wendy wrote:Not all johnson-figures have coplanar vertex figures, such as oxx3ooo&#xt. The vertices at the bases of the tetrahedra do not have a planar verf.
3,3,3: tetrahedron, elongated triangular pyramid, triangular dipyramid, elongated triangular dipyramid, augmented tridiminished icosahedron
3,3,4: square pyramid
3,3,5: pentagonal pyramid
3,4,4: triangular prism, elongated triangular pyramid, gyrobifastigium, augmented triangular prism
3,4,6 (chiral): triangular cupola
3,4,8 (chiral): square cupola
3,4,10 (chiral): pentagonal cupola
3,5,5: metabidiminished icosahedron, tridiminished icosahedron [fits in 2 different ways], augmented tridiminished icosahedron, bilunabirotunda
3,5,10 (chiral): pentagonal rotunda
3,6,6: truncated tetrahedron, augmented truncated tetrahedron [fits in 2 different ways]
3,8,8: truncated cube, augmented truncated cube [fits in 4 different ways, 2 of them chiral], biaugmented truncated cube
3,10,10: truncated dodecahedron, augmented truncated dodecahedron [fits in 10 different ways, 6 of them chiral], parabiaugmented truncated dodecahedron [fits in 4 different ways, 2 of them chiral], metabiaugmented truncated dodecahedron [fits in 20 different ways, 16 of them chiral], triaugmented truncated dodecahedron [fits in 10 different ways, 6 of them chiral]
4,4,4: cube, elongated square pyramid [fits in 3 different ways]
4,4,5: pentagonal prism, elongated pentagonal pyramid, augmented pentagonal prism [fits in 3 different ways, 2 of them chiral], biaugmented pentagonal prism
4,4,6: hexagonal prism, elongated triangular cupola [fits in 2 chiral ways], augmented hexagonal prism [fits in 4 chiral ways], parabiaugmented hexagonal prism, metabiaugmented hexagonal prism [fits in 2 chiral ways]
4,4,7: heptagonal prism
4,4,8: octagonal prism, elongated square cupola [fits in 2 chiral ways]
4,4,9: enneagonal prism
4,4,10: decagonal prism, elongated pentagonal cupola [fits in 2 chiral ways], elongated pentagonal rotunda [fits in 2 chiral ways]
4,5,10 (chiral): diminished rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron [fits in 5 different ways], bigyrate diminished rhombicosidodecahedron [fits in 5 different ways], parabidiminished rhombicosidodecahedron, metabidiminished rhombicosidodecahedron [fits in 5 different ways], gyrate bidiminished rhombicosidodecahedron [fits in 10 different ways], trigyrate rhombicosidodecahedron [fits in 5 different ways]
4,6,6: truncated octahedron
4,6,8 (chiral): truncated cuboctahedron
4,6,10 (chiral): truncated icosidodecahedron
5,5,5: dodecahedron, augmented dodecahedron [fits in 9 different ways], parabiaugmented dodecahedron [fits in 3 different ways], metabiaugmented docecahedron [fits in 15 different ways, 6 of them chiral], triaugmented dodecahedron [fits in 5 different ways]
5,6,6: truncated icosahedron
3,3,3,3: octahedron, square pyramid, elongated square pyramid, gyroelongated square pyramid, elongated square dipyramid, gyroelongated square pyramid, augmented triangular prism [fits in 2 different ways], biaugmented triangular prism [fits in 4 different ways], triaugmented triangular prism [fits in 2 different ways], augmented pentagonal prism [fits in 2 different ways], biaugmented pentagonal prism [fits in 4 different ways], augmented hexagonal prism [fits in 2 different ways], parabiaugmented hexagonal prism [fits in 2 different ways], metabiaugmented hexagonal prism [fits in 4 different ways], triaugmented hexagonal prism [fits in 2 different ways], augmented sphenocorona [fits in 4 different ways]
3,3,3,3 (skew1): triangular dipyramid
3,3,3,3 (skew2): pentagonal dipyramid
3,3,3,3 (skew3): snub disphenoid [fits in 2 different ways]
3,3,3,3 (skew4): sphenomegacorona [fits in 2 different ways]
3,3,3,4: square antiprism, gyroelongated square pyramid
3,3,3,4 (skew1, chiral): augmented triangular prism, biaugmented triangular prism
3,3,3,4 (skew2, chiral): sphenocorona, augmented sphenocorona
3,3,3,5: pentagonal antiprism, gyroelongated pentagonal pyramid, metabidiminished icosahedron [fits in 3 different ways, 2 of them chiral], tridiminished icosahedron, augmented tridiminished icosahedron, triangular hebesphenorotunda
