quickfur wrote:Klitzing wrote:I think you got the right feeling, quickfur, both with the relation of the 2 specific finds (according to my second idea) to sidpith and to stawros. But both weren't exact.
[...]
The important point here is for all that stuff to work: that the wedge angle of 2n-gon||n-p and that of n-gon||2n-p is the same!
[...]
Interesting! So I got 2n-gon||n-prism mixed up with n-gon||2n-prism. It's interesting that they have exactly the same wedge angle. It makes sense in retrospect, but it's not something I would've expected just from being told their structure.
quickfur wrote:More thoughts about CRFs with duoprismic symmetry: suppose you have a polychoron with a ring of n-prisms and an orthogonal ring of m-prisms, with some lacing edges between them. I'm thinking that if the polychoron is CRF, then it must correspond with some augmentation of some underlying m/d,n/e-duoprism...
Klitzing wrote:
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x4x x4x
x4x x4o x4o x4x
x4x x4o x4o x4x
x4x x4x
Klitzing wrote:quickfur wrote:More thoughts about CRFs with duoprismic symmetry: suppose you have a polychoron with a ring of n-prisms and an orthogonal ring of m-prisms, with some lacing edges between them. I'm thinking that if the polychoron is CRF, then it must correspond with some augmentation of some underlying m/d,n/e-duoprism...
Not in that generality, but in my today discussed 2 cases I already ponted that out:
You take some tower of prisms, glue some shapes onto its sides, bend that into 4D, and if you're lucky, the outer shape again displays only squares. Thus this torus of squares would correspond to the supporting duoprism (having been provided explicitely in the discussed cases). And those prisms of this very duoprism, which are parallel to the starting ones of that original tower, then will be the augmented ones.
But, btw., a thing looks surprising so:
When I made a lace city display of some polytope from 4D and above so far, then always the representations of the perp space components at some para space position would have themselves a smaller circumradius, when laying farer out. But here this is not the case, so. Instead rather to the contrary, cf. e.g.:Klitzing wrote:
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x4x x4x
x4x x4o x4o x4x
x4x x4o x4o x4x
x4x x4x
The perp space x4x clearly has a larger circumradius than the perp space x4o. But the para space position of those x4x here is farer out than that of the x4o. - Crude things happen!
--- rk
Marek14 wrote:Well, one possibility is to have the inner-most rings that are not on a single torus. A simple case would be a ring of cupolas joined by their n-gons and 2n-gons (two basic forms are available, with ortho- and gyro- joins between the cupolae). After all, isn't the first J92 polychoron we found, which contains 4 J92's in a ring, something similar? Then you also get an additional degree of freedom: by having two different "joins" in the ring, the dichoral angle can be different for each.
BTW - is there a name for the shape you cut off (in two copies) from a 600-cell to get a gap?
quickfur wrote:[...]
This suggests that perhaps non-trivial duoprismic CRFs are possible with heterogenous rings...
o5x
x5o x5o
f5o
o5f o5f
o5x o5x
f5o f5o
o5f o5f
x5o x5o
f5o f5o
o5f
o5x o5x
x5o
x5o x5o
o5f
o5x o5x
x5x x5x
f5o x5f f5o
o5x F5o F5o o5x
o5F o5F
f5x f5x
x5x f5f f5f x5x
x5o o5F V5o o5F x5o
F5o o5V o5V F5o
f5f f5f
o5f x5F o5f
x5f F5x F5x x5f
x5o V5o V5o x5o
x5F x5F
F5o F5o
x5x f5f f5f x5x
o5F o5F
F5x F5x
o5x o5V o5V o5x
f5x x5F x5F f5x
f5o F5x f5o
f5f f5f
o5F V5o V5o o5F
o5x F5o o5V F5o o5x
x5x f5f f5f x5x
x5f x5f
F5o F5o
x5o o5F o5F x5o
o5f f5x o5f
x5x x5x
x5o x5o
f5o
o5x o5x
quickfur wrote:student91 wrote:[...]
I think there are other ways to close it up, just found a new one (I hope): (whatever)|| o3F3o || x3x3f || F3o3x || x3f3x || o3x3x. let's hope it works
(for whatever you can either use o3x3x or f3x3o || x3o3o.)
That should be the third thing you're talking about (in fact there are two, because you can insert two "whatevers". the one with o3x3x should be the most interesting, as it has two sets of thawro's )
Unfortunately, this tower is non-CRF. The x3f3x layer overhangs the F3o3x, so there are non-CRF lacing edges from x3x3f to x3f3x (edge length 1/phi). (Unless I made a mistake again.... it's rather late and I'm not thinking very clearly... but I did check the difference in coordinates between F3o3x and x3f3x, they are all unit edges, but x3x3f to x3f3x has 1/phi edges. )
student91 wrote:quickfur wrote:student91 wrote:[...]
I think there are other ways to close it up, just found a new one (I hope): (whatever)|| o3F3o || x3x3f || F3o3x || x3f3x || o3x3x. let's hope it works
(for whatever you can either use o3x3x or f3x3o || x3o3o.)
