily in face-normal directions, but using the full symmetry of the starting figures).
[...]
--- rk
student91 wrote:Klitzing wrote:[...]In this context it should be noted that the transitions of the yesterday provided pics of "true" transitions from faceted ike to bilbiro resp. from (differently) faceted ike to thawro (as well as the there not yet provided further one from even different faceted ike to pocuro) all are partial in Alicia's sense only: they use a subset of faces, which have to be expanded, not a subsymmetry! It only surpasses Alicia's setup, that expansions would not be normal to the corresponding "limits" but here rather orthogonal to the axial symmetry of the objects.
In contrast in my partial expansions, eg. from oct to esquidpy (J15) even uses still all faces, but breaks the former symmetry instead! (Whereas in the following steps of that very sequence, i.e. from esquidpy to squobcu (J28), resp. from squobcu to sirco it then would be "partial" in the sense of student91 again only (subset of faces moving non-necessarily in face-normal directions, but using the full symmetry of the starting figures).
[...]
--- rk
I'm not sure what you ment here, my expansions clearly are with respect to a full subsymmetry. It is because of this subsymmetry, that we get more than one set of vertices, together making a full set of faces and stuff. A expansion of ike with a subset of faces could be x3o5o=>(-x)3x5o=>o3x5o. you take the x5o-faces that are hidden in ike (the faces of the great dodecahedron) and move them outwards. My partial expansions use a subsymmetry, so ike x3o5o has a axial subsymmetry, fox2xfo2oxf&#zx. when you expand this, you first change it in fox2xfo2of(-x)&#zx, then you add x to the right group of nodes. This makes two sets of fox2xfo(2xoF)&#zx move outwards (two pentagons together with two triangles). This clearly is according to axial subsymmetry. I don't think it differs that much from oct=>esquidpy, as that is qoo2oqo2ooq&#zx, expanded. The "limits" that are moving are qoo2oqo(2ooq)&#zx, which is a square pyramid. Thus oct=>esquidpy is, just as ike=>bilbiro, an expansion according to axial subsymmetry, if I'm right. not that the whole partial expansion chain of oct=>sirco can be described in axial subsymmetry: qoo2oqo2ooq&#zx => qoo2oqo2xxw&#zx => qoo2xwx2xxw&#zx => wxx2xwx2xxw&#zx. So this is a chain of partial expansions according to my definition. this is different from ike=>bilbiro in that that one doesn't give a chain of expansions. It might very well be that my definition is the same as yours.
student91 wrote:As far as I got it, in that part she was describing contractions of octahedral polytopes according to tetrahedral subsymmetry, resp. contractions of tesseractic polytopes according to demitesseractic symmetry. She describes:Klitzing wrote:[...]
While re-reading it now once more, I even found that Alicia herself used the term "partial operations" (page 15 of the pdf) when refering to the application of expansion to the triangular "limits" (faces) only in the transition from cuboctahedron towards truncated cube. Thus my term of "partial Stott expansion / contraction / operation" already got a precursor. But still, my idea would have been new. As she there is still using full symmetry again, "partial" there was just refering to a subset of "limits" (facets).
½e_{2}O=tT, that is o4o3x (O) = o3x3o => x3x3o (½e_{2}O), or o4o3x => s4o3x with snub-nodes
½c_{0}c_{0}e_{1}C=T, that is x4o3o (C) => x4x3o (e_{1}C)=> o4x3o (c_{0}e_{1}C)= x3o3x => o3o3x (½c_{0}c_{0}e_{1}C)
½c_{0}c_{0}e_{1}C_{8}=C_{16}, that is, x4o3o3o (C_{8})=> x4x3o3o (e_{1}C_{8})=> o4x3o3o (c_{0}e_{1}C_{8}) = x3o3x*b3o => x3o3o*b3o (½c_{0}c_{0}e_{1}C_{8}).
Thus she describes full Stott-contractions, but according to a subsymmetry. This part thus treats the overlap between tetrahedral and octahedral symmetry, resp. that of tessic and demitessic symmetry.