3,3,3,6: hexagonal antiprism, gyroelongated triangular cupola [fits in 2 different ways]
3,3,3,7: heptagonal antiprism
3,3,3,8: octagonal antiprism, gyroelongated square cupola [fits in 2 different ways]
3,3,3,9: enneagonal antiprism
3,3,3,10: decagonal antiprism, gyroelongated pentagonal cupola [fits in 2 different ways], gyroelongated pentagonal rotunda [fits in 2 different ways]
3,3,4,4: triangular orthobicupola
3,3,4,4 (skew1): elongated triangular pyramid, elongated triangular dipyramid
3,3,4,4 (skew2): elongated square pyramid, elongated square dipyramid, square orthobicupola
3,3,4,4 (skew3): elongated pentagonal pyramid, elongated pentagonal dipyramid
3,3,4,4 (skew4): pentagonal orthobicupola
3,3,4,4 (skew5): sphenocorona
3,3,4,4 (skew6): sphenomegacorona
3,3,4,4 (skew7): hebesphenomegacorona [fits in 2 chiral ways]
3,3,4,4 (skew8): disphenocingulum
3,4,3,4: cuboctahedron, triangular cupola [fits in 2 different ways], elongated triangular cupola [fits in 2 different ways], gyroelongated triangular cupola [fits in 2 different ways], triangular orthobicupola [fits in 2 different ways], elongated triangular orthobicupola [fits in 2 different ways], elongated triangular gyrobicupola [fits in 2 different ways], gyroelongated triangular bicupola [fits in 4 chiral ways], augmented truncated tetrahedron [fits in 2 different ways]
3,4,3,4 (skew1, chiral): gyrobifastigium
3,4,3,4 (skew2, chiral): square gyrobicupola
3,4,3,4 (skew3, chiral): pentagonal gyrobicupola
3,3,4,5 (skew1, chiral): pentagonal gyrocupolarotunda
3,3,4,5 (skew2, chiral): augmented pentagonal prism, biaugmented pentagonal prism [fits in 2 different ways]
3,4,3,5 (skew, chiral): pentagonal orthocupolarotunda, bilunabirotunda, triangular hebesphenorotunda
3,3,4,6 (skew1, chiral): augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism [fits in 2 different ways], triaugmented hexagonal prism
3,3,4,6 (skew2, chiral): triangular hebesphenorotunda
3,4,3,6 (skew, chiral): augmented truncated tetrahedron
3,4,3,8 (skew, chiral): augmented truncated cube, biaugmented truncated cube
3,4,3,10 (skew, chiral): augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron [fits in 5 different ways], triaugmented truncated dodecahedron [fits in 5 different ways]
3,3,5,5: pentagonal orthobirotunda
3,3,5,5 (skew1): augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented docecahedron [fits in 5 different ways, 4 of them chiral], triaugmented dodecahedron [fits in 5 different ways, 4 of them chiral]
3,3,5,5 (skew2): augmented tridiminished icosahedron
3,5,3,5: icosidodecahedron, pentagonal rotunda [fits in 4 different ways], elongated pentagonal rotunda [fits in 4 different ways], gyroelongated pentagonal rotunda [fits in 4 different ways], pentagonal orthocupolarotunda [fits in 4 different ways], pentagonal gyrocupolarotunda [fits in 4 different ways], pentagonal orthobirotunda [fits in 4 different ways], elongated pentagonal orthocupolarotunda [fits in 4 different ways], elongated pentagonal gyrocupolarotunda [fits in 4 different ways], elongated pentagonal orthobirotunda [fits in 4 different ways], elongated pentagonal gyrobirotunda [fits in 4 different ways], gyroelongated pentagonal cupolarotunda [fits in 8 chiral ways], gyroelongated pentagonal birotunda [fits in 8 chiral ways], bilunabirotunda, triangular hebesphenorotunda [fits in 2 different ways]
3,4,4,4: rhombicuboctahedron, square cupola, elongated square cupola [fits in 3 different ways, 2 of them chiral], gyroelongated square cupola, square orthobicupola, square gyrobicupola, elongated square gyrobicupola [fits in 3 different ways, 2 of them chiral], gyroelongated square bicupola [fits in 2 chiral ways], augmented truncated cube, biaugmented truncated cube
3,4,4,4 (skew1, chiral): elongated triangular cupola, elongated triangular orthobicupola, elongated triangular gyrobicupola
3,4,4,4 (skew2, chiral): elongated pentagonal cupola, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda
