General Approach--can 3D methods be generalized?

[Klitzing]

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...

You definitely should have a look here that! Showing up plenty of cases with octs <--> esquidpy <--> squobcu <--> sirco transitions within different dimensions and different geometries.

--- rk

[ytrepus]

Thank you everyone for the replies -- I think this is a really good discussion.

I still struggle to visualize a lot of this stuff, so can't really contribute in a massive massive way, but a few thoughts I had:

- If we focus an algorithm (as a first step) on identifying non-composite CRF, we don't have to worry about all of the million/billion-odd 600-cell family diminishings or duoprism augmentations clogging it up. By non-composite, I mean solids that cannot be sliced by a hyperplane into two distinct CRF figures. The 3D analogy would be slicing an octahedron down the center into two square pyramids, or in 4D, slicing a 16-cell into two octahedral pyramids.

- To find more interesting cases first, maybe an algorithm could be constructed in a way to start with, say a sphenocorona cell. We could try to figure out all possible non-composite figures containing a sphenocorona as a starting point, and then do the same for each of the Johnson solids and uniform polyhedra in turn, finishing up on the most difficult and prolific cases of tetrahedra and square pyramids, cupolas, etc. This is to try to find crown jewels that aren't derived from cutting Archimedeans, which we would already have on the list (see below). Obviously for all the reasons given above, constructing such an algorithm is not trivial, and may be impossible due to the systems of quadratics that someone mentioned above. But it eliminates the issues with millions of diminishings and also finds high-reward shapes earlier.

- Once we have a good number of non-composites, we start identifying the possible cells where the dichoral angles allow for combining, augmenting, gyrating, etc., building up from the smallest figures into the larger ones. Hopefully we'd get an inventory of, say 500 or fewer 4D figures (maybe that is too optimistic!) that we could use as building blocks, and then start to augment and combine them in different ways. Then it just becomes a combinatorial problem and only simple angle calculations are necessary. The hard part is over in step 1. Challenges at this stage would be volumes of augmentations (rather than diminishings since we are building from the ground up), and identifying repeats.

Do we know out of the 4D Archimedeans and polyhedral prisms etc, which ones are composite and what they are composed of? Can we start to fill out a chart like the below? And then start to look at augmentations and gyrations and any other operations specific to 4D? I think it would be worthwhile to compile an inventory specifically of non-composite solids. Theeeen we say how many CRFs are composed of no more than 2 non-composite solids, then 3, then 4, and keep building on these until we can't augment anymore.

[1] tetrahedron
composite: no

[2] cube
composite: no

octahedron
composite: yes
composition: two [3] square pyramids joined at {4} - {4}

[4] dodecahedron
composite: no

icosahedron
composite: yes
composition:
1) core 1: [5] pentagonal antiprism with two [6] pentagonal pyramids, one joined to each {5} face
2) core 2: [7] tridiminished icosahedron with three pentagonal pyramids, one joined to each {5} face

cuboctahedron
composite: yes
composition: two [8] triangular cupolas joined {6} - {6}, with {3} and {4} forming a common edge (gyro)

icosidodecahedron
composite: yes
composition: two [9] pentagonal rotundas joined {10} - {10}, with {3} and {5} forming a common edge (gyro)

[10] truncated tetrahedron
[11] truncated cube
[12] truncated octahedron
[13] truncated dodecahedron
[14] truncated icosahedron
composite: no

rhombicuboctahedron
composite: yes
composition: [15] octagonal prism with two [16] square cupolas, one joined to each {8} face, square edges lining up (ortho)

rhombicosidodecahedron
composite: yes
composition:
1) core 1: [17] parabidiminished rhombicosidodecahedron with two pentagonal cupolas, one joined to each {10} face, {3} meeting {4}
2) core 2: [18] tridiminished rhombicosidodecahedron with three pentagonal cupolas, one joined to each {10} face, {3} meeting {4}

[19] truncated cuboctahedron
[20] truncated icosidodecahedron
[21] snub cuboctahedron
[22] snub icosidodecahedron
n-gonal prisms - [23] triangular, [24] pentagonal, [25] hexagonal, heptagonal (not used), octagonal (already counted), enneagonal (not used), [26] decagonal
n-gonal antiprisms - [27] square, pentagonal (already counted), [28] hexagonal, heptagonal (not used), [29] octagonal, enneagonal (not used), [30] decagonal
composite: no

other non-composites:
[31] snub disphenoid
[32] snub square antiprism
[33] sphenocorona
[34] sphenomegacorona
[35] hebesphenomegacorona
[36] disphenocingulum
[37] bilunabirotunda
[38] triangular hebesphenorotunda

And to get started in 4D:

[1] 5-cell
composite: no

[2] tesseract
composite: no

16-cell
composite: yes
composition: two [3] octahedral pyramids joined oct-oct

24-cell
composite: ?

120-cell
composite: no?

600-cell
composite: yes
composition: ?

Archimedeans:

rectified 5-cell
composite: ?

rectified tesseract
composite: ?

rectified 16-cell=24-cell

rectified 24-cell
composite: ?

rectified 120-cell
composite: ?

rectified 600-cell
composite: ?

truncated 5-cell
composite: ?

truncated tesseract
composite: ?

truncated 16-cell
composite: ?

truncated 24-cell
composite: ?

truncated 120-cell
composite: ?

truncated 600-cell
composite: ?

cantellated 5-cell
composite: ?

cantellated tesseract
composite: ?

cantellated 16-cell = rectified 24-cell

cantellated 24-cell
composite: ?

cantellated 120-cell
composite: ?

cantellated 600-cell
composite: ?

runcinated 5-cell
composite: ?

runcinated tesseract / 16-cell
composite: ?

runcinated 24-cell
composite: ?

runcinated 120-cell / 600-cell
composite: ?

bitruncted 5-cell
composite: ?

bitruncated tesseract / 16-cell
composite: ?

