Multistory CRF polychora

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Multistory CRF polychora

Postby Marek14 » Tue Oct 28, 2014 11:54 am

Looking at the CRF discovery page, I noticed that we still don't have enumerated various "multistory" polychora formed by gluing segmentochora and similar together in one axial direction.
With Stella and its automatical calculation of dichoral angles I should be able to do this as long as I have models for the various shapes.

Basically, there are two types of polychora that can work in this regard. One type are those that have flat "top" and "bottom" cells and can be stacked upon each other. These include prisms, (n,4)-duoprisms, cupolas and rotundas. Next, there are polychora that only have the "bottom" cell and their top is subdimensional -- these can be only used at the "end" of the resulting polychoron. These include pyramids, pseudopyramids, bicupolic rings, (n,3)-duoprisms and biantiprismatic rings. And there are probably some segmentochora that don't belong to any of these or that have more usable orientations.

There might be more, but these are the basic ones. Do we have a model archive somewhere, or at least the numbers for them? Or would I need to build models for all of these?
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Re: Multistory CRF polychora

Postby student91 » Tue Oct 28, 2014 3:23 pm

Hi marek, good that you're talking about that. Frankly what you are suggesting is basically (a wiser set of) what my school project is about. In my school project, I try to enumerate all possible CRFs that can be constructed by glueing segmentochora to each other. This project has to be finished around Christmas, so I am quite busy lately. This also means that I will have a conclusion which can be uploaded to the wiki. This school project also includes a comprehensible explanation of Stott-expansion etc, explanation of the 4th dimension and calculation of dichoral angles. As you might understand this would be very interesting to have uploaded to the wiki. Is there anyone besides student5 who can talk Duch on this forum? They might translate the project if they want.

Student91
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Re: Multistory CRF polychora

Postby Marek14 » Tue Oct 28, 2014 4:31 pm

Hm, if we build a database of models... Putting them together is relatively straightforward, after all, once the angles are known.
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Re: Multistory CRF polychora

Postby Klitzing » Tue Oct 28, 2014 5:00 pm

First of all:
what you call here "multistory" probably is what the original article on convex segmentochora introduced to be called "multistratic". So it might be wise to stick to the original naming and not to design every now and then completely independent new names for already known terms.


As far as I know, Great Stella already has a built in possibility to measure dihedral angles of bounding cells. Thus all this data could thus easily deduced, provided the "stacks" are produced therein.

Glueing 2 already designed polychora together is also no problem with Great Stella. Either you take the parts from the built-in examples, or you design them first and pull them then from memory.

Thus the only remaining question is - as I do not own a copy of Great Stella - whether or not Robert Webb already managed to add my set of convex segmentochora to the built-in examples. - At least I know that he once planed to do so.

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Re: Multistory CRF polychora

Postby Marek14 » Tue Oct 28, 2014 6:17 pm

Klitzing wrote:First of all:
what you call here "multistory" probably is what the original article on convex segmentochora introduced to be called "multistratic". So it might be wise to stick to the original naming and not to design every now and then completely independent new names for already known terms.


As far as I know, Great Stella already has a built in possibility to measure dihedral angles of bounding cells. Thus all this data could thus easily deduced, provided the "stacks" are produced therein.

Glueing 2 already designed polychora together is also no problem with Great Stella. Either you take the parts from the built-in examples, or you design them first and pull them then from memory.

Thus the only remaining question is - as I do not own a copy of Great Stella - whether or not Robert Webb already managed to add my set of convex segmentochora to the built-in examples. - At least I know that he once planed to do so.

--- rk


Actually, no, I used "multistory" on purpose to acknowledge that it can be glued from various parts that are NOT necessarily bistratic. The number of "stories" is the number of elementary polychora glued together, but it doesn't need to correspond with the number of hyperplanes for vertices.
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Re: Multistory CRF polychora

Postby Klitzing » Tue Oct 28, 2014 7:22 pm

Okay, you then need to list first all elementary objects first - here: elementary wrt. gluebility (a.k.a. exterior blendings).

You are right indeed: even if 2 parts make up a bistratic stack, a next one needs not to be attached either onto the top or bottom face only, it well might be attached anywhere else...
So this then might get a little bit harder task, as you have to keep track of all possible orientations of each derived part additionally!

--- rk
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Re: Multistory CRF polychora

Postby Marek14 » Tue Oct 28, 2014 8:29 pm

Klitzing wrote:Okay, you then need to list first all elementary objects first - here: elementary wrt. gluebility (a.k.a. exterior blendings).

You are right indeed: even if 2 parts make up a bistratic stack, a next one needs not to be attached either onto the top or bottom face only, it well might be attached anywhere else...
So this then might get a little bit harder task, as you have to keep track of all possible orientations of each derived part additionally!

--- rk


Yeah. Of course, the thing is that I don't actually need a complete list right out of bat; it can be gradually extended. Segmentochora are one part of it, and even with just those we can probably find interesting combinations. The rotundas come next.

Ideally, it might be possible to find something like pocuro. Pocuro is just one of possible blends, but it's related to J91 and J92 which have some parts with identical geometry to pocuro's "equator" -- that's probably one reason why it appears in CRF's.

So a good analogue of pocuro would have geometry that could be slightly modified to reach a J91/J92 type of crown jewel.
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Re: Multistory CRF polychora

Postby Klitzing » Wed Nov 05, 2014 2:35 pm

pocuro, just like bilbiro and thawro are expanded kaleido-facetings of the icosahedron. It is this relation which lets them show up lately in the CRF polychora quest - amongst other icosahedron derivatives (peppy, pap, mibdi, and teddi).

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