Quickfur's renders

[quickfur]

Here's the next 4D Catalan, as promised:



This is what the Polytope Wiki calls the Triangular Antitegmatic Hexacontatetrachoron, the dual of the runcinated tesseract. It's obviously the direct analogue of the deltoidal icositetrahedron, the dual of the rhombicuboctahedron.

Interestingly enough, the projection envelope of this 4D baby is exactly the deltoidal icositetrahedron, having exactly the same proportions. This leads to a conjecture that all duals of x4o...o3x have this projective relationship with each other. I haven't worked out the proof yet, nor have I checked the 5D analogue yet, but I'm fairly confident this is the case. I find this remarkable, because most of the 4D Catalans (that I've checked so far) don't have such direct proportions with their 3D analogues. It seems that x4o...o3x is one of those cases where there's a direct analogy across dimensions -- if this conjecture is true. It's also remarkable that this polychoron has 3 different edge lengths, whereas the deltoidal icositetrahedron has only 2 different edge lengths, yet in the projection the edge length proportions line up nicely. So this projective relationship isn't as trivial as it may seem. In any case, it's an interesting observation.

[wendy]

that's right. with the runcinate xP..Qx, the dual is an antigum along the line between the vertices of xP...Qo and its dual. so the antitegum is based on what is the ...
I use the Conway term strombiate.

[mr_e_man]

Next Catalan?

[quickfur]

Oops, I actually had it ready 2 weeks ago, but got busy and totally forgot to post it on the website. I'll get around to it sometime today, hopefully.

[quickfur]

OK, got it set up faster than expected. Here it is:



This is the square antitegmatic hecatontetracontatetrachoron (what a mouthful!), the dual of the runcinated 24-cell. This time round I actually have two series of structural renders, one the traditional layer-by-layer cell breakdown, and a bonus one showing the Hopf fibration structure of a subset of the cells. Very interesting, check it out!

[quickfur]

This month's POM (polytope of the month) is the dual of the uniform pentagonal prism, the (non-Johnson) pentagonal bipyramid:



Nothing special in itself, but it does feature as a cell in a 4D Catalan which will be posted next month. So stay tuned!

[Klitzing]

That one then ought be the dual of the rectified 600-cell, cf. https://bendwavy.org/klitzing/incmats/o3m3o5o.htm, which also is known as "pentagonal-bipyramidal heptacosiicosachoron".
In fact yourself provided a first render in here already, cf. http://hi.gher.space/forum/viewtopic.php?p=16141#p16141, however the pic itself seems lost since (but then was saved and still is shown on my page, hehe).

--- rk

[quickfur]

Very nice! So you already know what's coming.

[quickfur]

And here it is:



This is the joined 120-cell, sporting 720 pentagonal bipyramids. It is the dual of the rectified 600-cell, one of the rare semiregular polychora.

[quickfur]

Hmm, apparently I neglected to post this month's POM:



This is, of course, the famous rhombic triacontahedron, which is one of the projection images of the castellated rhombicuboctahedral prism, the first CRF we discovered that sported bilunabirotunda cells in a non-trivial manner. The order-5 vertices correspond to the apices of the castellated prism's pentagonal pyramid cells, and the order-3 vertices correspond with the tetrahedral cells.

[Keiji]

Thank you for going to the effort to continue to post these after all these years!

I'm hoping to spend a little more time reading up on these topics again. What would you say were the most interesting discoveries of the last 5 years or so?

[quickfur]

Hey Keiji, good to have you active here again, it's been a while!

Sad to say, I have hardly been keeping up with discoveries lately. Mostly just doing what I can to keep my website updating in a semi-regular fashion (has been better since last year when I made a large enough buffer of Catalan solid entries that I could continue posting monthly in spite of often having no time at all to do anything 4D or 3D related). I'm dreading the day my buffer runs out, then I might have to go on hiatus again.

//

Anyway, since I'm at it, might as well include the latest entry on my website, that I haven't gotten around to posting here until now:



This is the so-called "bidecachoron", the dual of the bitruncated 5-cell. (Not perfectly happy with the name, I got the name from the Polytope Wiki, apparently coined by Bowers. But for lack of a better name, it will have to suffice for now.) Interestingly enough, the projection envelope is a cube, and the layout of cells in it embodies the two ways one could inscribe a tetrahedron in a cube. I'm sure there's a rational reason behind it all, of course, but it's just an interesting coincidence that struck me when I was doing these renders.

[username5243]

Just to provide some context: That name was chosen more out of analogy with a different family than the Catalans - the family of convex vertex-transitive polychora formed from the convex hull of two inversely oriented 5-cell truncates, so you have names like the bitruncatodecachoon rfo the hull of two opposite truncated 5-cells, etc. As you can tell from the coordinates listed on your site, the dual of the bitruncated 5-cell is just the convex hull of the compound of two dual 5-cells.

A full investigation into these (along with the 24-cell family analogs of them) was done by those of us on the Discord/wiki a few years back. Interestingly, for some there are several related forms based on taking the hull of two similar polychora, but with the edge lengths varied - any Wythoffian uniform polychoron with more than one ringed node in the CD has infinite non-uniform (but still isogonal) variations formed by adjusting edge lengths, and in some cases (such as that of the cantellated 5-cell) taking the hull of two of these variants will give results with different types of cells, rather than just similar edge-length variations. There are in fact a total of 24 of these isogonal polychora with doubled 5-cell symmetry and up to 240 vertices, and each has a corresponding one with doubled 24-cell symmetry. The 5-cell ones actually have a page on the Bowers polychoron site now if you are at all interested - he also gets into even stranger chiral varations, because you can do this with variations of the (non-uniform but still isogonal) snub 5-cell as well.

[quickfur]

I've noticed that even among the 3D Catalans, the coordinates are essentially a union of the coordinates of 2 or more of xNo3o, oNx3o, and oNo3x, variously scaled. I.e. the Catalans are the convex hull of various compounds of the regular and quasi-regular polyhedra of their respective families.

Apparently in 4D this is also the case. I surmise it's true in all dimensions.

[username5243]

I've noticed that even among the 3D Catalans, the coordinates are essentially a union of the coordinates of 2 or more of xNo3o, oNx3o, and oNo3x, variously scaled. I.e. the Catalans are the convex hull of various compounds of the regular and quasi-regular polyhedra of their respective families.

Apparently in 4D this is also the case. I surmise it's true in all dimensions.

Correct, at least if the original polytope is Wythoffian. And it is easy to figure out which specific vertex sets are involved, based on the facets of the original. As you know the facets of a Wythoffian uniform polytope can be fonud from the CD by removing one of the nodes and all adjacent edges. However, sometimes this will yield a lower-dimensional element instead - take x4x3x3o, where removing the third node gives x4x o, representing only an octagonal face between two great rhombicuboctahedra. Its dual then does not use the o4o3o3o rectate with that node (that is o4o3x3o) but does use the others.

For non-Wythoffian cases of course things get more complicated, so this will not work for stuff like the snub 24-cell if you get around to its dual.

[quickfur]

Forgot to post this here, this is the latest 4D Catalan:



This is the joined 16-cell, the dual of the rectified tesseract. Interestingly enough, the projection envelope is a regular octahedron.