wendy wrote:Here is what i know of ursulatopes.
Ursulatopes are a kind of partially truncated fostrums, literally the prototype is xu&#ft. If the truncation happens at 1/F of the height from the large base, and the large base is rectified, the style of the figure is xfo&xt, a pentagon. The base figure is rectified.
The ursulation of a polygon gives xfoPoop&#t, where p is the shortchord of P (eg x=3, q=4, f=5, h=6, u=infinity. The largest value of P which leads to mostly unit-edge ursulate is P=10. A polytope with this size is x3o3o5o, so xfo3oox3ooo5ooo&#xt is in fact flat.
student91 wrote:[...]
so in summary all bilbiro-ings are weird stott-expansions, although "we" (I) don't know why. Also orthocupolarotunda-ing is the same procedure as bilbiro-ing. More complex weird stott-expansions such as D4.11 and thawro-ing are still unexplained as well. Furthermore I think this kind of expansion shouldn't be called "partial", because it's a full, combined expansion of both ike=>id and ike=>x5o3x. I guess I'll stick to "weird" expansions as I don't know any better name. lastly, I don't know what ike=>thawro should be seen as, all analogy's seem to fail. please help me out, I think we've got something important here. If you don't understand something of this post, please tell me, I will explain it as best as I can, in order to work this out
student91 wrote:[...]
so in summary all bilbiro-ings are weird stott-expansions, although "we" (I) don't know why. Also orthocupolarotunda-ing is the same procedure as bilbiro-ing. More complex weird stott-expansions such as D4.11 and thawro-ing are still unexplained as well. Furthermore I think this kind of expansion shouldn't be called "partial", because it's a full, combined expansion of both ike=>id and ike=>x5o3x. I guess I'll stick to "weird" expansions as I don't know any better name.
lastly, I don't know what ike=>thawro should be seen as, all analogy's seem to fail. please help me out, I think we've got something important here. If you don't understand something of this post, please tell me, I will explain it as best as I can, in order to work this out
x3o
u3o x3f F3o x3x
x3o x3f F3x u3f F3o x3o
F3o x3F X3o x3f
x3x u3f X3o F3f x3F F3x u3o
F3o x3F X3o x3f
x3o x3f F3x u3f F3o x3o
u3o x3f F3o x3x
x3o
Klitzing wrote:Stott expansion in that sense should be possible. But then you should do that consistently, i.e. in the same sense to all left sides of the perp space symbols:
- Code: Select all
x3o
u3o x3f F3o x3x
x3o x3f F3x u3f F3o x3o
F3o x3F X3o x3f
x3x u3f X3o F3f x3F F3x u3o
F3o x3F X3o x3f
x3o x3f F3x u3f F3o x3o
u3o x3f F3o x3x
x3o
x3o
o3x x3f F3o x3x
x3o x3f F3x u3f F3o x3o
F3o x3F X3o x3f
x3x u3f X3o F3f x3F F3x o3x
F3o x3F X3o x3f
x3o x3f F3x u3f F3o x3o
o3x x3f F3o x3x
x3o
In fact it happens to be nothing but sectioning ex at its equatorial hyperplane, and inserting prisms at this cut. - Whether in that orientation again some tetrahedra get dissected - as in the pentagonal orientation - I can not see at the moment. (And esp. that those then could be replaced by something CRF...)
--- rk
student5 wrote:... orthocup ...student91 wrote:... orthocupolarotunda ...
Klitzing wrote:Just want to throw in the OBSA of that thingy for a future neater reference: it's called pocuro (for Pentagonal OrthoCUpolaROtunda).
--- rk
student5 wrote:Klitzing wrote:Stott expansion in that sense should be possible. But then you should do that consistently, i.e. in the same sense to all left sides of the perp space symbols:
- Code: Select all
x3o
u3o x3f F3o x3x
x3o x3f F3x u3f F3o x3o
F3o x3F X3o x3f
x3x u3f X3o F3f x3F F3x u3o
F3o x3F X3o x3f
x3o x3f F3x u3f F3o x3o
u3o x3f F3o x3x
x3o
but would it be possible to "invert" x3o to (-x)3x before expanding? you'd then get o3x when expanded and the thawros, thus the following city.