bitruncated 24-cell
composite: ?

bitruncated 120-cell / 600-cell
composite: ?

cantitruncated 5-cell
composite: ?

cantitruncated tesseract
composite: ?

cantitruncated 16-cell = truncated 24-cell

cantitruncated 24-cell
composite: ?

cantitruncated 120-cell
composite: ?

cantitruncated 600-cell
composite: ?

runcitruncated 5-cell
composite: ?

runcitruncated tesseract
composite: ?

runcitruncated 16-cell
composite: ?

runcitruncated 24-cell
composite: ?

runcitruncated 120-cell
composite: ?

runcitruncated 600-cell
composite: ?

omnitruncted 5-cell
composite: ?

omnitruncated tesseract / 16-cell
composite: ?

omnitruncated 24-cell
composite: ?

omnitruncated 120-cell / 600-cell
composite: ?

snub 24-cell/snub demitesseract
composite: ?

grand antiprism
composite: ?

polyhedral prisms, antiprism prisms, duoprisms, etc...

Could someone help fill out the rest? Remember we don't want to get bogged down in the myriad ways to diminish the 120-cell family, just get down to the cores + components. Should be a simple rule e.g. non-adjacent icosahedral pyramids can be removed or whatever the components are, leaving possible cores of xxx or yyy. We shouldn't even mention the intermediate cases as long as they are composite.

Which of the segmentochora, ursachora, J91/J92 CRFs etc already discovered are composite and what are their components? Which are non-composite that would then go into the inventory of 'building blocks'?

Thanks a lot everyone.
Rob

[Marek14]

Well, many of the uniform polychora can be diminished.

First of all, any diminishing will slice through some vertices, dividing them. This means that you can't diminish polychora whose vertices are of degree 4 (there's nothing simpler to slice them into). These are:

5-cell
Tesseract
120-cell
Truncated 5-cell
Truncated tesseract
Truncated 24-cell
Truncated 120-cell
Bitruncated 5-cell
Bitruncated tesseract/16-cell
Bitruncated 24-cell
Bitruncated 120-cell/600-cell
Cantitruncated 5-cell
Cantitruncated tesseract
Cantitruncated 24-cell
Cantitruncated 120-cell
Cantitruncated 600-cell
Omnitruncated 5-cell
Omnitruncated tesseract/16-cell
Omnitruncated 24-cell
Omnitruncated 120-cell/600-cell

As for others:
16-cell: can be cut in two octahedral pyramids -- and note that octahedral pyramid itself can be also cut.
24-cell: you can cut off cube pyramids (eventually leading to tesseract, OR to some other maximal diminishings), or you can cut it in two octahedron || cuboctahedron.
600-cell: you can cut off icosahedral pyramids. There are also the hemi-600 cell cuttings which are more complex diminishings. Basically, 600-cell is the only uniform polychoron where you can connect vertices in ways that don't lie on surface of the vertex figure (icosahedron) -- grand antiprism is another example of such complex diminishing.
Truncated 16-cell: can be cut in two octahedron-truncated octahedron rotundas.
Truncated 600-cell: you can cut off tristratic polychora icosahedron || truncated icosahedron.
Rectified 5-cell: can be cut in triangular prismatic pyramid and a triangle || gyrated triangular prism.
Rectified tesseract: can be cut in two tetrahedron || truncated tetrahedron plus one tetrahedron || dual tetrahedron.
Rectified 24-cell: you can cut off segmentochora cuboctahedron || truncated octahedron.
Rectified 120-cell: you can cut off tristratic polychora icosidodecahedron || truncated dodecahedron
Rectified 600-cell: this is a complex case. First of all, you can cut off icosahedral caps, but you can do so at two different depths: either as segmentochora icosahedron || icosidodecahedron or as tristratic icosahedron || icosidodecahedron || rhombicosidodecahedron. And in addition to this, it's also possible to cut off pentagonal prismatic pyramids off the vertices. Both (or all three) approaches can be probably combined, I doubt there would be a single maximally diminished rectified 600-cell.
Cantellated 5-cell: first of all, it can be cut in two segmentochora, cuboctahedron || truncated tetrahedron and octahedron || truncated tetrahedron. However, you can also slice another segmentochoron: triangle || hexagonal prism, leaving tristratic triangular prism || hexagonal prism.
Cantellated tesseract: can be cut into two rhombicuboctahedron || truncated cube plus a truncated cube prism. There is also option to slice off square || octahedral prism.
Cantellated 24-cell: you can cut off its cuboctahedral cells, and you can do so in two different depths: either as cuboctahedron || truncated cube, or as tristratic cuboctahedron || truncated cube || truncated cuboctahedron.
Cantellated 120-cell: you can cut off segmentochora rhombicosidodecahedron || truncated dodecahedron. You can also cut off pentagon || decagonal prism.
Cantellated 600-cell: you can cut off its icosidodecahedral cells, and you can do so in two different depths: either as icosidodecahedron || truncated icosahedron, or as tetrastratic icosidodecahedron || truncated icosahedron || truncated icosidodecahedron.
Runcitruncated 5-cell: can be cut into truncated tetrahedron || truncated octahedron and tristratic cuboctahedron || truncated octahedron.
Runcitruncated tesseract: can be cut in two tristratic rhombicuboctahedron || truncated cuboctahedron and a truncated cuboctahedral prism.
Runcitruncated 16-cell: you can cut off segmentochora truncated cube || truncated cuboctahedron.
Runcitruncated 24-cell: you can cut off segmentochora truncated octahedron || truncated cuboctahedron.
Runcitruncated 120-cell: you can cut off tristratic thombicosidodecahedron || truncated icosidodecahedron.
Runcitruncated 600-cell: you can cut off truncated dodecahedron || truncated icosidodecahedron.
Runcinated 5-cell: can be cut in two tetrahedron || cuboctahedron.
Runcinated tesseract/16-cell: can be cut in two segmentochora cube || rhombicuboctahedron and a rhombicuboctahedral prism.
Runcinated 120-cell/600-cell: you can cut off segmentochora dodecahedron || rhombicosidodecahedron.
Runcinated 24-cell: you can cut off segmentochora octahedron || rhombicuboctahedron.

