Mrrl wrote:Of course, it's not uniform - it has rhombic ridges. There was error in my angles computations. Actually, their sum is exactly 180°, so the resulting polytope has cells that are unions of 4- and 3-pyramids.
But it's the best that we could find on this way. Ok, let's look in other directions...
quickfur wrote:I see.
I'm trying to derive coordinates for joining the square biantiprismic ring to itself by the square antiprism cell. One way of doing this leads to a non-convex polychoron, but the other way may yield a CRF. However, the coordinates are very ugly... the 2^(1/4) height of the square antiprism is making the algebra really unmanageable when I try to solve for uniform edge lengths.
Mrrl wrote:quickfur wrote:I see.
I'm trying to derive coordinates for joining the square biantiprismic ring to itself by the square antiprism cell. One way of doing this leads to a non-convex polychoron, but the other way may yield a CRF. However, the coordinates are very ugly... the 2^(1/4) height of the square antiprism is making the algebra really unmanageable when I try to solve for uniform edge lengths.
But it is what we've done just now!
Mrrl wrote:Is there any estimates of the number of diminished 600-cells? Now we counted only two of them, but what about all others? Should we make a list of them, or just write total number?
And... what is the bihedral angle of dodecaheral pyramid? Are there any augmented 120-cells, and is there a limitation for sets of faces (can we add pyramids two connected faces or not). In the latter case number of augmented 120-cells is the same as of diminished 600-cells
Mrrl wrote:So, we should remove dodecaheral pyramids and related polychorons from the list. It's very contrintuitive for me, but such is reality...
For diminished of 600-cell, we can select one vertex and remove one icosahedral pyramid. Then select another vertex that is not adjacent to the first and remove its pyramid... and even if they are adjacent, you get a polytope with two diminish-icosahedra cells??? We have to understand, what constraints are here. No 3 vertex in the single cell? Or something more strong? But it looks like we are very close to 2^120 CRFs!
quickfur wrote:Already done.
But what about symmetry though? Won't that reduce the number somewhat?
Marek14 wrote:Stepping away from all CRF polychora for a while, there is a related question: Which are all PRIMITIVE CRF polychora?
A primitive polychoron is one that cannot be decomposed into others. Once you have this list, you can make all remaining CRF polychora by just augmentation. For example, out of all uniform + johnson polyhedra, octahedron is not primitive since it decomposes into two square pyramids, icosahedron is not primitive, as it can be diminished, cuboctahedron can be split in two triangular cupolas, rhombicuboctahedron splits into octahedral prism + two square cupolae, icosidodecahedron splits into two pentagonal rotundas and rhombicosidodecahedron can be diminished in various ways. From Johnson solids, all the elongated and gyroelongated bodies, all dipyramids and bicupolae, gyrobifastigium, anything augmented (including augmented tridiminished icosahedron and augmented sphenocorona) and all forms of rhombicosidodecahedron except for parabidiminished and tridiminished are non-primitive. So the list gets cut down quite a lot.
On the other hand, some augmented analogues seem possible for non-regular uniform polychora. Is it possible, for example, to augment truncated-tetrahedral cells of truncated pentachoron with truncated tetrahedron-tetrahedron copula? Augment truncated cubes in truncated tesseract with truncated cube-cube cupolas?
And for uniform shape that includes decagonal prisms, maybe capping them with pentagonal prism||decagonal prism, and then saddint a pentagonal prism pyramid atop that?
I don't know the dichoral angles, so I'm not sure which of these are possible, but augmented uniforms seem to be a large possible group.
Edit: and rectified 600-cell. It's made of octahedra and icosahedra, I feel that there is a possibility of many diminished shapes containing square pyramids and various types of diminished icosahedra.
Edit2: what I'm really interested in are 4D analogues of the "hard" Johnson shapes at the end of the list which don't require any uniform polyhedron to construct. Hereby, I suggest to call any such polychoron to be found a "crown jewel".