- Code: Select all
x3o
o3x x3f F3o x3x
x3o x3f F3x u3f F3o x3o
F3o x3F X3o x3f
x3x u3f X3o F3f x3F F3x o3x
F3o x3F X3o x3f
x3o x3f F3x u3f F3o x3o
o3x x3f F3o x3x
x3o
In fact it happens to be nothing but sectioning ex at its equatorial hyperplane, and inserting prisms at this cut. - Whether in that orientation again some tetrahedra get dissected - as in the pentagonal orientation - I can not see at the moment. (And esp. that those then could be replaced by something CRF...)
--- rk
in this picture, I guess it is visible that quite some tetrahedra are dissected, if I understand it correctly, all visible tetrahedra do not lie flat, the have a point sticking out to either side, but they'd become, depending on dichoral angle, an elongated pentagonal bipyramid, or 2 pentagonal pyramids with a prism in between
student91 wrote:... I found another way to partially stott-expand the ike, this time into a orthocupolarotunda!! ugly drawing:
the pentagons are black, red, yellow, blue and green. those should be moved according to the pink arrows, every pentagon has five arows at its vertices that are somewhat parralel. the central vertex should be pulled apart into a pentagon, the 1-further-vertices should be placed atop the ones next to them, and the next vertices should be pulled apart int an edge. the last vertex (not drawn) should be pulled into a pentagon as well,completing the cupola.
maybe the occurence of the orthocupolarotunda's in the bilbiro'd o5x3o3x isn't that random.
I think this is the last such partial stott-contraction of ike
quickfur wrote:Skimmed over your post, will come back and read it in more detail later, but wanted to just comment on your last paragraph:student91 wrote:[...]
so in summary all bilbiro-ings are weird stott-expansions, although "we" (I) don't know why. Also orthocupolarotunda-ing is the same procedure as bilbiro-ing. More complex weird stott-expansions such as D4.11 and thawro-ing are still unexplained as well. Furthermore I think this kind of expansion shouldn't be called "partial", because it's a full, combined expansion of both ike=>id and ike=>x5o3x. I guess I'll stick to "weird" expansions as I don't know any better name.
In my mind, it's a "partial" Stott expansion because we're expanding along some symmetry axes but not others (since a full Stott expansion will correspond to adding an x to an o-node in the CD diagram, and that just produces a uniform polychoron). Obviously, there are several layers of expansions going on with these bilbiro'd and thawro'd polychora, since we have some 3D elements that are being (partially or fully) expanded, but only to a selected subset of them, rather than everywhere (which yields a uniform polychoron).lastly, I don't know what ike=>thawro should be seen as, all analogy's seem to fail. please help me out, I think we've got something important here. If you don't understand something of this post, please tell me, I will explain it as best as I can, in order to work this out
This is just my gut feeling, since I haven't studied this rigorously yet, but it seems that what we're seeing here is a kind of modified, or "complex" Stott expansion / contraction (depending on which way you look at it), where some elements are identified with a non-convex element with the same vertices, and the Stott expansion is applied to (parts of) the non-convex structure rather than the usual convex structure. Technically, there is no Stott expansion (in the traditional sense) that can transform an ike into a bilbiro; the way we did it was by substituting parts of ike with parts of the great dodecahedron and applying Stott expansion to that instead, while simultaneously keeping other parts of ike as-is. So in a sense, it's really a pseudo-ike, not ike itself, that's transforming into a bilbiro; it's a kind of "frankenstein monster" double of ike, which is not really ike but a hybrid of ike and great dodecahedron. It just so happens that great dodecahedron has the same vertices as ike, so this hybridization "works", and seemingly it's ike itself that's transforming into bilbiro. But actually, it's the frankensteinian pseudo-ike that is undergoing the transformation.
