Mrrl wrote:quickfur wrote:[...] Also, I'm more confident now that it is possible to runcinate this rotunda, you will get the same count of rectified dodecahedra, but with pentagonal prisms between them, and the tetrahedra around the top cell become cuboctahedra, and the tetrahedra around the bottom cell become triangular cupola, and the bottom cell is expanded into great rhombicosidodecahedron. It is essentially the diminishing of a cantellated 600-cell (CD diagram is o5x3o3x), some of the cuboctahedra are cut into triangular cupolae.
Yes, I also think that it exists. More, you can cut it in one more level and get great rhombicosidodecahedron|||truncated icosahedron rotunda It has 12 pentagonal rotunda, 30 pentagonal prisms and 40 triangular cupola as side cells
[...] Like augmentations of something by side cell of segmentotope (that can't be used as base because other side is not parallel to this cell), or, better, combining two segmentotopes by cells (at least one of which is not a base) that have some bichoral angles >90 (like we tried to do with two biantiprasmatic rings). All augmentations can be done after that.
quickfur wrote:[...] I'm going to try to compute coordinates and render them, it shouldn't be too hard since all I need is to find the hyperplane equation parallel to the top of the rotunda, and find depth for cutting.
<(3+sqrt(5))/2, 0, 0, ±1>
<(3+sqrt(5))/2, 0, ±1, 0>
<(3+sqrt(5))/2, ±1, 0, 0>
<(3+sqrt(5))/2, ±(-1+sqrt(5))/4, ±1/2, ±(1+sqrt(5))/4>
<(3+sqrt(5))/2, ±1/2, ±(1+sqrt(5))/4, ±(-1+sqrt(5))/4>
<(3+sqrt(5))/2, ±(1+sqrt(5))/4, ±(-1+sqrt(5))/4, ±1/2>
<(2+sqrt(5))/2, 0, ±(-1+sqrt(5))/4, ±(5+sqrt(5))/4>
<(2+sqrt(5))/2, ±(-1+sqrt(5))/4, ±(5+sqrt(5))/4, 0>
<(2+sqrt(5))/2, ±(5+sqrt(5))/4, 0, ±(-1+sqrt(5))/4>
<(2+sqrt(5))/2, ±(-1+sqrt(5))/4, ±(1+sqrt(5))/4, ±(1+sqrt(5))/2>
<(2+sqrt(5))/2, ±(1+sqrt(5))/4, ±(1+sqrt(5))/2, ±(-1+sqrt(5))/4>
<(2+sqrt(5))/2, ±(1+sqrt(5))/2, ±(-1+sqrt(5))/4, ±(1+sqrt(5))/4>
<(2+sqrt(5))/2, ±(1+sqrt(5))/4, ±1, ±(3+sqrt(5))/4>
<(2+sqrt(5))/2, ±1, ±(3+sqrt(5))/4, ±(1+sqrt(5))/4>
<(2+sqrt(5))/2, ±(3+sqrt(5))/4, ±(1+sqrt(5))/4, ±1>
<(3+3*sqrt(5))/4, 0, ±1/2, ±(3+sqrt(5))/4>
<(3+3*sqrt(5))/4, ±1/2, ±(3+sqrt(5))/4, 0>
<(3+3*sqrt(5))/4, ±(3+sqrt(5))/4, 0, ±1/2>
<(3+3*sqrt(5))/4, ±(1+sqrt(5))/4, ±(1+sqrt(5))/4, ±(1+sqrt(5))/4>
Marek14 wrote:Post OFF files too, please
quickfur wrote:Marek14 wrote:Post OFF files too, please
Here:
Icosidodecahedral rotunda
Runcinated icosidodecahedral rotunda
Sorry, my .off convertor doesn't know how to import povray colors, so you're still getting the randomized per type color assignments.
quickfur wrote:Hmph. You are using a pre-release version of scons, you know that, right? I've tested on SCons 1.0.0 and everything builds fine. But no matter. I've made a different version of the SConscript specifically for you, so install the new sources here:
http://eusebeia.dyndns.org/polyview/polyview-3.0pre-3568.tar.gz
and instead of running 'scons', run 'scons -f SConscript.0.97'. (You only need to do this in the main directory; i've removed the offending rule from the proglib subdir because that file isn't even used anymore anyway, just leftover junk from long ago.)