Grand antiprism and snub 24-cell are both types of 600-cell diminishings.

[Klitzing]

- If we focus an algorithm (as a first step) on identifying non-composite CRF, we don't have to worry about all of the million/billion-odd 600-cell family diminishings or duoprism augmentations clogging it up. By non-composite, I mean solids that cannot be sliced by a hyperplane into two distinct CRF figures. The 3D analogy would be slicing an octahedron down the center into two square pyramids, or in 4D, slicing a 16-cell into two octahedral pyramids.

- Once we have a good number of non-composites, we start identifying the possible cells where the dichoral angles allow for combining, augmenting, gyrating, etc., building up from the smallest figures into the larger ones. Hopefully we'd get an inventory of, say 500 or fewer 4D figures (maybe that is too optimistic!) that we could use as building blocks, and then start to augment and combine them in different ways. Then it just becomes a combinatorial problem and only simple angle calculations are necessary. The hard part is over in step 1. Challenges at this stage would be volumes of augmentations (rather than diminishings since we are building from the ground up), and identifying repeats.

Hey Rob, this is exactly the way once Victor Zalgaller 1969 managed to proove the completeness of the set of Johnson solids in 3D (a brief outline and reference to the russian paper can be found e.g. here): he first identified the set of elemental 28 (besides of prisms and antiprisms), and then constructed therefrom all other convex ones.

For reference here is his set of 28:

1. M1 = tet = x3o3o
2. M2 = squippy = J1
3. M3 = peppy = J2
4. M4 = tricu = J3
5. M5 = squacu = J4
6. M6 = pecu = J5
7. M7 = teddi = J63
8. M8 = bilbiro = J91
9. M9 = pero = J6
10. M10 = tut = x3x3o
11. M11 = tic = o3x4x
12. M12 = tid = o3x5x
13. M13 = tedrid = J83
14. M14 = pabidrid = J80
15. M15 = doe = o3o5x
16. M16 = toe = x3x3x = x3x4o
17. M17 = girco = x3x4x
18. M18 = grid = x3x5x
19. M19 = ti = x3x5o
20. M20 = thawro = J92
21. M21 = hawmco = J89
22. M22 = waco = J86
23. M23 = wamco = J88
24. M24 = dawci = J90
25. M25 = snadow = J84
26. M26 = snic = s3s4s
27. M27 = snid = s3s5s
28. M28 = snisquap = J85

--- rk

[Klitzing]

Well, many of the uniform polychora can be diminished. ...

Well, Marek, Rob's (and Zalgaller's) approach were not so much to consider whether and how polytopes might be dissected, but what would be exactly the set of maximal diminishings each.

This first task of mostly mere collections would be quite managable, I think.
The second step would then be to set up an algorithm to search for all elemental (= non-divisible) CRFs (which, as a check should at least reproduce the set of step one).

I think this step 2 would be large enough to stop here, at least with respect to any complete enummeration. So, the full set of CRFs then would be derived in a third step, by iterated adjoins of the formers.

--- rk

[Marek14]

The question is how many distinct maximal diminishings exists for 600-cell symmetry group. Then there's additional wrinkle of "nonintuitive" diminishings of 600-cell like grand antiprism...
First we should probably enumerate subsymmetric convex facetings of icosahedron to see what is possible.

[Klitzing]

A complete enumeration of what I defind to be the first step is desirable (for cross-check) but not necessary, esp. in those cases with huge sets of elementals. The second step (as I defined it) then should produce all these too in an automated way...

--- rk

[quickfur]

I spent 2 hours typing up a response last night, only to hit the wrong button and close the browser window, thus losing everything I typed. So here's the short(er) version of my reply:

Firstly, I have already begun research in this direction for about a year or two, maybe a bit more, at least among the uniform polychora. I called them "maximal diminishings": basically, the smallest CRF pieces that can be obtained by deleting vertices from uniform polychora, such that no more vertices can be deleted from them without (1) becoming non-CRF, and (2) changing their edge length. (2) is basically to prevent trivialities like deleting vertices from an 8,8-duoprism to make a tesseract (of different edge length), whereas an 8,8-duoprism really should be considered a minimal diminishing. I'll post the list that I found sometime later today maybe -- I'm reasonably confident that I have the full list for the 5-cell uniforms and the 16-cell/tesseract uniforms. I started on the 24-cell uniforms and 120-cell uniforms, but I'm not sure I have all of the 24-cell ones, and definitely sure I don't have all the 120-cell ones because there are too many ways to diminish those things. But many of the CRFs I posted before the discovery of the BT polytopes were discovered as a result of this research into minimal diminishings.

Second, there may not be a unique minimal diminishing for a given CRF, or at least, not a minimal CRF one. For example, the tetraaugmented J92 rhombochoron can be diminished into the J92 rhombochoron or D4.4.2, but the latter two (AFAIK) do not have a common CRF core.

Thirdly, our current definition of composability seems to encompass only naïve cut-n-paste operations; we should consider whether or not Stott expansion should be considered as part of our set of fundamental operations, because it allows derivations where no obvious cut-n-paste operation would work, such as the Stott expansion of the castellated rhombicosidodecahedral prism (D4.3.1) to its expanded brother D4.3.2. Since D4.3.2 is trivially derived from D4.3.1, it seems better if we don't have to include D4.3.2 among the set of fundamental CRFs; so this requires adding Stott expansion to our set of fundamental operations.