About this D4.11, I was thinking, could it be derived from the 600-cell instead of the snub 24-cell? This would mean, that if you insert teddi-pyramids in the teddi's, bilbiro-pseudopyramids in the bilbiro's, and maybe a tetrahedron in the cuboctahedra, does it become a convex shape? this shape would certainly be non-convex if you do it the described way, but my hope is that when you do it this way, the peppi's of the bilbiro-pseudopyramid and the teddi-pyramid would produce a connecting edge, just as what happens when you place two hemi-600-cells together.Similarly, things like D4.11 are produced technically not from the snub 24-cell itself, but from a "frankenstein double" of snub 24-cell where the icosahedra are hybrids of ike and great dodecahedron. The coincidence of vertices means that these "pseudo-ike"'s can be interpreted either way.
I hope we don't have to go consider all the non-convex uniforms . Instead, I hope we only have to consider those that have a negative-node representation. e.g. the great dodecahedron can be written as o5x3(-x) "=" o5o3x, and the stellated dodecahedron as f5(-x)3o "=" o5o3x. I'm not able to do such a thing for the great stellated dodecahedron and the great icosahedron, probably because they are based on cuts that are deeper than one edge-length of the convex hull. Anyway, I thought as well this weird stott-expansion/contraction was the grid you were looking for, and that's why I said this might be something importantNow the interesting part about some of the bilbiro'd and thawro'd polychora that we found, is that most of our constructions were by deleting some vertices or substituting some other vertices. So it's more like a partial (pseudo) Stott contraction than a Stott expansion. Also, this happens to a hybridized "frankensteinian" version of the polychoron instead of the "pure" convex version. For example, we produced thawro's in o5x3o3o by shrinking some vertex layer (sorry I can't remember the exact symbol off-hand, it was something like x5f3f) to x5x3x. This could be understood as the substitution of o5x3o3o with parts of some non-convex uniform with the same vertices, where the Stott contraction of the non-convex unform produces a x5x3x cross-section. Since the vertices of the non-convex and o5x3o3o are the same, the vertices can be interpreted either way, which gives us the flexibility to apply the Stott expansion/contraction to them, partly as if they are part of o5x3o3o and partly as if they are part of the non-convex polychoron.
This somewhat ties in with my previous observation about something fishy going on with 4D and the golden ratio, in the sense that the numbers somehow keep working out "coincidentally", sometimes in unexpected ways. This phenomenon can be seen to be less strange if we understand that given, say, some 600-cell family convex uniform, there are entire regiments (to use Bowers' term) of non-convex uniforms with the same vertices, each of which is related to other non-convex uniforms of various relative sizes with different vertices via Stott expansion/contraction/faceting/stellating. I suspect that many of the CRFs we have been finding are actually CRF pieces of some of the non-convex uniforms, or some combination of multiple non-convex uniforms. The fact that there are so many non-convex uniforms, and they are related to each other in many ways via various Stott contractions/expansions and stellations/facetings, means that there are many combinations of vertices that will have unit edge lengths (e.g., the relation between a uniform and its Stott-expanded version is a unit edge length difference), so they form some kind of pentagonal grid in 4D space that we can cut many CRFs out of. The coincidence of vertices of many non-convex uniforms with convex uniforms means that many of these pieces may have an underlying non-convex derivation, but they can be reinterpreted as convex, and therefore many of them become CRF.
Klitzing wrote:Both for the bilbiro-2-ike(faceting) and the thawro-2-ike(faceting) transitions I've designed already some pics. They will be part of my next incmats webpage update. But I want to show them here already before.
bilbiro-2-ike(faceting) is clearly a true Stott expansion (not in the sense of Dynkin symbols but in the sense of Stott, incorporating the full symmetry of the object). It was only the false additional application of the convex hull, which resulted in the full symmetrical ike. And with respect to that one then it would be just a subsymmetry. But not so for the to be used here ike faceting.
thawro-2-ike(faceting) in the same sense is a true Stott expansion. Again it is just a matter of selecting the correct ike faceting, to which it shall apply.
