Elementary Polychora

[ndl]

Has anyone ever tried to compose a set of elementary crf polychora? That would seem to be a lot easier than trying to count up all of the millions of augmentations and diminishings.

[wendy]

Klitzing has a list of layers (segmentotopes), for building these. There has been a couple of partial stott expansions, such as by student91 etc, that extend it.

[quickfur]

I've tried to study the maximal diminishings of the uniform polychora -- i.e., start with each of the uniforms, and find all CRF subsets of vertices where (1) the edge length remains the same, (2) you cannot delete any more vertices without making it non-CRF. These would constitute the "elementary" CRFs: those that you cannot construct by gluing together smaller pieces.

I haven't covered all of the uniform polychora yet -- there are too many in the 120-cell family like you said -- but I did find quite a few interesting maximal CRF diminishings. Many of these from the 5-cell family also happen to be among Klitzing's segmentochora -- including a particularly interesting one with two hexagonal prisms sharing a square face. There are also certain symmetric diminishings of the 24-cell that emphasize different 24-cell subsymmetries, like the metatridiminishings that show various combinations of 3,6- / 6,6-duoprism chiral symmetries. A particularly interesting one from the 120-cell family is the diminished 600-cell thin wedge, produced by bisecting the 600-cell with two hyperplanes a narrow angle apart and replacing non-unit edges with pentagonal pyramids. It consists of a pentagonal antiprism sandwiched between two conjoined pentagonal rotundae, with a bunch of pentagonal pyramids and tetrahedra filling in the gaps.

However, even this reduced set of CRFs is non-trivial to find. There's no straightforward enumeration algorithm that I know of that we could use to generate all combinations, from which we can cull non-CRFs to obtain the result. Keep in mind that crown jewels would also belong to this category, which tells you how difficult it is to actually enumerate all elementary CRFs!

[ndl]

Yes, I guess it would be difficult to enumerate even an elementary set. Does anyone have .off files for the ones that have been found or at least a good way of generating them? I only found a few on the wiki (some of them are very interesting) but not too much to start with. Most of the segmentatopes are not there. Thanks

[quickfur]

I have the .def files for all of the ones I discovered, along with a few found by the others here. I can readily convert them to Stella4D's .off format on request. You can look through the various CR discovery threads, everywhere I posted images that means I have the .def files. Just let me know which ones you want converted and I'll post them.

Though the ones posted on the wiki represent a good chunk of the models I have, so don't expect too much more. For the segmentochora, ask Klitzing, he may have the .off files handy as well.

For the ones I don't have, if there's a straightforward construction for them I can generate them relatively easily, depending on the overall complexity of the polytope. Stuff that's derived from the 120-cell family polytopes will take longer because I'll have to compute the coordinates and stuff.

[quickfur]

BTW, according to the definition of elementary that I gave above, even relatively complex things like bidex and crown jewels like the castellated rhombicosidodecahedral prism are included as well. Some of these I've posted on my website with images (if you want the .off files, just ask).

[ndl]

Most of your crown jewels already have .off files on the wiki, and yes some of them are very interesting. I guess what I really want is an easy way to cut pieces off of uniforms, which is a feature stella4d does not have.
If someone has .off files for all the non-uniform segmentachora I'll certainly take those (uniform ones are already included in stella).

[ndl]

Maybe also if you have .off for those dodecahedron diminishings of the 600-cell. Thanks

[quickfur]

Maybe also if you have .off for those dodecahedron diminishings of the 600-cell. Thanks

Here they are:

600-cell diminishing with 3 dodecahedra (.off file)

600-cell diminishing with 3 dodecahedra and 1 antiprism (.off file)

Note that these are likely not to be elementary, since I did not try very hard to find further diminishings beyond making it nice and symmetric. They can probably be diminished further if you try.

[Klitzing]

Yes, I guess it would be difficult to enumerate even an elementary set. Does anyone have .off files for the ones that have been found or at least a good way of generating them? I only found a few on the wiki (some of them are very interesting) but not too much to start with. Most of the segmentatopes are not there. Thanks

Segmentotopes in general - and segmentochora in special - indeed where designed to be very easily "visualizable". In fact those are stacks of 2 parallel polytopal layers, there bases. And those bases then are being laced with very primitive lateral polytopes. In fact segmentatopes are the direct lift to 4D (and beyond) of the 3D pyramids, prisms, antiprisms, and cupolae.
--- rk

[ndl]

Is there any program that can take a set of 4D coordinates and create the convex hull? Stella needs the ridges to be enumerated which is very tedious and even then it doesn't always work without also enumerating the cells.

