quickfur wrote:I'm wondering, is the vertex figure of the snub 24-cell (snub demitesseract ) a CRF tridiminished icosahedron? 'cos if it is, then wouldn't it imply that the truncated snub 24-cell and the rectified snub 24-cell must also be CRF, even though they are not uniform? And the Stott expansion would be CRF too (I believe this is prismatorhombato snub 24-cell?).
quickfur wrote:I can't believe this CRF has been sitting under our noses all this time, and we never thought of it!![...]
the symmetry is of course the snub 24-celll symmetry, maybe this symmetry is then more related to the pyritohedral symmetry than I thougt.It has a very interesting structure where pairs of truncated tetrahedra (green) share an edge, interspersed by tridiminished icosahedra (brown) that apparently link things together in ... a 4D equivalent of pyritohedral symmetry? The yellow cell in the center is, of course, a truncated icosahedron (aka buckyball ).
[...]
apacs<0, 0, 2*phi, 2*phi^2>
apacs<1, 1, phi^3, phi^3>
epacs<0, phi^2, phi^3, 2+phi>
epacs<1, phi, 2*phi^2, phi^2>
student91 wrote:quickfur wrote:I can't believe this CRF has been sitting under our noses all this time, and we never thought of it!![...]
That's not entirely true, it was already discovered. If you read all subsequent pages, you will read that it can be seen as an ico-diminished o5o3x3x. In fact every 120-uniform can have such a ico-diminishing if it allows a shallow cut. You've already listed the 120-uniforms with this property, in the "number of CRF's topic." Every such ico-diminishing can also be seen as a stott-expansion of the respective polytope, then you just apply the diminishing before you apply the expansion. This is all that gets, very long-windedly, explained in all the subsequent pages.the symmetry is of course the snub 24-celll symmetry, maybe this symmetry is then more related to the pyritohedral symmetry than I thougt.It has a very interesting structure where pairs of truncated tetrahedra (green) share an edge, interspersed by tridiminished icosahedra (brown) that apparently link things together in ... a 4D equivalent of pyritohedral symmetry? The yellow cell in the center is, of course, a truncated icosahedron (aka buckyball ).
[...]
quickfur wrote:[..] In fact, I may have unconsciously gotten the idea from your original post.[...]
quickfur wrote:I'm wondering, is the vertex figure of the snub 24-cell (snub demitesseract ) a CRF tridiminished icosahedron? 'cos if it is, then wouldn't it imply that the truncated snub 24-cell and the rectified snub 24-cell must also be CRF, even though they are not uniform? And the Stott expansion would be CRF too (I believe this is prismatorhombato snub 24-cell?).
quickfur wrote:I can't believe this CRF has been sitting under our noses all this time, and we never thought of it!! So, I did a Stott expansion on the snub 24-cell's edges, and produced the truncated snub 24-cell, confirmed to be CRF. Here's a mugshot of this new pretty:
It has a very interesting structure where pairs of truncated tetrahedra (green) share an edge, interspersed by tridiminished icosahedra (brown) that apparently link things together in ... a 4D equivalent of pyritohedral symmetry? The yellow cell in the center is, of course, a truncated icosahedron (aka buckyball ).
This beauty has 24 truncated icosahedra, 96 tridiminished icosahedra, and 120 truncated tetrahedra.
Edit: uploaded .def and .off files to the wiki page Truncated snub demitesseract.
Edit 2: Hmm, apparently I forgot about the tetrahedra of the second kind in the snub 24-cell, that produces analogous truncated tetrahedra in this CRF, that sits between 4 other truncated tetrahedra.