3,4,4,5 (chiral): gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron [fits in 5 different ways], trigyrate rhombicosidodecahedron [fits in 5 different ways], paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron [fits in 5 different ways], bigyrate diminished rhombicosidodecahedron [fits in 10 different ways], gyrate bidiminished rhombicosidodecahedron [fits in 5 different ways]
3,4,4,5 (skew, chiral): elongated pentagonal rotunda, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, elongated pentagonal orthobirotunda, elongated pentagonal gyrobirotunda
3,4,5,4: rhombicosidodecahedron, pentagonal cupola, elongated pentagonal cupola, gyroelongated pentagonal cupola, pentagonal orthobicupola, pentagonal gyrobicupola, pentagonal orthocupolarotunda, pentagonal gyrocupolarotunda, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, gyroelongated pentagonal bicupola [fits in 2 chiral ways], gyroelongated pentagonal cupolarotunda [fits in 2 chiral ways], augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron [fits in 5 ways, 4 of them chiral], triaugmented truncated dodecahedron [fits in 5 ways, 4 of them chiral], gyrate rhombicosidodecahedron [fits in 10 different ways, 6 of them chiral], parabigyrate rhombicosidodecahedron [fits in 6 different ways, 4 of them chiral], metabigyrate rhombicosidodecahedron [fits in 20 different ways, 16 of them chiral], trigyrate rhombicosidodecahedron [fits in 10 different ways, 6 of them chiral], diminished rhombicosidodecahedron [fits in 9 different ways, 6 of them chiral], paragyrate diminished rhombicosidodecahedron [fits in 7 different ways, 4 of them chiral], metagyrate diminished rhombicosidodecahedron [fits in 35 different ways, 32 of them chiral], bigyrate diminished rhombicosidodecahedron [fits in 25 different ways, 22 of them chiral], parabidiminished rhombicosidodecahedron [fits in 3 different ways, 2 of them chiral], metabidiminished rhombicosidodecahedron [fits in 15 different ways, 12 of them chiral], gyrate bidiminished rhombicosidodecahedron [fits in 20 different ways, 16 of them chiral], trigyrate rhombicosidodecahedron [fits in 5 different ways, 2 of them chiral]
3,3,3,3,3: icosahedron, pentagonal pyramid, elongated pentagonal pyramid, gyroelongated pentagonal pyramid [fits in 6 different ways], pentagonal dipyramid, elongated pentagonal dipyramid, augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented dodecahedron [fits in 5 different ways], triaugmented dodecahedron [fits in 5 different ways], metabidiminished icosahedron [fits in 5 different ways]
3,3,3,3,3 (skew1): gyroelongated square pyramid, gyroelongated square pyramid
3,3,3,3,3 (skew2): biaugmented triangular prism, triaugmented triangular prism
3,3,3,3,3 (skew3): snub disphenoid
3,3,3,3,3 (skew4): snub square antiprism
3,3,3,3,3 (skew5): sphenocorona, augmented sphenocorona [fits in 2 chiral ways]
3,3,3,3,3 (skew6): sphenocorona, augmented sphenocorona [fits in 2 different ways]
3,3,3,3,3 (skew7, chiral): augmented sphenocorona
3,3,3,3,3 (skew8): sphenomegacorona
3,3,3,3,3 (skew9): sphenomegacorona
3,3,3,3,3 (skew10): hebesphenomegacorona [fits in 2 different ways]
3,3,3,3,3 (skew11): hebesphenomegacorona
3,3,3,3,3 (skew12): disphenocingulum
3,3,3,3,4: snub cube [fits in 2 chiral ways]
3,3,3,3,4 (skew1, chiral): gyroelongated triangular cupola, gyroelongated triangular bicupola [fits in 2 chiral ways]
3,3,3,3,4 (skew2, chiral): gyroelongated square cupola, gyroelongated square bicupola [fits in 2 chiral ways]
3,3,3,3,4 (skew3, chiral): gyroelongated pentagonal cupola, gyroelongated pentagonal bicupola [fits in 2 chiral ways], gyroelongated pentagonal cupolarotunda [fits in 2 chiral ways]
3,3,3,3,4 (skew4): snub square antiprism
3,3,3,3,4 (skew5, chiral): augmented sphenocorona
3,3,3,3,4 (skew6, chiral): sphenomegacorona
3,3,3,3,4 (skew7, chiral): hebesphenomegacorona
3,3,3,3,4 (skew8, chiral): disphenocingulum
3,3,3,3,5: snub dodecahedron [fits in 2 chiral ways]
3,3,3,3,5 (skew, chiral): gyroelongated pentagonal rotunda, gyroelongated pentagonal cupolarotunda [fits in 2 chiral ways], gyroelongated pentagonal birotunda [fits in 2 chiral ways]
ytrepus wrote:[...]
