student91 wrote:After reading everything carefully, and looking up what a voroni-cell is, I think I understood your post.
Am I right when I say voroni-cells look a bit like the dual cells of s3s4o3o, connected to the center of s3s4o3o to make them 4D-bodies?
I think this polytope is the coolest of the 24-diminished .5.3.3.'s. I mean, it has J83's!!
quickfur wrote:I may have discovered a crown jewel candidate(!!!). Well, I'm not 100% sure it exists or is CRF just yet, but I'm posting here for you bright minds out there to verify whether it's possible.
Marek's post about the bilunabirotunda (J91) inspired me to investigate possible CRF combinations of this shape.
[...]
... the result should be CRF, and would be a tetrastratic polychoron with two x5o3x's, 30 J91's, 12 pentagonal pyramids, and 20 tetrahedra.
Can someone verify whether this shape actually exists, and is CRF?
P.S. The lace tower of this polychoron should be x5o3x || o5o3A || B5o3o || o5o3A || x5o3x where A and B are as-yet unknown scalars. The height of the tower should be exactly the Golden Ratio (chord of the pentagonal faces of the J91's).
o....3o....5o.... & | 120 * * | 2 2 1 1 0 | 1 2 1 2 2 2 | 1 1 1 2
.o...3.o...5.o... & | * 24 * | 0 0 5 0 1 | 0 0 0 5 0 5 | 0 1 0 5 verf = ox5oo&#f
..o..3..o..5..o.. | * * 20 | 0 0 0 6 0 | 0 0 0 0 6 3 | 0 0 2 3 verf = f x3o
------------------------+-----------+--------------------+---------------------+-----------
x.... ..... ..... & | 2 0 0 | 120 * * * * | 1 1 0 0 1 0 | 1 0 1 1
..... ..... x.... & | 2 0 0 | * 120 * * * | 0 1 1 1 0 0 | 1 1 0 1
oo...3oo...5oo...&#x & | 1 1 0 | * * 120 * * | 0 0 0 2 0 1 | 0 1 0 2
o.o..3o.o..5o.o..&#x & | 1 0 1 | * * * 120 * | 0 0 0 0 2 1 | 0 0 1 2
.o.o.3.o.o.5.o.o.&#x | 0 2 0 | * * * * 12 | 0 0 0 0 0 5 | 0 0 0 5
------------------------+-----------+--------------------+---------------------+-----------
x....3o.... ..... & | 3 0 0 | 3 0 0 0 0 | 40 * * * * * | 1 0 1 0
x.... ..... x.... & | 4 0 0 | 2 2 0 0 0 | * 60 * * * * | 1 0 0 1
..... o....5x.... & | 5 0 0 | 0 5 0 0 0 | * * 24 * * * | 1 1 0 0
..... ..... xo...&#x & | 2 1 0 | 0 1 2 0 0 | * * * 120 * * | 0 1 0 1
x.o.. ..... .....&#x & | 2 0 1 | 1 0 0 2 0 | * * * * 120 * | 0 0 1 1
ooooo3ooooo5ooooo&#xt | 2 2 1 | 0 0 2 2 1 | * * * * * 60 | 0 0 0 2
------------------------+-----------+--------------------+---------------------+-----------
x....3o....5x.... & | 60 0 0 | 60 60 0 0 0 | 20 30 12 0 0 0 | 2 * * * srid
..... oo...5xo...&#x & | 5 1 0 | 0 5 5 0 0 | 0 0 1 5 0 0 | * 24 * * peppy
x.o..3o.o.. .....&#x & | 3 0 1 | 3 0 0 3 0 | 1 0 0 0 3 0 | * * 40 * tet
xFoFx ..... xofox&#xt | 8 4 2 | 4 4 8 8 2 | 0 2 0 4 4 4 | * * * 30 bilbiro
Klitzing wrote:quickfur wrote:I may have discovered a crown jewel candidate(!!!). [...]
P.S. The lace tower of this polychoron should be x5o3x || o5o3A || B5o3o || o5o3A || x5o3x where A and B are as-yet unknown scalars. The height of the tower should be exactly the Golden Ratio (chord of the pentagonal faces of the J91's).