288 * * | 1 2 2 0 0 0 | 2 1 2 2 1 0 0 | 1 1 2 1 "head" of {5}
* 288 * | 0 0 2 1 1 0 | 0 0 2 1 2 1 0 | 2 0 1 1 "arms" of {5}
* * 288 | 0 0 0 1 1 2 | 0 0 2 0 2 1 1 | 2 0 1 1 "legs" of {5}
------------+-------------------------+-------------------------+------------
2 0 0 | 144 * * * * * | 2 0 2 0 0 0 0 | 1 1 2 0
2 0 0 | * 288 * * * * | 1 1 0 1 0 0 0 | 0 1 1 1
1 1 0 | * * 576 * * * | 0 0 1 1 1 0 0 | 1 0 1 1
0 1 1 | * * * 288 * * | 0 0 0 0 2 1 0 | 2 0 0 1
0 1 1 | * * * * 288 * | 0 0 2 0 0 1 0 | 2 0 1 0
0 0 2 | * * * * * 288 | 0 0 1 0 1 0 1 | 1 0 1 1
------------+-------------------------+-------------------------+------------
6 0 0 | 3 3 0 0 0 0 | 96 * * * * * * | 0 1 1 0
3 0 0 | 0 3 0 0 0 0 | * 96 * * * * * | 0 1 0 1
2 2 2 | 1 0 2 0 2 1 | * * 288 * * * * | 1 0 1 0
2 1 0 | 0 1 2 0 0 0 | * * * 288 * * * | 0 0 1 1
1 2 2 | 0 0 2 2 0 1 | * * * * 288 * * | 1 0 0 1
0 3 3 | 0 0 0 3 3 0 | * * * * * 96 * | 2 0 0 0
0 0 3 | 0 0 0 0 0 3 | * * * * * * 96 | 0 0 1 1
------------+-------------------------+-------------------------+------------
12 24 24 | 6 0 24 24 24 12 | 0 0 12 0 12 8 0 | 24 * * * ti
12 0 0 | 6 12 0 0 0 0 | 4 4 0 0 0 0 0 | * 24 * * tut (full sym)
6 3 3 | 3 3 6 0 3 3 | 1 0 3 3 0 0 1 | * * 96 * tut (axial sym)
3 3 3 | 0 3 6 3 0 3 | 0 1 0 3 3 0 1 | * * * 96 teddi
quickfur wrote:And of course, the rectified snub 24-cell is also CRF:
It has 24 icosidodecahedra, 96 tridiminished icosahedra, 120 octahedra (divided into 24 octahedra which are surrounded only by other octahedra, and 96 octahedra that touch the icosidodecahedra).
Its coordinates are:
- Code: Select all
apacs<0, 0, 2*phi, 2*phi^2>
apacs<1, 1, phi^3, phi^3>
epacs<0, phi^2, phi^3, 2+phi>
epacs<1, phi, 2*phi^2, phi^2>
I'll upload the software models to the wiki page: rectified snub demitesseract
144 * | 4 4 0 0 | 2 2 2 4 2 0 0 | 1 1 2 2 "head" of {5}
* 288 | 0 2 2 2 | 0 0 2 1 4 1 1 | 2 0 1 2 "arms" + "legs" of {5}
--------+-----------------+-------------------------+------------
2 0 | 288 * * * | 1 1 0 1 0 0 0 | 0 1 1 1
1 1 | * 576 * * | 0 0 1 1 1 0 0 | 1 0 1 1
0 2 | * * 288 * | 0 0 0 0 2 1 0 | 2 0 0 1
0 2 | * * * 288 | 0 0 1 0 1 0 1 | 1 0 1 1
--------+-----------------+-------------------------+------------
3 0 | 3 0 0 0 | 96 * * * * * * | 0 1 1 0
3 0 | 3 0 0 0 | * 96 * * * * * | 0 1 0 1
1 2 | 0 2 0 1 | * * 288 * * * * | 1 0 1 0
2 1 | 1 2 0 0 | * * * 288 * * * | 0 0 1 1
1 4 | 0 2 2 1 | * * * * 288 * * | 1 0 0 1
0 3 | 0 0 3 0 | * * * * * 96 * | 2 0 0 0
0 3 | 0 0 0 3 | * * * * * * 96 | 0 0 1 1
--------+-----------------+-------------------------+------------
6 24 | 0 24 24 12 | 0 0 12 0 12 8 0 | 24 * * * id
6 0 | 12 0 0 0 | 4 4 0 0 0 0 0 | * 24 * * oct (tet sym)
3 3 | 3 6 0 3 | 1 0 3 3 0 0 1 | * * 96 * oct (3-ap sym)
3 6 | 3 6 3 3 | 0 1 0 3 3 0 1 | * * * 96 teddi
Klitzing wrote:quickfur wrote:[...]
Edit 2: Hmm, apparently I forgot about the tetrahedra of the second kind in the snub 24-cell, that produces analogous truncated tetrahedra in this CRF, that sits between 4 other truncated tetrahedra.
There are no tetrahedra within tisadi!
Klitzing wrote:[...]
truncated sadi = tisadi (found in 2004 by Andrew Weimholt)
rectified sadi = risadi (found in 2004 by Andrew Weimholt)
Stott expanded sadi = prissi (found in 2005 by me)
[...]
Marek14 wrote:Hm, looking on in Stella, it seems to me that there might be a way to split the icosidodecahedra and octahedra in rectified snub 24-cell, as it's orbiform, so both of it vertex figures of this shape are fairly tame.
Also, the runcinated version might have some cuts going through the icosahedra...
Klitzing wrote:quickfur wrote:And of course, the rectified snub 24-cell is also CRF:
It has 24 icosidodecahedra, 96 tridiminished icosahedra, 120 octahedra (divided into 24 octahedra which are surrounded only by other octahedra, and 96 octahedra that touch the icosidodecahedra).