--- rk
quickfur wrote:So it's a kind of substitution + Stott-expansion operation, which differs from the usual expansion processes in the initial substitution step, even though the expansion itself isn't fundamentally different.
Klitzing wrote:[...]quickfur wrote:So it's a kind of substitution + Stott-expansion operation, which differs from the usual expansion processes in the initial substitution step, even though the expansion itself isn't fundamentally different.
Yep. And it often is that this substitution process itself is what breaks symmetry. - Partial Stott expansion / contraction then is different as well: that one would transform only along a subsymmetry of the to be transformed figure. And the symmetry of that substituted figure clearly differs in general from the symmetry of the non-substituted figure! So even if the total aggregated transformation seems to act with respect to a subsymmetry only, it still might contain normal (non-partial) Stott transformations only.
[...]
quickfur wrote:Klitzing wrote:[...]quickfur wrote:So it's a kind of substitution + Stott-expansion operation, which differs from the usual expansion processes in the initial substitution step, even though the expansion itself isn't fundamentally different.
Yep. And it often is that this substitution process itself is what breaks symmetry. - Partial Stott expansion / contraction then is different as well: that one would transform only along a subsymmetry of the to be transformed figure. And the symmetry of that substituted figure clearly differs in general from the symmetry of the non-substituted figure! So even if the total aggregated transformation seems to act with respect to a subsymmetry only, it still might contain normal (non-partial) Stott transformations only.
[...]
This makes me wonder... are there CRFs which can be derived by full-symmetric Stott expansion? Obviously, the uniform polychora themselves are examples of this, but I'm thinking of something that's no longer uniform, e.g., is it possible to Stott-expand the bitruncated 24-cell o3x4x3o to get a CRF with augmented 24-cell symmetry? By this I mean, push out the truncated cubes radially and fill in the gaps with other CRF polyhedra. The result will not be uniform (since otherwise we'd have known about it before ), but can it still be CRF (or made CRF with some suitable modifications)?
wendy wrote:The {3,3,5} ursulate xfo3oox5ooo&#xt, as well as xfo3oox5xxx&#xt are like the decagon surrounded by petagons, indde flat.
o5o3o3o5/2o
o5/2o5o3o5/2o
o5o5/2o5o5/2o
o5o3o5/2o5o
o3o5o5/2o3o
o3o3o3o3o5/2*c
o3o3o3o3/2o5*c
...
pt || gaghi = ox5oo5/2oo3oo&#x
ragaghi || tigaghi = ox5xx5/2oo3oo&#x
rigfix || sirgaghi = ox5oo5/2xx3oo&#x
gofix || quipdohi = ox5oo5/2oo3xx&#x
pt || sishi = ox5/2oo5oo3oo&#x
rofix || sirsashi = ox5/2oo5xx3oo&#x
fix || padohi = ox5/2oo5oo3xx&#x
pt || sidtixhi = ox3oo3oo3oo5/2*b&#x
pt || gidtixhi = ox3oo3oo3/2oo5*b&#x
...
wendy wrote:There are pentagonal tilings in 2, 3, and 4 dimensions, these are infinitely dense, but just involve numbers of the form a+b\phi.
The 3d tiling centres around the group o5o5/2oAo. In 4D, there are many dynkin symbols for it, but o5/2o3o3z3o3o and o5o3o3o5/2o are examplers of it.
Of the 15 polytopes of the {5,3,3} type, all but the smallest and largest can be laced together in a flat tower. The ursulated {3,3,5} connects the x3o3o5o and o3x3o5o.
quickfur wrote:...
Also, do you have a page listing the various node symbols (x, f, F, q, z, etc.) and their corresponding values? I know a few, but the rest elude me. I've also been thinking if it may not be amiss to use numerical notation for them instead, since they are after all an oblique coordinate system!
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