... I guess nowadays, we won't really call this partial expansion resp. contraction.
Klitzing wrote:My point was that both, the faceting of ike, which I showed up, and your vertex representation of ike within that subsymetry (it should read rather fox2xfo2o(-x)f&#zx, I suppose) already break the global symmetry before we even apply Stott expansion.
As you correctly point out, the transitions areBut you can see that you (first line) need to start with an assymmetric version, while I (second line) start with the full symmetrical one. That is the tiny, but crucial difference.
- fox2xfo2o(-x)f&#zx + (+o)2(+o)2(+x) = fox2xfo2xoF&zx
- qoo2oqo2ooq&#zx + (+o)2(+o)2(+x) = qoo2oqo2xxw&#x
(My subsequent chain does not contribute to the argument. In fact, at the start of the second transition, the former symmetry already has been broken.)
--- rk
Klitzing wrote:[...]... I guess nowadays, we won't really call this partial expansion resp. contraction.
In fact, this exactly is, what partial (=subsymmetrical) expansion is about!
--- rk
student91 wrote:Recently I've found another partial expansion of ex (one that gives bilbiro's): take the .5.3.-representation of ex:
o3o5o||x3o5o||o3o5x||f3o5o||o3x5o||f3o5o||o3o5x||x3o5o||o3o5o
...
This time we move the layers themselves apart. This gives us:
o3o5o||x3o5o||o3o5x||o3x5o||f3o5o||o3x5o||o3o5x||x3o5o||o3o5o
...
... Such 3d-representations follow pretty straightforward from what we know from lace-cities and lace-towers. In 3D, we have lace-cities with single nodes. This is a 2D-representation with 1D-subsymmetry, and 2+1=3D. We also have lace-towers: 1D-representations with 2D-subsymmetry (again, 1+2=3D). (we could also take 0D-representations with 3D-subsymmetry (the original dynkin-diagram), and 3D-representations with 0D-subsymmetry (all it's vertices placed in space), but these are limiting representations that don't add anything). in 4D we have 2D-representations with 2D-subsymmetry, and 1D-representations with 3D-subsymmetry. 3D-representations with 1D-subsymmetry would have nodes (x,f,F etc.) placed in a 3D space.
student91 wrote:Recently I've found another partial expansion of ex (one that gives bilbiro's): take the .5.3.-representation of ex:
o3o5o||x3o5o||o3o5x||f3o5o||o3x5o||f3o5o||o3o5x||x3o5o||o3o5o
As I've explained before, on the equator are hidden ikes in .2.2.-orientation. We've already expanded these ikes two ways (one gave castellated x5o3x-prism, one gave bilbiro'd o5o3x3o). These two ways are both done by adding an x to all parts of the polytope(see my first post on this topic). The third one should be done differently. This time we move the layers themselves apart. This gives us:
o3o5o||x3o5o||o3o5x||o3x5o||f3o5o||o3x5o||o3o5x||x3o5o||o3o5o
The bilbiro's can be revealed by deleting corresponding vertices of the id's.
Maybe we want to add x to some representation of ex that would give us this. The representation we would need is 3-dimensional. Such 3d-representations follow pretty straightforward from what we know from lace-cities and lace-towers. In 3D, we have lace-cities with single nodes. This is a 2D-representation with 1D-subsymmetry, and 2+1=3D. We also have lace-towers: 1D-representations with 2D-subsymmetry (again, 1+2=3D). (we could also take 0D-representations with 3D-subsymmetry (the original dynkin-diagram), and 3D-representations with 0D-subsymmetry (all it's vertices placed in space), but these are limiting representations that don't add anything). in 4D we have 2D-representations with 2D-subsymmetry, and 1D-representations with 3D-subsymmetry. 3D-representations with 1D-subsymmetry would have nodes (x,f,F etc.) placed in a 3D space. The case we want has: 1 node at the center with value 2f, 12 nodes placed at the vertices of an ike with value F, 20 nodes placed at the vertices of a doe with value f, another 12 nodes on the vertices of a (bigger) ike with value x, and finally 30 vertices with value o. First we change all x's in (-x)'s, then we add an x to everything, and we get the thing I described. (if we don't change x's in (-x)'s, we get an elongated ex, oxofoofoxo3ooooxxoooo5ooxooooxoo&#xt).