[quickfur]

Is there any program that can take a set of 4D coordinates and create the convex hull? Stella needs the ridges to be enumerated which is very tedious and even then it doesn't always work without also enumerating the cells.

I wrote a program for doing precisely this.

Unfortunately, it's Linux-only (and maybe other Posix-based OSes -- I've never tested that). If you're not afraid of compiling source code yourself and have access to a Posix-compatible shell (Cygwin is available for Windows), I'm happy to share the source code.

[Klitzing]

Well, there is a concept called Quickhull, which even works within R^n, cf. http://www.cise.ufl.edu/~ungor/courses/fall06/papers/QuickHull.pdf.
And there is a software called Qhull available, which implements that concept for n=4, cf. http://www.qhull.org/.

I for one implemented the concept of the paper way back in the 1990th as a Fortran program. But I have it no longer available. I never tried that mentioned one so.

--- rk

[quickfur]

The problem with Quickhull is that it splits degenerate polytopes (i.e., those with non-simplex facets) into simplicial polytopes. It's very useful for computing optimization problems, but annoying to use for geometrical analysis where you don't want the facets to be subdivided into simplices.

To do the kind of analysis we do here, you need a convex hull algorithm that preserves "degenerate" facets. So far, the only algorithm I found with this property is called the "double-description method", which is implemented by Komei Fukuda in his cddlib, and more recently, an improved version of the algorithm was implemented in the C++ library called Skeleton.

But either way, convex hull alone will not be sufficient to give you what you want: because it only computes the H-representation of the polytope, whereas Stella4D requires an enumeration of all surtopes. So you need to implement a face enumeration algorithm using the incidence matrix returned by the convex hull algorithm. Again, there are several face enumeration algorithms out there; some of them are very inefficient because they extract the face lattice by computing LP problems for each face. However, most of the computation is redundant because the convex hull incidence matrix already contains the information you need to reconstruct the entire face lattice. One such algorithm is described in the paper by Kaibel, et al, which I have implemented in my program to output the complete face lattice after the convex hull algorithm is completed.

[ndl]

I don't have so much experience with linux based stuff (or programming in general really) but I would love to try. Thanks

[quickfur]

How comfortable are you with installing and running stuff from the command prompt / shell? If you're at least somewhat comfortable with that, then maybe we have something to go on. Otherwise I'm afraid you might find yourself quite lost.

Also, I should warn you that I haven't used Windows in any serious way for about 15-20 years, so I will be unable to help you if you're using Windows and run into difficulties. If you already have a Linux / Posix system running, though, that would make things a lot simpler, depending.

For the current version of my code, you will need to install: (1) Python, (2) a build system called SCons (which is based on Python), (3) Komei Fukuda's libcdd (if your OS doesn't already have a package for this, I can send you the source code, but then you'll also need a working C compiler and probably Make and a bunch of other tools to compile it), (4) a working D compiler (dlang.org/download).

[ndl]

Yes, this is definitely beyond me but thanks anyway. Maybe I'll try to start with qhull and see what that does.

[ndl]

The problem with Quickhull is that it splits degenerate polytopes (i.e., those with non-simplex facets) into simplicial polytopes. It's very useful for computing optimization problems, but annoying to use for geometrical analysis where you don't want the facets to be subdivided into simplices.

To do the kind of analysis we do here, you need a convex hull algorithm that preserves "degenerate" facets. So far, the only algorithm I found with this property is called the "double-description method", which is implemented by Komei Fukuda in his cddlib, and more recently, an improved version of the algorithm was implemented in the C++ library called Skeleton.

But either way, convex hull alone will not be sufficient to give you what you want: because it only computes the H-representation of the polytope, whereas Stella4D requires an enumeration of all surtopes. So you need to implement a face enumeration algorithm using the incidence matrix returned by the convex hull algorithm. Again, there are several face enumeration algorithms out there; some of them are very inefficient because they extract the face lattice by computing LP problems for each face. However, most of the computation is redundant because the convex hull incidence matrix already contains the information you need to reconstruct the entire face lattice. One such algorithm is described in the paper by Kaibel, et al, which I have implemented in my program to output the complete face lattice after the convex hull algorithm is completed.