Nice find, quickfur! Yes, indeed it is possible.
[...]
Accordingly it could be rewritten as xFoFx3ooooo5xofox&#xt = srid || pseudo F-ike || pseudo f-doe || pseudo F-ike || srid.
Coordinates then are:
[...]
Note that here the dihedral angle between the srids and the bilbiros (i.e. at the squares) would be 90 degrees.
Further it should be noted (what also becomes evident from the matrix) that there are no edges connecting the tropal vertices to the equatorial ones!
Btw., as you have 2 srids, the count of the therefrom emanating tets (trigonal pyramids) and peppies (pentagonal ones) gets doubled too (pointing inward from either side). - Those counts you got wrong in your mail. Cf. the above cited bit. - Only the count of the bilbiros remains undoubled, as those connect both opposite basal srids.
Thus, corrected, the cell count here is (as can be read off directly from the Matrix too):
2x srid + 24x peppy + 40x tet + 30x bilbiro.
So, having provided to you even the coordinates, we soon may get some nice pictures of that fellow?
[...]
Marek14 wrote:Hm, maybe "lunatic srid prism"?
Marek14 wrote:Hm, maybe "lunatic srid prism"?
quickfur wrote:Marek14 wrote:Hm, maybe "lunatic srid prism"?
Oh, btw, I keep forgetting you want stuff in Stella4D format, so here's the file for this lunatic prism.
student91 wrote:I think so too. J92 has C3v-symmetry, so it should be viewed as xfox3oxFx&#xt. (J91 has D2h-symmetry, so that one can be seen as either xfofx(2)oxfxo&#xt, ofxfo(2)oxFxo&#xt or xFoFx(2)xofox&#xt. the last one does work, the other ones haven't been tested) If this one is extended to xfox3oxFx3oooo&#xt, we have oooo(2)xfox&#x, not giving a facet, xfox3ox&#xt, giving J92, and oxFx3oooo&#xt, giving a non-CRF.
Marek14 wrote:quickfur wrote:Marek14 wrote:Hm, maybe "lunatic srid prism"?
Oh, btw, I keep forgetting you want stuff in Stella4D format, so here's the file for this lunatic prism.
Thanks!
I'm trying to investigate the cuts, but it doesn't seem to have any obvious diminishings.
I wonder if the pentagonal pyramids could be augmented -- the height would be relatively small...
I tried to make a VRML file, not sure if it worked...
student91 wrote:That thing is just super awesomely cool.
I mean, ursachora are pretty cool, and this one is like 20 times cooler.
Speaking of which, it has some similarity with ursachora. Ursachora are based on the xfo3oox-buildup of a trid.ike, and this one is based on the xFoFx(2)xofox-buildup of the bilunabirotunda. Therefore, as the ursachora are possible with other symmetries as well, this one might be too, so e.g. xFoFx3ooooo3xofox&#xt might be possible as well, and maybe expanded ones, so xFoFx3xxxxx5xofox&#xt too. I'm not sure about this, and just extrapolated the ursachora to this one. Of course I couldn't've done this extrapolation without your awesome discovery. It's real awesome
[...]
Marek14 wrote:Hm, J92 also has a part with o5x3o shape, but the rest of it would probably pose problems...
Marek14 wrote:For the third possibility, could two be glued together to restore CRF properties?
quickfur wrote:OTOH, J92 does look like you should be able to stick four of them around a tetrahedron and get something analogous to one of the 120-cell uniforms out of it. Not sure how the rest of the shape will close up, but that's a start
quickfur wrote:Now, the all-important question is, what should we call this polychoron? The only name I could think of is triaconta-bilunabirotunda, but that just doesn't seem to sit very well.
quickfur wrote:[...]
I'm not so sure about the possibility with other symmetries, because a degree-3 edge is rigid, and we have 3 J91's around an edge. So it may not close up if we have only 4 J91's, I think. (In fact, it was this rigidity that initially led me to consider icosahedral symmetry, because I was originally thinking of tetrahedra or square pyramids for those J91 triangles around the top/bottom edges, but I wasn't sure if the angles will match up properly. At least with icosahedral symmetry there's precedent for a 5-fold symmetry around that edge, which allows CRF pentagonal pyramids to fill the gap.)