Finally, a few random observations for @ytrepus: J91 (bilunabirotunda) and J92 (triangular hebesphenorotunda) have been found to be derivable via Stott expansion of certain (admittedly non-obvious) facetings of the icosahedron. So they are, in a sense, "less fundamental" than, say, the snub disphenoid, for which there is no such derivation (at least as far as we can tell). While in 3D there are very few such cases, such derivations are quite numerous in 4D, namely the class of BT polychora (or at least a significant subset thereof).

[quickfur]

Alright, maybe I'll just post my maximal diminishings results piecemeal instead of all at once. I'll start with the 5-cell family, which I'm reasonably sure is complete. (Or, if it's not, at least it's a good starting point. ) I'll go by CD symbol:

o3o3o3x (5-cell): no diminishing is possible without becoming subdimensional.

o3o3x3o (rectified 5-cell): there are two maximal diminishings:
- the bidiminished rectified 5-cell (same as segmentochoron K4.8.2);
- the triangular prism pyramid (segmentochoron K4.7.2) -- the piece that's cut off each time the rectified 5-cell is diminished.

o3o3x3x (truncated 5-cell): no CRF diminishings possible without becoming subdimensional.

o3x3o3x (cantellated 5-cell):
- triangle||hexagonal_prism (K4.45.1): a monostratic cap.
- octahedron||truncated_tetrahedron: another possible monostratic cap to cut off.
- the bi-(K4.45.1)-diminished cantellated 5-cell: equivalently, this is to delete the vertices of two triangles from the polytope, resulting in a CRF with two adjacent hexagonal prisms. Described in this post.

o3x3x3o (bitruncated 5-cell): no known CRF diminishings.

o3x3x3x (cantitruncated 5-cell): no known CRF diminishings.

x3o3o3x (runcinated 5-cell):
- K4.24 (tetrahedron||triangular_cupola)
- K4.25 (trigon||triangular_cupola) - the piece that's cut off twice to make K4.24 (if it's only cut off once, the result is the non-minimal tetrahedron||cuboctahedron).

x3o3x3x (runcitruncated 5-cell):
- truncated tetrahedron||truncated octahedron: a monostratic cap.
- diminished runcitruncated 5-cell: what's left after cutting of this monostratic cap.

x3x3x3x (omnitruncated 5-cell): no known CRF diminishings.

[quickfur]

Tesseract/16-cell family:

o4o3o3x (16-cell):
- maximal diminishing is the square pyramid pyramid (square_pyramid||point): a 16-cell can be cut into 4 of these (the octahedral pyramid mentioned by @ytrepus is actually non-minimal).

o4o3x3o (same as 24-cell: I'll post this later).

o4o3x3x (truncated 16-cell):
- Octahedral rotunda (bisected truncated 16-cell).

o4x3o3o (rectified tesseract):
- tetrahedron||truncated tetrahedron: a monostratic cap
- parabidiminished rectified tesseract: what's left after two such caps, lying on parallel hyperplanes, are cut off
- metabidiminished rectified tesseract: what's left after two non-parallel caps are cut off (to be confirmed: I haven't actually constructed this to verify its existence).

o4x3o3x (same as o3o4x3o; will post this later in 24-cell family).

o4x3x3o (bitruncated tesseract): no known CRF diminishings.

x4o3o3o (tesseract): no known CRF diminishings. (Well, you could alternate it to make a 16-cell, but that changes the edge length so doesn't qualify under my working definition of maximal diminishing.)

x4o3o3x (runcinated tesseract): can be stratified into several CRF layers:
- K4.73 (square orthobicupolic ring)
- square cupola prism (K4.69)
- 8,4-duoprism

x4o3x3o (cantellated tesseract): this one has a whole bunch of diminishings, not 100% sure I got all the maximal ones.
- x4o3x || truncated_cube
- truncated cube prism
- square magnabicupolic ring (octagonal prism || square)
- 1,1,(5,8)-tetradiminished cantellated tesseract (basically corresponding to a maximal tetradiminishing of the 24-cell)
- 1,(1,2,3),6-pentadiminished cantellated tesseract (corresponding to a maximal pentadiminishing of the 24-cell)
- 8,8-duoprism (aka octadiminished cantellated tesseract).

x4o3x3x (runcitruncated 16-cell):
- rhombicuboctahedron rotunda.
- great rhombicuboctahedron prism.
(Basically, x4o3x3x stratifies into two copies of the rotunda + the x4x3x prism in the middle.)

x4x3o3o (truncated tesseract): no known CRF diminishing

x4x3o3x (runcitruncated tesseract):
- x4x3x || x4x3o: monostratic cap
- bidiminished x4x3o3x: basically what's left after two parallel caps are cut off from the polytope.

x4x3x3o (cantitruncated tesseract): no known CRF diminishing.

x4x3x3x (omnitruncated tesseract): no known CRF diminishing.

[Marek14]

quickfur: Would you agree that it's not possible to diminish polychoron with all vertices of degree 4? The logic is that vertex figure of a vertex of diminishing is diminishing of the vertex figure of the original vertex (which must be convex, but, of course, is not required to be CRF) and vertices of degree 4 have tetrahedron as their verf, which can't be diminished.

[ytrepus]

Ah cool, thanks everyone for the replies -- looking forward to going through in more detail later, and trying to visualize some of this stuff.

Just a quick note on the Stott expansions -- think we'll have to consider both the examples you mentioned as 'non-composite'--for the purposes of an algorithm, the Stott expansion is more like other polyhedral operations like cantellation, truncation, snubification etc rather than a cut and paste in the way that would need to be considered for any simple algorithm to work. The program would have to identify cells that can be used to augment, and it wouldn't find Stott expansions in the same way.

Thanks again

[quickfur]

quickfur: Would you agree that it's not possible to diminish polychoron with all vertices of degree 4? The logic is that vertex figure of a vertex of diminishing is diminishing of the vertex figure of the original vertex (which must be convex, but, of course, is not required to be CRF) and vertices of degree 4 have tetrahedron as their verf, which can't be diminished.