Keiji wrote:[...] That's very kind of you, but unfortunately the proglib script still gives the same error.
Marek14 wrote:[...] EDIT: It seems to me that the runcinated cupola could be further split into two shapes: if the cuboctahedra are cut in half, you can replace half of them, the 12 top pentagonal prisms and the icosidodecahedron with a truncated icosahedron.
quickfur wrote:Yes, mrrl already realized this (see his post earlier in this topic). The result can be considered a rhombicosidodecahedral rotunda. So we have at least 3 rotundae in 4D.
Mrrl wrote:Coordinates of 4.109. It is some kind of 4-ring (there are point, square, octagon, gyrated square in 4 parallel planes):
0,0,0,-1
±1,±1,1,0
±sqrt(2),0,-1,0
0,±sqrt(2),-1,0
±(1+sqrt(2)),±1,0,1
±1,±(1+sqrt(2)),0,1
It has two square cupolae, two square pyramids and 8 triangular prisms as cells.
I wonder what happens if we replace 3-rd and 4-th lines by single ±1,±1,-1,0 (rotate the second square back). Is there a chance for it to remain CRF? Or some edges will be of length sqrt(2).
quickfur wrote:Just tested it. I got uneven edge lengths, 2 and 3.2906575520 (unidentified).
quickfur wrote:Looks like Keiji's gyro square cupolic ring, with the square antiprism replaced by a gyrated pyramid that forms prisms with the original square pyramids. Interesting construction... I wonder if analogous structure exists for pentagonal cupolic rings?
4.24 tetrahedron || trigonal cupola
height: sqrt(5/8) = 0.790569
circumradius: 1
comments: kind of diminished half-of runcinated-
pentachoron (tetrahedron as
"tetrahedron - trigon" and trigonal cupola as
"cuboctahedron - trigonal cupola")
cells: 2 tetrahedra + 6 trigonal prisms +
2 trigonal cupolae
4.16 square pyramid || gyrated cube
height: sqrt(sqrt(8)-1)/2 = 0.676097
circumradius: sqrt((4+sqrt(2))/7) = 0.879465
other names: -
comments: kind of diminished cubic antiprism
(square pyramid as "octahedron - square
pyramid" and cube as "cube - square")
cells: 4+4+4 tetrahedra + 1+1+4 square
pyramids + 1 square antiprism + 1 cube
Marek14 wrote:I realized that I can use the section function in Stella to systematically go through all convex uniform polychora and identify the possible cupolas/rotundas/pyramids, as well as diminishings.
[...] truncated tetrahedron||truncated octahedron (truncated tetrahedron cut off prismatorhombated pentachoron)
the rest of the body gives a shape made of 1 cuboctahedron, 4 truncated tetrahedra, 6 hexagonal prisms, 4 triangular prisms, 4 triangular cupolas and 1 truncated cuboctahedron. I suspect it might be possible to glue these two parts of prismatotruncated pentachoron together the "wrong way", so that the the triangular cupolas will be joined to hexagonal prisms and truncated tetrahedra to triangular cupolas instead of combining the cupolas to form cuboctahedra. Not sure if the joins won't blend (into augmented truncated tetrahedra and/or elongated triangular cupolas), but either way, it should be a possible shape!
[...] Sorry for hard-to-read list, I was writing as I was discovering
Note that some small rhombated polychora have two different ways of diminishing. Small rhombated 120-cell, in particular, can have either rhombicosidodecahedra removed, which removes 30 triangular prisms and 20 octahedra, diminishes 12 rhombicosidodecahedra around it, and adds 1 great icosidodecahedron, or it can have a pentagon removed, which removes five triangular prisms, cuts in half 5 octahedra, diminishes two rhombicosidodecahedra and adds a decagonal prism. And both types of diminishing can be used at once! That will be a lot of shapes.