Keiji wrote:[...]
(If the stauro- form exists, that would be a casper, and if the pyro- form exists, that would be a capper. But I have a feeling those wouldn't exist for the same reason that you don't get trigonal or square rotundae in 3D.)
Keiji wrote:quickfur wrote:Now, the all-important question is, what should we call this polychoron? The only name I could think of is triaconta-bilunabirotunda, but that just doesn't seem to sit very well.
Now this is a task for Keiji
It's very similar to a birotunda, having 5 layers of vertices just like our 3D birotundae.
However, it doesn't have a contour for the center layer, rather it has a kind of castellated edge like the dodecahedron does.
The ends are rhodoperihedra.
This leads me to call it the castellated rhodoperihedral birotunda - or for short, a carper!
(If the stauro- form exists, that would be a casper, and if the pyro- form exists, that would be a capper. But I have a feeling those wouldn't exist for the same reason that you don't get trigonal or square rotundae in 3D.)
quickfur wrote:the J91's are at 90° to the x5o3x's, btw!
quickfur wrote:But yeah, I think the trigonal/square versions are not CRF, because the 3 J91's around the vertex are rigid, which in turn fixes the dihedral angle of the triangular faces of the pentagonal pyramids, so they have to be pentagonal pyramids and not tetrahedra or square pyramids, otherwise things won't close up.
quickfur wrote:But yeah, I think the trigonal/square versions are not CRF, because the 3 J91's around the vertex are rigid, which in turn fixes the dihedral angle of the triangular faces of the pentagonal pyramids, so they have to be pentagonal pyramids and not tetrahedra or square pyramids, otherwise things won't close up.
student91 wrote:quickfur wrote:But yeah, I think the trigonal/square versions are not CRF, because the 3 J91's around the vertex are rigid, which in turn fixes the dihedral angle of the triangular faces of the pentagonal pyramids, so they have to be pentagonal pyramids and not tetrahedra or square pyramids, otherwise things won't close up.
Note that the ursachora have similar edges (3 tridiminished icosahedra placed face-to-face, and a pyramid filling the gap). This one also works out.
Marek14 wrote:Well, you can't place J92's around octahedron or icosahedron since that would be a "shard" of o5x3o4o or o5x3o5o, which are hyperbolic.
Surrounding a tetrahedron is a shard of o5x3o3o. The only other uniform polychoron that uses icosidodecahedra is o5x3o3x.
This would lead into cuboctahedron surrounded by 4 J92's, 4 other cuboctahedra (or possibly triangular cupolas/gyrobicupolas, if those would fit better) and 6 pentagonal prisms. This is a second form that would have to be checked if it can be completed.
quickfur wrote:Marek14 wrote:Well, you can't place J92's around octahedron or icosahedron since that would be a "shard" of o5x3o4o or o5x3o5o, which are hyperbolic.
Quite so.Surrounding a tetrahedron is a shard of o5x3o3o. The only other uniform polychoron that uses icosidodecahedra is o5x3o3x.
This would lead into cuboctahedron surrounded by 4 J92's, 4 other cuboctahedra (or possibly triangular cupolas/gyrobicupolas, if those would fit better) and 6 pentagonal prisms. This is a second form that would have to be checked if it can be completed.
What's wrong with putting 4 J92's around a tetrahedron? It would appear that we should be able to at least insert 4 more tetrahedra into the result (i.e. following the pattern of o5x3o3o), which then leaves only the outer surface of hexagons, squares, and triangles, to be closed up. Seems like it should be possible? Or am I missing something obvious?
quickfur wrote:student91 wrote:quickfur wrote:But yeah, I think the trigonal/square versions are not CRF, because the 3 J91's around the vertex are rigid, which in turn fixes the dihedral angle of the triangular faces of the pentagonal pyramids, so they have to be pentagonal pyramids and not tetrahedra or square pyramids, otherwise things won't close up.
Note that the ursachora have similar edges (3 tridiminished icosahedra placed face-to-face, and a pyramid filling the gap). This one also works out.