I'm not sure I understand your reasoning here. A tesseract has vertices of degree 4, yet we can make a truncated tesseract (or, for that matter, a rectified tesseract). Of course, this isn't the same as a diminishing, but I'm not sure I see the connection between vertices of degree 4 and the impossibility of a diminishing. A tetrahedral bipyramid, for example, can have a vertex of degree 4 diminished (the result is a 5-cell).

[Marek14]

quickfur: Would you agree that it's not possible to diminish polychoron with all vertices of degree 4? The logic is that vertex figure of a vertex of diminishing is diminishing of the vertex figure of the original vertex (which must be convex, but, of course, is not required to be CRF) and vertices of degree 4 have tetrahedron as their verf, which can't be diminished.

I'm not sure I understand your reasoning here. A tesseract has vertices of degree 4, yet we can make a truncated tesseract (or, for that matter, a rectified tesseract). Of course, this isn't the same as a diminishing, but I'm not sure I see the connection between vertices of degree 4 and the impossibility of a diminishing. A tetrahedral bipyramid, for example, can have a vertex of degree 4 diminished (the result is a 5-cell).

Vertex of degree 4 can be removed, but any sort of diminishing will reduce the degree of vertices adjacent to the removed one. So, if all vertices are degree 4, which is minimal, no more diminishing is possible.

[Klitzing]

(answering this post of quickfur:)

at 1)   I knew. That was one of the reasons, why I brought it in.

at 2)   known too, but this is not the point. We just want to get a compilation of to be estimated outcomes for the automatized research for cross-check. So it is completely unimportant how this listing is derived, whether just one or severals are obtained by potential dissectings of each individual known CRF (including Wythoffians).

at 3)   my first knee-jerk would be: no, as the Zalgaller way of prooving just listed the non-dissectables and then proceeded by iterated glueing.
In an afterthought however it might be a way to decrease the total count of "elementaries" once more. - But then one ought to have in mind, that, before applying the iterated glueing step, exactly this benefit then has to be undone again, as the full roll-out of both, contracted and expanded ones, are to be checked for possible glueings. I don't think that an on-the-fly generation of the expanded versions each would be a wise thing, so. - Thus it seems a rather temporary benefit only, admissible perhaps when having in mind some publication of all elementary ones, provided this computer aided research outcome would become too large for that purpose.

--- rk

[quickfur]

Ah cool, thanks everyone for the replies -- looking forward to going through in more detail later, and trying to visualize some of this stuff.

Shameless self-plug: http://eusebeia.dyndns.org/4d/vis/vis

Just a quick note on the Stott expansions -- think we'll have to consider both the examples you mentioned as 'non-composite'--for the purposes of an algorithm, the Stott expansion is more like other polyhedral operations like cantellation, truncation, snubification etc rather than a cut and paste in the way that would need to be considered for any simple algorithm to work. The program would have to identify cells that can be used to augment, and it wouldn't find Stott expansions in the same way.
[...]

True, while Stott expansions are easy to construct programmatically (in fact, that is how I derived the coordinates for the 120-cell family uniform polychora that I posted on my website), they are non-trivial to be recognized as a Stott expansion if you're given an already-constructed polytope. (I.e., given a runcinated tesseract, it's not trivial for a program to know that it can be derived from a tesseract via Stott expansion.) So perhaps we will have to consider them as non-composite.

In any case, the way I see it is, restricting the search to only non-composite CRFs is only useful if it is easy to implement -- that is, there is an easy way to determine if a particular cell complex is composite or not, or somehow detect and eliminate candidates that will be composite without having to completely construct them in the first place. Otherwise, it doesn't save us anything -- the program will still have to go through all possible combinations, only to discard some of them afterwards.

Now, assuming we always cut CRFs into smaller CRFs, this means that a given polytope is composite if there exists a subset of co-hyperplanar vertices that (1) are the vertices of a 3D CRF (it can be Johnson, Archimedean, or Platonic, or a prism/antiprism), and (2) the convex hulls of the vertices on either side of the hyperplane are CRF. Since the brute force algorithm will be constructing CRF candidates from ground up, this is equivalent to saying that whenever we add a new vertex to the cell complex, we have to check if it is co-hyperplanar with some subset of the current vertices, such that this hyperplane does not lie on the surface of the (partially-constructed) polytope. If it is, then we have to determine if it forms a CRF with the vertices on that hyperplane; and if it does, we have to check if the vertices remaining on either side form smaller CRF fragments. I'm not confident how much effort this will save us, actually -- checking for coplanarity with existing vertices seems particularly troublesome because, programmatically, you can't tell without checking almost every possible subset of the existing vertices, which is an exponential algorithm.

Perhaps another approach to brute forcing is a breadth-first search: instead of starting with some chosen initial cell and then trying to close it up, we maintain a queue of partially-constructed cell complexes, and in each iteration of the program, we remove the cell complex at the head of the queue, and construct all possible further CRF extensions of it. If any of them result in a closed cell complex, we know we have a CRF, so output that. Otherwise, we put the incomplete fragment back into the queue. The queue itself will be a priority queue sorted by increasing total dichoral angles. The idea is that the head of the queue will always represent the fragment with the smallest dichoral angles, and therefore the most likely to close up soon, so that if it does close up, it will represent the smallest CRF constructible from that cell complex. Fragments with large dichoral angles generally tend to result in large polytopes, so the completion of those are postponed until we have exhausted the smaller CRFs first. The only potential killer of this idea is that it will require exorbitant amounts of storage space for the queue, because as the cell complexes grow in size, they will also grow exponentially in number.

[quickfur]

[...]
Vertex of degree 4 can be removed, but any sort of diminishing will reduce the degree of vertices adjacent to the removed one. So, if all vertices are degree 4, which is minimal, no more diminishing is possible.

Oooh I see. OK, that makes sense.