I would guess finding the non-composite solids is the hard part, but then combining them must be relatively straightforward. Is there any particular reason that these approaches couldn't be modified appropriately for 4d and then an algorithm used to identify every possibility, using the uniforms + Johnson solids as the only possible cells? Obviously there will be many more possibilities, but processing power is much higher than the 60s when the 3d cases were catalogued. Or am I massively oversimplifying here? Just from browsing it seems like the current effort is a little bit random (I am not in any way criticizing it!), rather than concentrating on developing an exhaustive algorithm to find a complete set.
[...]
Marek14 wrote:As for separating interesting CRF's from uninteresting, one way could be cell counts. The diminishings of 600-cell, for example, would all consist of tetrahedra and icosahedra and all diminishings by the same number would have identical cell counts for both. On the other hand, unique CRF's would also have unusual cell counts, either by amounts of certain cells or by the presence of unusual cells.
[...]
quickfur wrote:Marek14 wrote:As for separating interesting CRF's from uninteresting, one way could be cell counts. The diminishings of 600-cell, for example, would all consist of tetrahedra and icosahedra and all diminishings by the same number would have identical cell counts for both. On the other hand, unique CRF's would also have unusual cell counts, either by amounts of certain cells or by the presence of unusual cells.
[...]
But in order to get the cell counts in the first place, you'll have to construct the polytope first. Which means 99% of your computational power is spent constructing and then discarding these polytopes. OTOH there doesn't seem to be a way around this, since potentially a crown jewel could have parts that locally look identical to a 600-cell diminishing, but with other parts that are unique, so you wouldn't want to discard something just because one part looks like a 600-cell diminishing. But perhaps it's OK to discard such constructions, since the brute force algorithm will eventually get to cell combinations representing the more interesting part of the same polytope.
Which brings up another point: how to tell if a given polytope has already been constructed before. This seems to be reducible to the graph isomorphism problem, which is NP-complete. Which means large polytopes like the "uninteresting" 600-cell diminishings will waste most of the computational time just because we keep having to determine whether or not they're identical to something previously constructed. Combinational explosion is a nasty beast to tame.
student91 wrote:But what if we only checked for polytopes which would give CVP 3 polytopes? Then we would either prove CVP3's don't exist (except for prisms and other trivial things) or we would find loads of things that are "interesting."
I'm not sure how these skew verfs work yet, but I do know for sure that non-skew verfs that are CVP3 will need CVP3 cells. therefore as a start, we could search for things with cells that are CVP 3.
Marek14 wrote:[...]
Here's an interesting question: what is the upper limit for number of vertices of CRF polychoron? I would guess that there will be none (except very large members of infinite families) that will surpass 14,400 of omnitruncated 600-cell, just as there's no Johnson solid with more than 120 vertices of truncated icosidodecahedron.
But 600-cell diminishings in particular are limited to no more than 120 vertices, so perhaps the comparing wouldn't be that time-intensive.
Marek14 wrote:student91 wrote:But what if we only checked for polytopes which would give CVP 3 polytopes? Then we would either prove CVP3's don't exist (except for prisms and other trivial things) or we would find loads of things that are "interesting."
I'm not sure how these skew verfs work yet, but I do know for sure that non-skew verfs that are CVP3 will need CVP3 cells. therefore as a start, we could search for things with cells that are CVP 3.
Problem here is that every CVP3 Johnson solid has skew verfs. Only snub cube and snub dodecahedron don't (are they CVP3?).
Marek14 wrote:quickfur: Do we really have to check coordinates for isomorphism, though? Wouldn't the graph structure be enough?
Alternate way is to make an algorithm that would create a "canonic orientation" for a polychoron...
quickfur wrote:Marek14 wrote:quickfur: Do we really have to check coordinates for isomorphism, though? Wouldn't the graph structure be enough?
Alternate way is to make an algorithm that would create a "canonic orientation" for a polychoron...
Well, you said to use vertices to determine equivalence, so I thought you meant checking vertex coordinates? Or did I misunderstand what you meant?
In any case, you should be aware that it is known that the complexity of a polytope can be exponential w.r.t. the number of vertices, so having up to 120 vertices may actually be quite intractible! Not to mention, each non-adjacent 600-cell diminishing has corresponding counterparts in other uniform members in the 600-cell family, so we're looking at potentially the same number of diminishings applied to a uniform with many more vertices than merely 120.