That should be the third thing you're talking about (in fact there are two, because you can insert two "whatevers". the one with o3x3x should be the most interesting, as it has two sets of thawro's )
Unfortunately, this tower is non-CRF. The x3f3x layer overhangs the F3o3x, so there are non-CRF lacing edges from x3x3f to x3f3x (edge length 1/phi). (Unless I made a mistake again.... it's rather late and I'm not thinking very clearly... but I did check the difference in coordinates between F3o3x and x3f3x, they are all unit edges, but x3x3f to x3f3x has 1/phi edges. )
oops, that's due to an error of mine, the height between F3o3x and x3f3x should've been sqrt(1/8)/phi instead of sqrt(1/8)/(phi^2). This is fixed by using x3f3o instead of x3f3x. This gives (whatever) || o3F3o || x3x3f || F3o3x || x3f3o || o3x3o. This time it should work.
quickfur wrote: D4.8 (maybe this should be renamed to D4.8.1?)
Keiji wrote:quickfur wrote: D4.8 (maybe this should be renamed to D4.8.1?)
Done, and while we're renumbering discovery indices, can we shift the D4.9's up by one, so D4.9.0 becomes D4.9.1, D4.9.1 becomes D4.9.2 and D4.9.2 becomes D4.9.3? Not really a fan of having that .0 there. Hopefully it won't have too much impact given that they don't have wiki pages yet.
Marek14 wrote:I'd suggest modifying the definition of "augmented" a bit -- wiki says that augmentation is done with pyramids, but even in 3D you have Johnson solids which are augmented with cupolas (augmented truncated cube, for example).
EDIT: D4.6's page doesn't mention perhaps the most interesting cell it has -- 24 pentagonal orthocupolarotundas.
quickfur wrote:[...]EDIT: D4.6's page doesn't mention perhaps the most interesting cell it has -- 24 pentagonal orthocupolarotundas.
What, really? Where do they occur?
student91 wrote:quickfur wrote:[...]EDIT: D4.6's page doesn't mention perhaps the most interesting cell it has -- 24 pentagonal orthocupolarotundas.
What, really? Where do they occur?
After thinking a bit, I concluded marek was right, the pentagonla cupola's and the half id's are at an 180 degrees dichoral angle, making them fuse together, resulting in orthocupolarotunda's!! that's indeed really cool.
the pentagonal cupola's that are inserted with the pentagonal prisms, are part of an x3x3o5x. this means those cupola's are comparable with the hemiated id's, and thus the rotunda and the cupola are comparably aligned, making them fuse together .
wendy wrote:It kind of like looks like a goldfish bowl in this view
Klitzing wrote:wendy wrote:It kind of like looks like a goldfish bowl in this view
Not too surprising in view of its construction:
chopping off the poles of a spherical thing, and cutting out some equatorial stratum, with some further local structure change in that region...
Btw., quickfur, your new pics are fascinating as ever! Looks indeed like a gem, quite appealing, by highlighting all that special stuff. - Only that the normal parts inbetween, within my personal view, are too less visible. I'd like to have the edges at least to remain visible, in order to be at least somehow guided what shall be filled inbetween.
--- rk
quickfur wrote:The problem with showing all the edges, is that there are too many of them. ...
Originally, when I first started rendering 4D polytopes, my dream was to show everything in one image, but it become obvious pretty soon that it won't work because of the clutter. That's what led me to do layer-by-layer renderings on my website; it's the only way to show the entire structure in a comprehensible way.
Klitzing wrote:[...] But I got some idea for that point (and you would have to implement it, haha):
So far you display all edge as bold struts, and in that other pic you just omitted several edges - as those would become too dominant for being only guide lines for subordinate structures.
So what would you think about a 3-state logic for edges?--- rk
- shown as struts (as in both of your pic versions - as far as those were displayed at all)
- shown as mere thin lines (perhaps additionally with applicable coloring, so that you might can fade those out and in rather easily, just by adapting that color a bit)
- not shown (as in the former pic in most cases)
quickfur wrote:Keiji wrote:quickfur wrote: D4.8 (maybe this should be renamed to D4.8.1?)
Done, and while we're renumbering discovery indices, can we shift the D4.9's up by one, so D4.9.0 becomes D4.9.1, D4.9.1 becomes D4.9.2 and D4.9.2 becomes D4.9.3? Not really a fan of having that .0 there. Hopefully it won't have too much impact given that they don't have wiki pages yet.
It was numbered .0 because it does not actually have any bilbiro or thawro cells, so it doesn't fit very well with the current definition of BT polytopes. Arguably, the definition of BT polytopes is a bit arbitrary... we should be basing the classification on the operations used to derive these things, rather than the occurrence of bilbiro or thawro cells, which is only incidental, not fundamental. Case in point, D4.4 arguably doesn't really belong in this category, because it isn't obvious how it is derived from any existing uniform polychora by the "bilbiro-ing" and "thawro-ing" operations, so it seems to stand apart from the others it's currently lumped with.
I'm not against renumbering, though, but if we're going to do that, we'll need to fix the references to those numbers in previous posts in this thread, otherwise we will lose referential consistency for when we come back to read this thread in the future.
D4.4 clearly can't be obtained by simple cut-n-paste operations. For the same reason, I think the D4.3, and the D4.8's may be reconsidered as well.the wiki wrote:Crown jewels are a catch-all term for unusual CRF polytopes with unique structures that cannot be obtained from the uniform polytopes or other simpler CRFs by simple "cut-and-paste" operations.
quickfur wrote:I know this nomenclature is still a bit rough around the edges, so suggestions for improvements are welcome,
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