I took a basic look at qhull, it seems to give the vertices of all the cells in 4D properly not just simplicies. The problem is as you were saying it doesn't give it surtopes. I wonder if someone has the interest and ability if they could join your algorithm with qhull to make an option for enumerating surtopes.

[quickfur]

In theory, you should be able to compute the surtopes yourself using the list of vertices and list of facets returned by qhull. Basically, once you have a list of, say, vertex numbers that belong to each facet, then you just take all possible intersections of the vertex sets with each other and you'll have the vertex sets of the surtopes.

Of course, this is very tedious to do by hand, so you'll probably want a computer program to do it for you. Also, the dimensionality of the resulting vertex sets may not be obvious; e.g., if you get a vertex set with 4 vertices, it may not be immediately obvious whether that's a tetrahedron (3D facet) or a square (2D face). So you'll need to construct the entire lattice of set intersections in order to find out which set lies at which dimensionality. Again, this is possible to do by hand in theory, but if you're working with large vertex sets (like 4D polytopes tend to have), then you really want a computer program to do this for you.

[ndl]

I asked Robert Webb on his Stella forum and he said if you just list the vertices Stella will figure out everything for you, which I did and it worked just fine. So I guess Stella has it's own convex hull creation algorithm.

[quickfur]

I asked Robert Webb on his Stella forum and he said if you just list the vertices Stella will figure out everything for you, which I did and it worked just fine. So I guess Stella has it's own convex hull creation algorithm.

Oh nice. Yeah that probably means Stella4D comes with a built-in convex hull algorithm. So that ought to save you a lot of the trouble of doing it yourself.

[ndl]

Back to the original topic of this post:
Considering diminishings of the 600-cell, there are many CRF diminishings that are not attainable by slicing off CRF portions of the original, rather extra little pieces have to be trimmed off. For example, a dodecahedral diminishing requires removing 12 additional edges that penetrate the pentagonal faces and converting the five*12 tetrahedra into pentagonal pyramids.
Now the question is, what is considered an elementary diminishing? If I have a polychoron that the only way to further diminish it would be to make one of these dimishings would one still call it elementary?

[quickfur]

I think the idea is not so much how the process of diminishing is done, but whether the resulting piece is something constructible from gluing smaller pieces together, or it's "irreducible" and serves as a building block to build other CRFs out of.

For example, you can bisect a 600-cell with a hyperplane that's exactly halfway between two opposite vertices. The bisection itself does not yield a CRF piece: it produces some half-length edges that results in non-CRF cells around them. However, you can delete these edges and "patch up" the gaps with pentagonal pyramids, which would result in a CRF, which we have named the "hemi-600-cell". You can glue two hemi-600-cells together and you'd get the 600-cell's vertices, but not all of its cells, because the pentagonal pyramids would be concave, which is not CRF. However, you can fill in the concave gaps with pentagonal-pyramid pyramids (aka pentagon || line_segment), and you will obtain the original 600-cell. So one could argue that there isn't a direct diminishing (as in, a single cutting hyperplane) that produces the hemi-600-cell from the 600-cell, but OTOH the 600-cell can be decomposed into two hemi-600-cells plus a number of pentagonal-pyramid pyramids, all of which are CRF.

Of course, the hemi-600-cell itself isn't elementary because it can be further diminished in a similar way, ultimately you get the so-called 600-cell lunes or lunae, and IIRC 10 of these lunae plus a bunch of pentagonal pyramid pyramids would reconstitute the 600-cell. The elementary piece would be the diminishing of the smallest luna, which is a kind of wedge made of two pentagonal rotundae and a pentagonal antiprism and a bunch of pentagonal pyramids. (The luna itself has an additional vertex over the pentagonal antiprism; so that can be cut off to obtain the maximal diminishing.) So this final piece might be considered one of the elementary CRFs obtainable from the 600-cell.

Also, it's possible to have many different elementary CRFs that you can reduce the 600-cell to, which may not have any commonality with each other. AFAIK, the grand antiprism is one of the diminishings of the 600-cell that cannot be cut any further without producing non-CRF pieces (I could be wrong, though). At least, bidex (bi-24-cell-diminished 600-cell; i.e., delete the vertices of two inscribed 24-cells from the 600-cell) is definitely one of these -- it cannot be reduced to the lunae AFAIK; it serves as an elementary piece from which (some of the) other diminishings of the 600-cell can be constructed.

[ndl]

I think the idea is not so much how the process of diminishing is done, but whether the resulting piece is something constructible from gluing smaller pieces together, or it's "irreducible" and serves as a building block to build other CRFs out of.