The difference there is that the number of tridiminished icosahedra around an edge is flexible. Here, the number of J91's around the 5.3.5.3 vertex can only be 3, because otherwise it becomes hyperbolic (the angle of the two pentagons is too wide to fit more than 3 J91's around that vertex). So the dichoral angles between the J91's is fixed, which means that if you continue the pattern of 3 J91's around the 5.3.5.3 vertices, it will eventually form an icosahedral framework of J91's, which means the top/bottom edges will have 5 J91's surrounding it.
Now, it may be possible to have other numbers of J91's around that edge, but it means that you cannot also have the repeating pattern of 3 J91's around the 5.3.5.3 vertices anymore, but you'll need some other arrangement of cells to close the shape up.
student91 wrote:quickfur wrote:student91 wrote:quickfur wrote:But yeah, I think the trigonal/square versions are not CRF, because the 3 J91's around the vertex are rigid, which in turn fixes the dihedral angle of the triangular faces of the pentagonal pyramids, so they have to be pentagonal pyramids and not tetrahedra or square pyramids, otherwise things won't close up.
Note that the ursachora have similar edges (3 tridiminished icosahedra placed face-to-face, and a pyramid filling the gap). This one also works out.
The difference there is that the number of tridiminished icosahedra around an edge is flexible. Here, the number of J91's around the 5.3.5.3 vertex can only be 3, because otherwise it becomes hyperbolic (the angle of the two pentagons is too wide to fit more than 3 J91's around that vertex). So the dichoral angles between the J91's is fixed, which means that if you continue the pattern of 3 J91's around the 5.3.5.3 vertices, it will eventually form an icosahedral framework of J91's, which means the top/bottom edges will have 5 J91's surrounding it.
Now, it may be possible to have other numbers of J91's around that edge, but it means that you cannot also have the repeating pattern of 3 J91's around the 5.3.5.3 vertices anymore, but you'll need some other arrangement of cells to close the shape up.
Ooh, you ment that edge!!, After a quick reflection I see what you mean, and I understand why it's impossible . I apologise for being so stubborn. the way I concluded It was impossible is by looking at the edge with the tetahedron and two bilunabirotunda's. This edge fixes the bichoral angle for the pentagon, and thus 5 is the only possibility .
Marek14 wrote:quickfur wrote:Surrounding a tetrahedron is a shard of o5x3o3o. The only other uniform polychoron that uses icosidodecahedra is o5x3o3x.
This would lead into cuboctahedron surrounded by 4 J92's, 4 other cuboctahedra (or possibly triangular cupolas/gyrobicupolas, if those would fit better) and 6 pentagonal prisms. This is a second form that would have to be checked if it can be completed.
What's wrong with putting 4 J92's around a tetrahedron? It would appear that we should be able to at least insert 4 more tetrahedra into the result (i.e. following the pattern of o5x3o3o), which then leaves only the outer surface of hexagons, squares, and triangles, to be closed up. Seems like it should be possible? Or am I missing something obvious?
Well, that was something which was already suggested. My suggestion was meant to be "in addition to", not "in place of" the previous one.
student91 wrote:Marek14 wrote:quickfur wrote:Surrounding a tetrahedron is a shard of o5x3o3o. The only other uniform polychoron that uses icosidodecahedra is o5x3o3x.
This would lead into cuboctahedron surrounded by 4 J92's, 4 other cuboctahedra (or possibly triangular cupolas/gyrobicupolas, if those would fit better) and 6 pentagonal prisms. This is a second form that would have to be checked if it can be completed.
What's wrong with putting 4 J92's around a tetrahedron? It would appear that we should be able to at least insert 4 more tetrahedra into the result (i.e. following the pattern of o5x3o3o), which then leaves only the outer surface of hexagons, squares, and triangles, to be closed up. Seems like it should be possible? Or am I missing something obvious?
Well, that was something which was already suggested. My suggestion was meant to be "in addition to", not "in place of" the previous one.
it seems that your polytope is an expansion of mine, so xfox3oxFx3oooo&#xt => xfox3oxFx3xxxx&#xt.
Users browsing this forum: No registered users and 0 guests