[Marek14]

Now, assuming we always cut CRFs into smaller CRFs, this means that a given polytope is composite if there exists a subset of co-hyperplanar vertices that (1) are the vertices of a 3D CRF (it can be Johnson, Archimedean, or Platonic, or a prism/antiprism), and (2) the convex hulls of the vertices on either side of the hyperplane are CRF. Since the brute force algorithm will be constructing CRF candidates from ground up, this is equivalent to saying that whenever we add a new vertex to the cell complex, we have to check if it is co-hyperplanar with some subset of the current vertices, such that this hyperplane does not lie on the surface of the (partially-constructed) polytope. If it is, then we have to determine if it forms a CRF with the vertices on that hyperplane; and if it does, we have to check if the vertices remaining on either side form smaller CRF fragments. I'm not confident how much effort this will save us, actually -- checking for coplanarity with existing vertices seems particularly troublesome because, programmatically, you can't tell without checking almost every possible subset of the existing vertices, which is an exponential algorithm.

Well, it's not really exponential, I think. You only need to check quartets of vertices. Four vertices generate a hyperplane to cut the polychoron, then you check which other vertices lie in there. If you remember which hyperplanes were already checked and which vertices were there, you can omit check for any quartet of vertices that is included in a checked hyperplane.

This algorithm would be quartic to quintic on vertex number (quartic polynom for number of quartets to check, then potentially one more degree for checking inclusion of all other vertices in the hyperplane).

Also, the last check you list ("we have to check if the vertices remaining on either side form smaller CRF fragments") might be superfluous. If the whole polychoron is CRF and there is a possible CRF cut, then there doesn't seem any possibility for non-CRF fragments.

[quickfur]

Now, assuming we always cut CRFs into smaller CRFs, this means that a given polytope is composite if there exists a subset of co-hyperplanar vertices that (1) are the vertices of a 3D CRF (it can be Johnson, Archimedean, or Platonic, or a prism/antiprism), and (2) the convex hulls of the vertices on either side of the hyperplane are CRF. Since the brute force algorithm will be constructing CRF candidates from ground up, this is equivalent to saying that whenever we add a new vertex to the cell complex, we have to check if it is co-hyperplanar with some subset of the current vertices, such that this hyperplane does not lie on the surface of the (partially-constructed) polytope. If it is, then we have to determine if it forms a CRF with the vertices on that hyperplane; and if it does, we have to check if the vertices remaining on either side form smaller CRF fragments. I'm not confident how much effort this will save us, actually -- checking for coplanarity with existing vertices seems particularly troublesome because, programmatically, you can't tell without checking almost every possible subset of the existing vertices, which is an exponential algorithm.

Well, it's not really exponential, I think. You only need to check quartets of vertices. Four vertices generate a hyperplane to cut the polychoron, then you check which other vertices lie in there. If you remember which hyperplanes were already checked and which vertices were there, you can omit check for any quartet of vertices that is included in a checked hyperplane.

This algorithm would be quartic to quintic on vertex number (quartic polynom for number of quartets to check, then potentially one more degree for checking inclusion of all other vertices in the hyperplane).

You're right, only quartets of vertices need to be checked. Still, that's a pretty hefty cost, considering that it's done for each new vertex added (or each time a cell complex is extended).

Also, the last check you list ("we have to check if the vertices remaining on either side form smaller CRF fragments") might be superfluous. If the whole polychoron is CRF and there is a possible CRF cut, then there doesn't seem any possibility for non-CRF fragments.

No, it's not superfluous. Remember that we're not considering CRF cuts per se, but just subsets of vertices that are coplanar. Consider the castellated x5o3x prism. The middle layer of vertices are (by definition!) co-hyperplanar, but obviously that's not a CRF cut, even though the vertices themselves have a CRF hull, because that bisects the bilbiro's (and bilbiro's have no CRF bisections).

[Marek14]

No, it's not superfluous. Remember that we're not considering CRF cuts per se, but just subsets of vertices that are coplanar. Consider the castellated x5o3x prism. The middle layer of vertices are (by definition!) co-hyperplanar, but obviously that's not a CRF cut, even though the vertices themselves have a CRF hull, because that bisects the bilbiro's (and bilbiro's have no CRF bisections).

Maybe the check could be for being CRF and having a unit edge?

[quickfur]

No, it's not superfluous. Remember that we're not considering CRF cuts per se, but just subsets of vertices that are coplanar. Consider the castellated x5o3x prism. The middle layer of vertices are (by definition!) co-hyperplanar, but obviously that's not a CRF cut, even though the vertices themselves have a CRF hull, because that bisects the bilbiro's (and bilbiro's have no CRF bisections).

Maybe the check could be for being CRF and having a unit edge?

Hmm. I need to think about this more carefully. Suppose V is the set of vertices. If a subset H of V lies on a single hyperplane, then we can partition V = A union H union B, where A, H, and B are disjoint. Suppose the convex hull of H is CRF. Does that guarantee that the hulls of A and B will be convex as well? Might there be a case where there is, say, a bilbiro cell that lies across A, H, and B, such that A or B by itself would end up with half a bilbiro cell which is non-CRF? (I'm not thinking about the castellated prism specifically, but CRFs in general. Is there a possible case where there might be some oblique orientation of a bilbiro cell across A, H, B such that H is CRF?)

While it seems intuitively tempting to assume that there is no such case, I have a hard time proving it rigorously. Do you have a proof?

[student91]

[...]
Hmm. I need to think about this more carefully. Suppose V is the set of vertices. If a subset H of V lies on a single hyperplane, then we can partition V = A union H union B, where A, H, and B are disjoint. Suppose the convex hull of H is CRF. Does that guarantee that the hulls of A and B will be convex as well? Might there be a case where there is, say, a bilbiro cell that lies across A, H, and B, such that A or B by itself would end up with half a bilbiro cell which is non-CRF? (I'm not thinking about the castellated prism specifically, but CRFs in general. Is there a possible case where there might be some oblique orientation of a bilbiro cell across A, H, B such that H is CRF?)