For example, you can bisect a 600-cell with a hyperplane that's exactly halfway between two opposite vertices. The bisection itself does not yield a CRF piece: it produces some half-length edges that results in non-CRF cells around them. However, you can delete these edges and "patch up" the gaps with pentagonal pyramids, which would result in a CRF, which we have named the "hemi-600-cell". You can glue two hemi-600-cells together and you'd get the 600-cell's vertices, but not all of its cells, because the pentagonal pyramids would be concave, which is not CRF. However, you can fill in the concave gaps with pentagonal-pyramid pyramids (aka pentagon || line_segment), and you will obtain the original 600-cell. So one could argue that there isn't a direct diminishing (as in, a single cutting hyperplane) that produces the hemi-600-cell from the 600-cell, but OTOH the 600-cell can be decomposed into two hemi-600-cells plus a number of pentagonal-pyramid pyramids, all of which are CRF.

Of course, the hemi-600-cell itself isn't elementary because it can be further diminished in a similar way, ultimately you get the so-called 600-cell lunes or lunae, and IIRC 10 of these lunae plus a bunch of pentagonal pyramid pyramids would reconstitute the 600-cell. The elementary piece would be the diminishing of the smallest luna, which is a kind of wedge made of two pentagonal rotundae and a pentagonal antiprism and a bunch of pentagonal pyramids. (The luna itself has an additional vertex over the pentagonal antiprism; so that can be cut off to obtain the maximal diminishing.) So this final piece might be considered one of the elementary CRFs obtainable from the 600-cell.

Also, it's possible to have many different elementary CRFs that you can reduce the 600-cell to, which may not have any commonality with each other. AFAIK, the grand antiprism is one of the diminishings of the 600-cell that cannot be cut any further without producing non-CRF pieces (I could be wrong, though). At least, bidex (bi-24-cell-diminished 600-cell; i.e., delete the vertices of two inscribed 24-cells from the 600-cell) is definitely one of these -- it cannot be reduced to the lunae AFAIK; it serves as an elementary piece from which (some of the) other diminishings of the 600-cell can be constructed.

What you are describing I think is classified as dissection, which can be done to many of the 3D "elementary" polyhedra (In fact all of the tetrahedral and octahedral uniforms can be dissected into smaller pieces). I did a science fair project on this in 12th grade, and only recently found out that these dissections had been enumerated in "Mathematical Models" by Cundy and Rollett (never read the book but saw it quoted in Stewart's book). So according to official Johnson definitions I think these dissections don't qualify to reduce, but agree with you that it makes more sense to look at it this way.
What I didn't realize is that all these diminishings could be dissected using CRF pieces, but it appears you are correct in that. Still getting my mind into 4D.
Again thanks for your help.

[quickfur]

Yep, dissection pretty much sums it up. Analogically speaking, this is in the same category as the "cut-and-paste" Johnson solids, including the Archimedean diminishings and augmentations.

So far, the only 4D CRFs we have found that corresponds with the "elementary" or "crown jewel" category of Johnson solids are the so-called EKF polychora, corresponding to J91 and J92, and the cube||icosahedron segmentochoron (K4.21), and the ursachora based on Wendy's construction. The ursachora are based on J61 and J62 but generalize them differently from merely a diminishing of a uniform polytope. So far we haven't found any non-trivial CRFs involving the other crown jewels among the Johnson solids, and based on Marek's proof of the rigidity of 4D (and higher) vertices, there may not exist any, except for the remote possibility of 4D (or higher) polytopes that contain one or more of those Johnson solids as surtopes. So far none are known, however.

[Klitzing]

… outside of the mere prisms of those, for sure.
--- rk

[quickfur]

:D … outside of the mere prisms of those, for sure.
--- rk

Well, yes, I think I did say "non-trivial", mostly meaning that prisms are excluded because they are trivial to construct.

[ndl]

I was working on dissecting the truncated pentachoron (tip) and I think I got it into the following pieces:

10 Pen
12 Trippy
6 4||tet

Based on the dissection of tut into 8 squippies and 7 tets.
Can anyone confirm I got this right?

[username5243]

According to Klitzing's site, tip can be decomposed into 6 raps and 10 pens. I assume here you're decomposing the raps into a 4||tet and 2 trip pyramids each. In that case I think the numbers check out.

[ndl]

Yes that is exactly how I was doing it. I did not see Klitzing had these on his site, thanks for directing me.