Mrrl wrote:[...]
3.2906575520 = sqrt(8+sqrt(8))
[...] I couldn't find pentagonal ring in the list, but there is triangular ring: [...]
Marek14 wrote:Of course, most of them will be known...
[...]
icosahedral rotunda (icosahedron cut from truncated 600-cell, not a segmentochoron: 1 icosahedron, 20 truncated tetrahedra, 12 pentagonal pyramids and 1 truncated icosahedron)
Marek14 wrote:truncated tetrahedron||truncated octahedron (truncated tetrahedron cut off prismatorhombated pentachoron)
the rest of the body gives a shape made of 1 cuboctahedron, 4 truncated tetrahedra, 6 hexagonal prisms, 4 triangular prisms, 4 triangular cupolas and 1 truncated cuboctahedron.
Marek14 wrote: I suspect it might be possible to glue these two parts of prismatotruncated pentachoron together the "wrong way", so that the the triangular cupolas will be joined to hexagonal prisms and truncated tetrahedra to triangular cupolas instead of combining the cupolas to form cuboctahedra. Not sure if the joins won't blend (into augmented truncated tetrahedra and/or elongated triangular cupolas), but either way, it should be a possible shape!
Marek14 wrote:Prismatotruncated tesseract has an "equatorial region" formed by great cuboctahedral prism and two caps, each of them being formed by 1 rhombicuboctahedron, 6 cubes, 12 hexagonal prisms, 8 truncated tetrahedra, 6 square cupolas and 1 great cuboctahedron.
Note that some small rhombated polychora have two different ways of diminishing. Small rhombated 120-cell, in particular, can have either rhombicosidodecahedra removed, which removes 30 triangular prisms and 20 octahedra, diminishes 12 rhombicosidodecahedra around it, and adds 1 great icosidodecahedron, or it can have a pentagon removed, which removes five triangular prisms, cuts in half 5 octahedra, diminishes two rhombicosidodecahedra and adds a decagonal prism. And both types of diminishing can be used at once! That will be a lot of shapes.
Mrrl wrote:Note that some small rhombated polychora have two different ways of diminishing. Small rhombated 120-cell, in particular, can have either rhombicosidodecahedra removed, which removes 30 triangular prisms and 20 octahedra, diminishes 12 rhombicosidodecahedra around it, and adds 1 great icosidodecahedron, or it can have a pentagon removed, which removes five triangular prisms, cuts in half 5 octahedra, diminishes two rhombicosidodecahedra and adds a decagonal prism. And both types of diminishing can be used at once! That will be a lot of shapes.
Ok, you have it for cantellated tesseract, simplex and 120-cell. This ridge-centered diminishing is very unexpected! Isn't there any example of edge-centered one?
Mrrl wrote:Marek14 wrote:[...]
icosahedral rotunda (icosahedron cut from truncated 600-cell, not a segmentochoron: 1 icosahedron, 20 truncated tetrahedra, 12 pentagonal pyramids and 1 truncated icosahedron)
This one is new [...]
Marek14 wrote:For the cube||icosahedron segmentochoron, how many different ways there are of gluing the two icosahedra together?
wendy wrote:[...] The order of replacing the x3o4o3x with peaks, applies to the x3o3o5o, since the 24 faces of the snub-24ch, correspond to three sets of the above, each of which can be appiculated in separate. One has eg, 12*3 + 4 = 40 different ways of apiculating the 24ch's icosahedra, which all lead to different johnson-figures,
quickfur wrote:And furthermore, one doesn't have to stick with the 24 apices of the snub 24-cell alone. One could, for example, diminish the 600-cell at an apex that lies between two of the snub 24-cell's icosahedra, leading to a different kind of diminishing. Adding in these other possibilities greatly expand the number of diminishings of the 600-cell (I think mrrl estimated the number to be close to a million).
Mrrl wrote:[...] No, these were closer to 2^100 but rectified 120-cell (using only one type of diminishing) beats them - there are proven 2^156 different CRFS with estimations up to 2^500
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