While it seems intuitively tempting to assume that there is no such case, I have a hard time proving it rigorously. Do you have a proof?

Proving it can be done. First, we could prove that for every 3D-CRF if there is such a H, this always works. case-by-case or something. then, in 4D, H is made of polygons. these polygons are either faces of V, or they are internal to a cell. if they are, this polygon is a partition H of the 3d-cell, and this cell can thus be bisected along this polygon as well. this means that you can validly cut along H. (It just occurred to me that the 3D case can be proven in a similar way, just that H won't have lines that are internal to an polygon, as these don't exist.

[quickfur]

But how do you prove that the polygons of H must be either an external face or internal to some cell? For example, an icosidodecahedral subset of the 600-cell's vertices has pentagonal faces, but these pentagons are neither a face of the 600-cell, nor internal to any single cell (in fact, they bisect 5-fold florets of tetrahedra in a non-CRF way).

[quickfur]

Continuing from my results of maximal CRF diminishings, here are the results I got so far from the 24-cell family (looking at my notes, it seems I didn't finish going through all the 24-cell uniforms yet, so this is definitely only a partial list):

o3o4o3x (24-cell):
- 1,1,(5,8)-tetradiminished 24-cell
- 1,(1,2,3),6-pentadiminished 24-cell
- square||cuboctahedron
- octahedron||hexagon (== triangle || gyro-3-cupola): the smallest 24-cell luna, IIRC
- cube pyramid (the monostratic cap that's cut off)
- tesseract (as octadiminished 24-cell).

o3o4x3o (rectified 24-cell):
- cuboctahedron||truncated octahedron: monostratic cap to be cut off each time
- metatridiminished rectified 24-cell

o3o4x3x (truncated 24-cell): no known CRF diminishings

o3x4o3x (cantellated 24-cell):
- bidiminished runcitruncated tesseract (x4x3o3x)
- metabathotridiminished cantellated 24-cell (maximality to be checked -- basically, there are two possible depths of diminishing; this one corresponds with 3 deeper cuts that produce a CRF with 3 great rhombicuboctahedra joined to each other in a triangular fashion)
- cuboctahedron||truncated_cube: the monostratic cap cut off with the shallow diminishing
- truncated_cube||x4x3x: the second strata that can be cut off underneath cuboctahedron||truncated_cube to make a bathodiminishing.

o3x4x3o (bitruncated 24-cell): no known CRF diminishings

o3x4x3x (cantitruncated 24-cell): no known CRF diminishings

x3o4o3x (runcinated 24-cell):
- metatridiminished runcinated 24-cell (not constructed explicitly, so not sure if this is minimal)
- ... (incomplete, there may be more minimal diminishings)

x3o4x3x: not investigated yet

x3x4x3x: not investigated yet (though seems unlikely to have CRF diminishings due to cell shapes)

//

Also, the 120-cell family is only barely explored, so I don't really have any claims to minimal diminishings beyond what is already commonly known. The most obvious ones are the 1/10 luna of the 600-cell, which can be diminished into a pentagonal birotundular ring (consisting of a pentagonal antiprism surrounded by concentric rings of pentagonal pyramids and tetrahedra, sandwiched between two pentagonal rotundae); bidex, which is scaliform, the grand antiprism which is uniform, and a few others (which may not be minimal -- that's yet to be investigated).

[student91]

Note that the article about the 600-cell cuts also has a table with a horrible number of "maximal cuts." Of course for the 600-cell, these are not truely maximal, but for some other 600-cell uniforms, they are. Anyway, It's a huge amount of them, meaning the 600-cell symmetry also thwarts this approach. Can't we then just search for CVP3's? I think that's much easier, and will either give a desired proof, or give interesting polytopes. I guess I'll investigate this when I'm done with the partial expansions, although I don't guarantee that. For the non-600-cell things, I think all work is usefull.

[student91]

But how do you prove that the polygons of H must be either an external face or internal to some cell? For example, an icosidodecahedral subset of the 600-cell's vertices has pentagonal faces, but these pentagons are neither a face of the 600-cell, nor internal to any single cell (in fact, they bisect 5-fold florets of tetrahedra in a non-CRF way).

I frankly don't know that. I think this case is rather case-specific, although the x5x3f-cut of the runcinated 600-cell behaves similarly. these weird cuts apparently only occur at things that look like %%%5%%%3%%%&#x. if this is true, it is intuitively proven, but I don't know if it is.

[quickfur]

Note that the article about the 600-cell cuts also has a table with a horrible number of "maximal cuts." Of course for the 600-cell, these are not truely maximal, but for some other 600-cell uniforms, they are. Anyway, It's a huge amount of them, meaning the 600-cell symmetry also thwarts this approach. Can't we then just search for CVP3's? I think that's much easier, and will either give a desired proof, or give interesting polytopes. I guess I'll investigate this when I'm done with the partial expansions, although I don't guarantee that. For the non-600-cell things, I think all work is usefull.

These are exactly the non-adjacent diminishings of the 600-cell that I quoted a few times. There's about 300 million of them. Now throw in adjacent CRF diminishings and various bisected/stratum-diminished 600-cells, and you'll easily increase that number by a few more orders of magnitude. I'm guessing there are at least trillions of CRF 600-cell diminishings, if not significantly more.

And now consider the fact that a significant subset of these diminishings have parallels in the other uniforms of the 600-cell family -- the cantellated 120-cell, for example, admits adjacent diminishings isomorphic to the 600-cell diminishings -- and we're talking about crazy amounts of CRFs just from the 600-cell family uniforms alone. And that's not even talking about potential bilbiro'ings and thawro'ings compounded into various diminished 600-cell uniforms. A lot of them will probably exhibit what locally looks like unique cell combinations, but when you build up the polytope and close it up, you discover it's yet another darned 600-cell family diminishing, thrice thawro'd and twice bilbiro'd, compressed, and then augmented with some arbitrary segmentochoron.