Marek14 wrote:Stepping away from all CRF polychora for a while, there is a related question: Which are all PRIMITIVE CRF polychora?
A primitive polychoron is one that cannot be decomposed into others. Once you have this list, you can make all remaining CRF polychora by just augmentation.
[...] On the other hand, some augmented analogues seem possible for non-regular uniform polychora. Is it possible, for example, to augment truncated-tetrahedral cells of truncated pentachoron with truncated tetrahedron-tetrahedron copula?
Augment truncated cubes in truncated tesseract with truncated cube-cube cupolas?
And for uniform shape that includes decagonal prisms, maybe capping them with pentagonal prism||decagonal prism, and then saddint a pentagonal prism pyramid atop that?
I don't know the dichoral angles, so I'm not sure which of these are possible, but augmented uniforms seem to be a large possible group.
Edit: and rectified 600-cell. It's made of octahedra and icosahedra, I feel that there is a possibility of many diminished shapes containing square pyramids and various types of diminished icosahedra.
Edit2: what I'm really interested in are 4D analogues of the "hard" Johnson shapes at the end of the list which don't require any uniform polyhedron to construct. Hereby, I suggest to call any such polychoron to be found a "crown jewel".
by Mrrl » Wed Nov 23, 2011 7:12 pm
by quickfur » Wed Nov 23, 2011 7:13 pm
Marek14 wrote:By "crown jewels" I mean things like triangular hebesphenorotunda. I don't see any simple way to finding this polyhedron if you start from uniforms. It exists, but it has no obvious relations to other uniforms or even Johnsons.
quickfur wrote:Marek14 wrote:By "crown jewels" I mean things like triangular hebesphenorotunda. I don't see any simple way to finding this polyhedron if you start from uniforms. It exists, but it has no obvious relations to other uniforms or even Johnsons.
So basically we have to pick three random Johnson solids with compatible faces, stick them together around a vertex, check if they have dihedral angle < 180°, then try all combinations of other solids to see if we can build a closed 4D shape out of it?
I think I'll start with more obvious combinations first, and then work my way to the more obscure ones. I consider it preliminary practice in building 4D shapes before the "real task" of building hard-to-guess CRFs.
Marek14 wrote:[...] I'd start with this: since augmentations of polytopes will be basically done by segmentochora, what are all segmentochora combinations that can be "chained" to form convex towers? (And still be used to augment other polytopes, obviously).
quickfur wrote:As should be obvious from this projection, you can insert a dodecahedral prism in between to make the elongated biantiprism, and you can cap either or both ends with icosahedral pyramids. So from this base shape you can make two augmented forms, an elongated form, and two augmented elongated forms.
Mrrl wrote:[...]
So now number of CRFs in finite families exceeds 2170/14400An I think that there are much more others!
Marek14 wrote:[...] So the question is whether the side angles are shallow enough to augment a 120-cell or runcinated 120-cell (only uniform polychora that feature dodecahedra) with this, since if they are, then each augment can be, on top of that, capped with icosahedral pyramid.
My gut tells me that 120-cell is too round for this sort of augmentation. Not sure about runcinated 120-cell. Are there other shapes that can be put into segmentochoric relationship with dodecahedron? How about a cube...?
quickfur wrote:Can someone confirm this?
Mrrl wrote:quickfur wrote:Can someone confirm this?
Probably, your program can:
(±1,±1,±1,sqrt(4*sqrt(2)-3))
(±sqrt(2),±sqrt(2),0,0)
(±sqrt(2),0,±sqrt(2),0)
(0,±sqrt(2),±sqrt(2),0)
Mrrl wrote:[...]
(±1,±1,±1,sqrt(4*sqrt(2)-3))
(±sqrt(2),±sqrt(2),0,0)
(±sqrt(2),0,±sqrt(2),0)
(0,±sqrt(2),±sqrt(2),0)
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