o3o
x3o o3f f3o o3x
o3o o3f f3x x3f f3o o3o
f3o o3F F3o o3f
o3x x3f F3o f3f o3F f3x x3o
f3o o3F F3o o3f
o3o o3f f3x x3f f3o o3o
x3o o3f f3o o3x
o3o
o3o
x3o o3f o3x
o3o o3f f3x x3f f3o
F3o
o3x x3f F3o f3f o3F
f3o o3F
o3o
x3o o3f o3x
o3o o3f f3x x3f f3o o3x
F3o o3o
x3f
o3x x3f F3o f3f o3F F3o o3f
f3o o3F f3f f3x x3o
o3x x3f F3o f3f o3F F3o o3f
x3f
F3o o3o
o3o o3f f3x x3f f3o o3x
x3o o3f o3x
o3o
Klitzing wrote:... Thus your inverse stacking o3x5o||f3o5o||o3x5o thus looks like having lots of broken lacing faces. - Or are you able to rearrange all these cell-bits somehow?
Indeed. However, because It's made by expanding the 600-cell, it's much more interesting than just a dodecaaugmented iddip (although that's a way you could have discovered it, but then you don't account for it's awesome hidden structure).Klitzing wrote:Ehh, not too difficult, after all: isn't this just iddip (oo3xx5oo&#x) augmented by 12 pippies (ox2ox5oo&#x)?
If I get this correctly, then we havethus all combined angles ought be convex - at least within this bistratic segment.
- angle at 4-gon between pip and trip of iddip = 142.622632 degrees
- angle at 4-gon between pip and squippy of pippy = 13.282526 degrees
- angle at 5-gon between pip and id of iddip = 90 degrees
- angle at 5-gon between pip and peppy of pippy = 18 degrees
(2f) (o5o3o) (2f) (o5o3o)
(F) (o5o3x) (F) (o5o3x)
(f) (x5o3o) (f) (x5o3o) (f)..(x..o)
(x) (o5o3f) (x) (o5o3f) (x)..(o..f) <=ike
(o) (o5x3o) "=" (o)(f5(-x)3x) (o)..(f..x)
expansion on (.)(x5.3.)-node
(2f) (o5o3o) (2f) (x5o3o)
(F) (o5o3x) (F) (x5o3x)
(f) (x5o3o) (f) (o5f3o)
(x) (o5o3f) (x) (x5o3f)
(o)(f5(-x)3x) (o)(F5(-x)3x) "=" (o)(x5x3o)
i.e. castellated x5o3x-prism
expansion on (.)(.5.3x)-node
(2f) (o5o3o) (2f) (o5o3x)
(F) (o5o3x) (F) (o5x3o)
(f) (x5o3o) (f) (x5o3x)
(x) (o5o3f) (x) (o5o3F)
(o)(f5(-x)3x) (o)(f5o3o)
i.e. "bilbiro'd o5o3x3o"
expansion on (x)(.5.3.)-node
(2f) (o5o3o) (F+f) (o5o3o)
(F) (o5o3x) (F+x) ox5o3x)
(f) (x5o3o) (F) (x5of3o)
(x) (o5o3f) (o) (o5o3f)
(o)(f5(-x)3x) (x)(f5(-x)3x)
i.e. my "new" discovery
student91 wrote:Indeed. However, because It's made by expanding the 600-cell, it's much more interesting than just a dodecaaugmented iddip (although that's a way you could have discovered it, but then you don't account for it's awesome hidden structure).Klitzing wrote:Ehh, not too difficult, after all: isn't this just iddip (oo3xx5oo&#x) augmented by 12 pippies (ox2ox5oo&#x)?