Its coordinates are:
- Code: Select all
apacs<0, 0, 2*phi, 2*phi^2>
apacs<1, 1, phi^3, phi^3>
epacs<0, phi^2, phi^3, 2+phi>
epacs<1, phi, 2*phi^2, phi^2>
I'll upload the software models to the wiki page: rectified snub demitesseract
Also known.
- Code: Select all
144 * | 4 4 0 0 | 2 2 2 4 2 0 0 | 1 1 2 2 "head" of {5}
* 288 | 0 2 2 2 | 0 0 2 1 4 1 1 | 2 0 1 2 "arms" + "legs" of {5}
--------+-----------------+-------------------------+------------
2 0 | 288 * * * | 1 1 0 1 0 0 0 | 0 1 1 1
1 1 | * 576 * * | 0 0 1 1 1 0 0 | 1 0 1 1
0 2 | * * 288 * | 0 0 0 0 2 1 0 | 2 0 0 1
0 2 | * * * 288 | 0 0 1 0 1 0 1 | 1 0 1 1
--------+-----------------+-------------------------+------------
3 0 | 3 0 0 0 | 96 * * * * * * | 0 1 1 0
3 0 | 3 0 0 0 | * 96 * * * * * | 0 1 0 1
1 2 | 0 2 0 1 | * * 288 * * * * | 1 0 1 0
2 1 | 1 2 0 0 | * * * 288 * * * | 0 0 1 1
1 4 | 0 2 2 1 | * * * * 288 * * | 1 0 0 1
0 3 | 0 0 3 0 | * * * * * 96 * | 2 0 0 0
0 3 | 0 0 0 3 | * * * * * * 96 | 0 0 1 1
--------+-----------------+-------------------------+------------
6 24 | 0 24 24 12 | 0 0 12 0 12 8 0 | 24 * * * id
6 0 | 12 0 0 0 | 4 4 0 0 0 0 0 | * 24 * * oct (tet sym)
3 3 | 3 6 0 3 | 1 0 3 3 0 0 1 | * * 96 * oct (3-ap sym)
3 6 | 3 6 3 3 | 0 1 0 3 3 0 1 | * * * 96 teddi
oxFxo3xooox3ofxfo&#xt
o....3o....3o.... & | 12 * * | 4 4 0 0 0 0 | 2 2 2 4 2 0 0 0 0 | 1 2 1 2 0 of tet-sym oct, verf = xx2xf&#x
.o...3.o...3.o... & | * 24 * | 0 2 2 1 1 0 | 0 0 2 1 2 1 2 1 0 | 0 1 2 1 1 of trip
..o..3..o..3..o.. | * * 12 | 0 0 0 2 0 2 | 0 0 0 0 4 0 0 1 1 | 0 0 2 2 0 of teddi bases, verf = xo2ox&#f
------------------------+----------+-------------------+------------------------+----------
..... x.... ..... & | 2 0 0 | 24 * * * * * | 1 1 0 1 0 0 0 0 0 | 1 1 0 1 0
oo...3oo...3oo...&#x & | 1 1 0 | * 48 * * * * | 0 0 1 1 1 0 0 0 0 | 0 1 1 1 0
.x... ..... ..... & | 0 2 0 | * * 24 * * * | 0 0 1 0 0 1 1 0 0 | 0 1 1 0 1 at trip bases
.oo..3.oo..3.oo..&#x & | 0 1 1 | * * * 24 * * | 0 0 0 0 2 0 0 1 0 | 0 0 2 1 0
.o.o.3.o.o.3.o.o.&#x | 0 2 0 | * * * * 12 * | 0 0 0 0 0 0 2 1 0 | 0 0 2 0 1 trip lacings
..... ..... ..x.. | 0 0 2 | * * * * * 12 | 0 0 0 0 2 0 0 0 1 | 0 0 1 2 0
------------------------+----------+-------------------+------------------------+----------
o....3x.... ..... & | 3 0 0 | 3 0 0 0 0 0 | 8 * * * * * * * * | 1 1 0 0 0 oct-oct
..... x....3o.... & | 3 0 0 | 3 0 0 0 0 0 | * 8 * * * * * * * | 1 0 0 1 0 oct-teddi
ox... ..... .....&#x & | 1 2 0 | 0 2 1 0 0 0 | * * 24 * * * * * * | 0 1 1 0 0
..... xo... .....&#x & | 2 1 0 | 1 2 0 0 0 0 | * * * 24 * * * * * | 0 1 0 1 0
..... ..... ofx..&#xt & | 1 2 2 | 0 2 0 2 0 1 | * * * * 24 * * * * | 0 0 1 1 0
.x...3.o... ..... & | 0 3 0 | 0 0 3 0 0 0 | * * * * * 8 * * * | 0 1 0 0 1
.x.x. ..... .....&#x | 0 4 0 | 0 0 2 0 2 0 | * * * * * * 12 * * | 0 0 1 0 1
.ooo.3.ooo.3.ooo.&#x | 0 2 1 | 0 0 0 2 1 0 | * * * * * * * 12 * | 0 0 2 0 0
..... ..o..3..x.. | 0 0 3 | 0 0 0 0 0 3 | * * * * * * * * 4 | 0 0 0 2 0
------------------------+----------+-------------------+------------------------+----------
o....3x....3o.... & | 6 0 0 | 12 0 0 0 0 0 | 4 4 0 0 0 0 0 0 0 | 2 * * * * oct (tet-sym)
ox...3xo... .....&#x & | 3 3 0 | 3 6 3 0 0 0 | 1 0 3 3 0 1 0 0 0 | * 8 * * * oct (as 3ap)
oxFxo ..... ofxfo&#xt | 2 8 4 | 0 8 4 8 4 2 | 0 0 4 0 4 0 2 4 0 | * * 6 * * bilbiro
..... xoo..3ofx..&#xt & | 3 3 3 | 3 6 0 3 0 3 | 0 1 0 3 3 0 0 0 1 | * * * 8 * teddi