I've thought about creating canonical orientations before... unfortunately, it turns out that it's equivalent to the graph isomorphism problem. So it does not give us any extra advantage over plain graph isomorphism itself. (If you think about it, creating a canonical layout of a graph will also solve the graph isomorphism problem, since once two graphs are in canonical orientation it's trivial to check for equivalence. Similarly, there's no essential difference between solving polytope equivalence vs. creating a canonical polytope orientation.)
Of course, just because graph isomorphism is NP-complete doesn't mean it's unsolvable; it just requires a lot of resources (both CPU power and time). Perhaps a distributed SETI-like project would be able to attack the problem effectively.
quickfur wrote:There's no need to prove anything, this is already a known result. Since we're dealing with CRFs, all edges are equal, and all cells are flat, so face lattice equivalence == polytope equivalence. But yes, you do need the entire face lattice (i.e., vertices, edges, polygons, cells); it's not enough to have only vertices and edges.
Marek14 wrote:quickfur wrote:There's no need to prove anything, this is already a known result. Since we're dealing with CRFs, all edges are equal, and all cells are flat, so face lattice equivalence == polytope equivalence. But yes, you do need the entire face lattice (i.e., vertices, edges, polygons, cells); it's not enough to have only vertices and edges.
I meant a proof whether the entire lattice is necessary under CRF limitations.
quickfur wrote:Marek14 wrote:quickfur wrote:There's no need to prove anything, this is already a known result. Since we're dealing with CRFs, all edges are equal, and all cells are flat, so face lattice equivalence == polytope equivalence. But yes, you do need the entire face lattice (i.e., vertices, edges, polygons, cells); it's not enough to have only vertices and edges.
I meant a proof whether the entire lattice is necessary under CRF limitations.
Oh I see. Hmm... I'm not sure, but I suspect in some cases you might need more than just vertices and edges? But in any case, if we're already constructing CRF candidates as cell complexes to begin with, then we should already be able to easily extract the face lattice, right?
quickfur wrote:I dunno, I think it amounts to the same thing. Working with cell complexes is equivalent to working with the vertices of the dual polytope, so I would expect the graph isomorphism problem to require about the same amount of effort. (Of course, there are cases where it's easier to with the dual vs. the original polytope, but if you amortize it across a large number of constructions, I'd expect both approaches to be roughly equal.)
Marek14 wrote:quickfur wrote:I dunno, I think it amounts to the same thing. Working with cell complexes is equivalent to working with the vertices of the dual polytope, so I would expect the graph isomorphism problem to require about the same amount of effort. (Of course, there are cases where it's easier to with the dual vs. the original polytope, but if you amortize it across a large number of constructions, I'd expect both approaches to be roughly equal.)
Though the question arises -- yes, there is huge amount of 600-cell diminishings, but what new things could the algorithm discover before even reaching their heights?
BTW, when I was looking at my list of vertex figures of 3D CRFs, I realized that pentagonal orthocupolarotunda contains vertices identical to those that appear in bilbro and thawro -- this would probably explain why this Johnson solid suddenly appeared in some figures.
Similarly, there should be a relationship between elongated square bipyramid and square orthobicupola, but of course we haven't found anything nontrivial that would contain these...
quickfur wrote:The elongated square bipyramid does occur in this cutie:
It's an augmentation of the cantellated 16-cell with octahedron||cuboctahedron.
I haven't tried constructing it yet, but I'm pretty sure a Stott expansion of this polychoron is possible, and will contain elongated square orthobicupola cells.
In general, the n-pyramid has a direct relationship with the n-cupola via Stott expansion perpendicular to its axis of symmetry, which means the dihedral angles between the bottom face and the triangular faces are identical in both the n-pyramid and the n-cupola, and their heights are also identical. This in turn means that there is a direct relationship between the 4D cube pyramid and the segmentochoron 8-prism||square, and between the square pyramid-pyramid and the square orthobicupolic ring, etc.. All of these are related via Stott expansions along their symmetry groups.
This also happens with 4D basis objects: a subsymmetrical Stott expansion of the 5-cell produces cuboctahedron||tetrahedron, for example. The resulting CRF retains the same dichoral angles between the cuboctahedron and tetrahedra as between the tetrahedra in the 5-cell; and it has the same height, etc.. Stott expansion is a very useful operator in producing more CRFs from existing CRFs.
Marek14 wrote:[...]
I was talking about unelongated square orthobicupola, though it's also an expansion of elongated square bipyramid, of course...
apacs<1, 1+√2, 1+√2, 1+√2>
apacs<0, √2, √2, 2+√2>
quickfur wrote:EDIT 2: This seems to be a Stott contraction of the cantellated 24-cell x3o4x3o.
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