If I get this correctly, then we havethus all combined angles ought be convex - at least within this bistratic segment.
- angle at 4-gon between pip and trip of iddip = 142.622632 degrees
- angle at 4-gon between pip and squippy of pippy = 13.282526 degrees
- angle at 5-gon between pip and id of iddip = 90 degrees
- angle at 5-gon between pip and peppy of pippy = 18 degrees
My version of it would also have the rest of the 600-cell glued on top of it (that way it is a partial expansion, and thus it fits in our paper). Look what happens when we glue the rest on top of it: o5o3o||o5o3x||x5o3o||o5x3o ++ o5x3o||o5o3f||o5x3o ++ etc.
The dichoral angle of x5o3o||o5x3o at o5x of id is 72. This means the peppy of o5x3o||o5o3f||... will merge with the gyroelongated peppy of o5o3x||x5o3o||o5o3x to make an ike. Thus this expansion also has ikes.
What's even more interesting, though hard to demonstrate, is that it hides bilbiro's in it's structure. These can be shown with my new notation:
(F)(x5o3o)+(x)(o5x3o)+(o)(o5o3f) "=" (F)(x5o3o)+(x)(f5(-x)3x)+(o)(o5o3f).
look at it this way now:
(F)(x...o)+(x)(f...x)+(o)(o...f)
That is a bilbiro (just think there to be 2's everywhere, then you have F2x2o+x2f2x+o2o2f = Fxo2xfo2oxf&#zx = bilbiro.)
The fact that we had to change id (o5x3o)=>(f5(-x)3x) means the bilbiro is internal, so when we delete vertices of the id's, we will be able to see the bilbiro's.
Now why this CRF is quite interesting, is because it is the last one in a small family.
Look at ex in my new notation:When you change o5x3o in f5(-x)3x, you can see the equatorial ikes in .2.2.-orientation, as highlighted above. As we know, fox2xfo2oxf&#zx can be expanded in 3 ways to get bilbiro's. (Fxo2xfo2oxf&#zx, fox2oFx2oxf&#zx and fox2xfo2xoF&#zx, the only difference in 3D is their orientation.)
- Code: Select all
(2f) (o5o3o) (2f) (o5o3o)
(F) (o5o3x) (F) (o5o3x)
(f) (x5o3o) (f) (x5o3o) (f)..(x..o)
(x) (o5o3f) (x) (o5o3f) (x)..(o..f) <=ike
(o) (o5x3o) "=" (o)(f5(-x)3x) (o)..(f..x)
In 4D, all these expansions can be done inside of ex:
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expansion on (.)(x5.3.)-node
(2f) (o5o3o) (2f) (x5o3o)
(F) (o5o3x) (F) (x5o3x)
(f) (x5o3o) (f) (o5f3o)
(x) (o5o3f) (x) (x5o3f)
(o)(f5(-x)3x) (o)(F5(-x)3x) "=" (o)(x5x3o)
i.e. castellated x5o3x-prism
- Code: Select all
expansion on (.)(.5.3x)-node
(2f) (o5o3o) (2f) (o5o3x)
(F) (o5o3x) (F) (o5x3o)
(f) (x5o3o) (f) (x5o3x)
(x) (o5o3f) (x) (o5o3F)
(o)(f5(-x)3x) (o)(f5o3o)
i.e. "bilbiro'd o5o3x3o"
- Code: Select all
expansion on (x)(.5.3.)-node
(2f) (o5o3o) (F+f) (o5o3o)
(F) (o5o3x) (F+x) ox5o3x)
(f) (x5o3o) (F) (x5of3o)
(x) (o5o3f) (o) (o5o3f)
(o)(f5(-x)3x) (x)(f5(-x)3x)
i.e. my "new" discovery
Similar families can be made with other symmetries, but unfortunately, uniforms other than the 600-cell only exceptionally produce CRF's this way.
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