So yeah, I second the motion to start with CVP3 constructions first. I think brute force search is going to be infeasible unless we find ways of greatly reducing the search space, probably via theorems about what constructions will/will not work.

[quickfur]

But how do you prove that the polygons of H must be either an external face or internal to some cell? For example, an icosidodecahedral subset of the 600-cell's vertices has pentagonal faces, but these pentagons are neither a face of the 600-cell, nor internal to any single cell (in fact, they bisect 5-fold florets of tetrahedra in a non-CRF way).

I frankly don't know that. I think this case is rather case-specific, although the x5x3f-cut of the runcinated 600-cell behaves similarly. these weird cuts apparently only occur at things that look like %%%5%%%3%%%&#x. if this is true, it is intuitively proven, but I don't know if it is.

Unfortunately, intuitively proven is not good enough. How do we know if a crown jewel of currently-unknown construction may also exhibit this effect?

As for your statement that all non-600-cell work is useful, consider the fact that my latest enumeration of duoprism augmentations, if indeed my program doesn't have hidden bugs, found almost 12 million augmentations with n-prism pyramids and 2n-prism||n-gon segmentochora. If we only search for minimal CRFs, then we don't have to worry about these, but still. Any initial construction of cell complexes may locally look unique, until you complete it and realize it's another one of those 12 million duoprism augmentations. (Not to mention, a good number of duoprisms are also augmentable with other pieces than n-prism pyramids and 2n-prism||n-gons, so if we're talking about the absolute total of CRF augmentations of duoprisms, we may be adding several orders of magnitude to the 12 million number. Combinatorial explosion is a nasty beast.)

Assuming, of course, that my program is actually correct. I'd appreciate it if somebody else independently verified my findings... (hint, hint, nudge, nudge). Especially since, being a programmer by career, I'm only too well aware that silly mistakes and not-so-obvious bugs creep into programs all the time.

[Klitzing]

Just a minor question, quickfur:

what do you refer by:

...
- 1,1,(5,8)-tetradiminished cantellated tesseract (basically corresponding to a maximal tetradiminishing of the 24-cell)
- 1,(1,2,3),6-pentadiminished cantellated tesseract (corresponding to a maximal pentadiminishing of the 24-cell)
...
- 1,1,(5,8)-tetradiminished 24-cell
- 1,(1,2,3),6-pentadiminished 24-cell
...

--- rk

[quickfur]

Just a minor question, quickfur:

what do you refer by:

...
- 1,1,(5,8)-tetradiminished cantellated tesseract (basically corresponding to a maximal tetradiminishing of the 24-cell)
- 1,(1,2,3),6-pentadiminished cantellated tesseract (corresponding to a maximal pentadiminishing of the 24-cell)
...
- 1,1,(5,8)-tetradiminished 24-cell
- 1,(1,2,3),6-pentadiminished 24-cell
...

--- rk

I knew that question would come up.

Basically, these numbers correspond with vertices on a 24-cell, and are assigned as follows. Take a 24-cell, and stratify it as point || cube || q-octahedron || cube || point. For the cube layers, we number the vertices thus:

Code: Select all

          1-----2
         /|    /|
        3-----4 |
        | 5---|-6
        |/    |/
        7-----8


For the q-octahedron layer, we use this labelling:

Code: Select all

            1
            | 3'
            |/
        2---+---2'
           /|
          3 |
            1'


Now, since a diminishing of the 24-cell always has at least one deleted vertex, we may simply assign that vertex to the first point. Thus the designation of the diminishing always begins with "1,...". Furthermore, only non-adjacent diminishings are admissible, so the second stratum (the first cube) is never diminished -- it is adjacent to the first point. So we may skip over the second stratum altogether. Thus, the second item in the designation refers to one of the q-octahedron vertices (which are all non-adjacent to each other, since they do not lie along a single cell, but are separated by the cube vertices on either side, so adjacent indices on the q-octahedron can be diminished). The third and fourth items, correspondingly, refer to the third cube, and the last point, if any. We also permute the vertices under 24-cell symmetry so that it gives us the smallest indices in the final designation.

Therefore, 1,1,(5,8) is to be understood as:
- 1,... : the first point is diminished
- ...,1,...: the top vertex of the q-octahedron is diminished
- ...,...,(5,8): the vertices marked 5 and 8 on the third cube are diminished.

Of course, when it comes to the cantellated tesseract x4o3x3o, the diminishings are not merely of single points, but of segmentochora 8-prism||4-gon. These segmentochora correspond with the square faces of the tesseract, and therefore corresponds with the cells of a 24-cell produced by augmenting the tesseract with cube pyramids, which in turn corresponds with the vertices of the dual 24-cell. Hence, 1,1,(5,8) here refers to 4 of the segmentochora of the corresponding positions to be deleted from x4o3x3o.

Similarly, 1,(1,2,3),6 is to be understood as:
- 1,...: the first point is diminished (this is always the case, since any diminishing can always be rotated under 24-cell symmetry to fall on the first point)
- ...,(1,2,3),...: the points marked 1, 2, and 3 on the q-octahedron are diminished;
- ...,...,6: the vertex marked 6 on the third cube is diminished.

The 1,1,(5,8)-tetradiminishing and the 1,(1,2,3),6-pentadiminishing are both maximal diminishings: the diminished vertices are such, that all of the remaining vertices are adjacent to one of them, so no more vertices can be deleted because that would introduce an adjacent diminishing which makes it non-CRF (we are not accounting for the 24-cell lunae here, since there is no corresponding construct for the x4o3x3o). In fact, these two diminishings are the only other maximal diminishings of the 24-cell besides the octadiminishing, which produces a tesseract (or resp., in the case of x4o3x3o, produces the 8,8-duoprism).