.x.x.3.o.o. .....&#x | 0 6 0 | 0 0 6 0 3 0 | 0 0 0 0 0 2 3 0 0 | * * * * 4 trip
Klitzing wrote:[...]
Just dreamed up the bilbiroing of risadi:
[...]
(I fear I might have lost track on what we already have found in all that CRFebruary. But suppose, that this is a new one, ain't it?)
--- rk
PS:
And furthermore, one probably might augment this structure with 6 line||bilbiro CRFs.
Question: would then the teddi-peppy adjoins become co-realmic, i.e. recombine to ikes?
student91 wrote:Klitzing wrote:[...]
Just dreamed up the bilbiroing of risadi:
[...]
(I fear I might have lost track on what we already have found in all that CRFebruary. But suppose, that this is a new one, ain't it?)
--- rk
PS:
And furthermore, one probably might augment this structure with 6 line||bilbiro CRFs.
Question: would then the teddi-peppy adjoins become co-realmic, i.e. recombine to ikes?
I'm afraid this structure was already found, as D9.1. This is a bilbiro-ing of o5o3x3o, with another diminishing, and because risadi is a ico-diminishing of o5o3x3o, its bilbiroing is quite analogous. Of course D9.1 has a different top section, but the interesting part is the same. This means, that when you augment it with the pseudopyramids, you indeed will get your ike back. What I think is more interesting, and what I already pointed out in this post, is that when you take D4.9.0 and delete a pentagonal pyramid, you get a pentagonal orthocupolarotunda.
Klitzing wrote:[...]
First of all, there is no "D9.1" at http://hddb.teamikaria.com/wiki/Bilbirothawroid.
quickfur wrote:FYI, the official index page is at discovery index; that's where you want to go to look up a D number. The bilbirothawroid page only contains a subset of the D numbers.
Klitzing wrote:quickfur wrote:FYI, the official index page is at discovery index; that's where you want to go to look up a D number. The bilbirothawroid page only contains a subset of the D numbers.
That's right the problem with that wiki! There are quite a lot of inter-links missing. If one does not know the URL of what one is looking for (as you obviously do), one usually cannot find that page at all!
--- rk
Klitzing wrote:[...]
First of all, there is no "D9.1" at http://hddb.teamikaria.com/wiki/Bilbirothawroid.
If you meant D4.9.1, then it definitely is different:
[...]--- rk
quickfur wrote:Klitzing wrote:quickfur wrote:FYI, the official index page is at discovery index; that's where you want to go to look up a D number. The bilbirothawroid page only contains a subset of the D numbers.
That's right the problem with that wiki! There are quite a lot of inter-links missing. If one does not know the URL of what one is looking for (as you obviously do), one usually cannot find that page at all!
--- rk
I know... so bug Keiji to send you an invite code for the wiki so that you can edit it yourself and fix these missing links.
student91 wrote:ah, I see, indeed different. I guess I was a bit too fast at reading your post.
It seems to be half of the B1 of the D4.8.x-series. I wonder if the procedure of gluing two of these together will result in true bilbiro's, i.e. do the vertices of the bilbiro's lay in the same "3-space"? If it doesn't, you've certainly shown a kind of bilbiroing that might be the key to the construction of D4.11. got to go now
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