## Isogonal Polychora Project: n-n duoprismatic compound swirls

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

### Isogonal Polychora Project: n-n duoprismatic compound swirls

So I decided to share an interesting family of polychora from my Isogonal Polychoron Project.

Start with a uniform n-n duoprism. Make a compound of these duoprisms, making sure that the n-n duoprisms are the same. Now let m be any number, representing the number of n-n duoprisms. Now, these can be inscribed in a uniform (n*m)-(n*m) duoprism if the individual n-gonal faces of the n-n duoprism are rotated by 2π/(n*m) radians or 360/(n*m) degrees with respect to each other. The compound will have m*n^2 vertices with directed n-antiprismatic m-swirl symmetry.

Next, take its convex hull and you will get an isogonal polychoron with n-gonal antiprisms and tetrahedra for cells. It cannot be made uniform in general, however. I'd call them the n-gonal duoprismatic m-swirlchoron with symbol n|m. Anyone got better names?

Here are some examples:
2|2 is the 16-cell and is regular with 16 regular tetrahedral cells. It is a compound of two orthogonal squares (2-2 duoprisms) placed in the xy and zw axes.
2|3 is the 6-6 duopyramid with 36 tetragonal disphenoid cells.
2|4 is the 8-8 duopyramid with 64 tetragonal disphenoid cells.
2|n is the n-n duopyramid with 4*n^2 tetragonal disphenoid (digonal antiprism) cells.

3|2 is the 3-3 duoantiprism with 12 triangular antiprism cells (in two perpendicular sets of 6 cells each) and 18 tetrahedra.
3|3 is the triangular duoprismatic triswirlchoron with 18 triangular antiprism cells (in two perpendicular sets of 9 cells each) and 54 tetrahedra.
3|4 is the triangular duoprismatic tetraswirlchoron with 24 triangular antiprism cells (in two perpendicular sets of 12 cells each) and 108 tetrahedra.
3|n is the triangular duoprismatic n-swirlchoron with 6n triangular antiprism cells (in two perpendicular sets of 3n cells each) and 9n^2-9n tetrahedra.

4|2 is the 4-4 duoantiprism, with 16 square antiprism cells (in two perpendicular sets of 8 cells each) and 32 tetrahedra.
4|3 is the square duoprismatic triswirlchoron, with 24 square antiprism cells (in two perpendicular sets of 12 cells each) and 96 tetrahedra.
4|4 is the square duoprismatic tetraswirlchoron, with 32 square antiprism cells (in two perpendicular sets of 16 cells each) and 192 tetrahedra.
4|n is the square duoprismatic n-swirlchoron with 8n square antiprism cells (in two perpendicular sets of 3n cells each) and 16n^2-16n tetrahedra.

n|m is the n-gonal duoprismatic m-swirlchoron with 2*n*m n-gonal antiprism cells (in two perpendicular sets of n*m) cells and (n*m)^2-n^2*m tetrahedra.
n|2 is the n-n duoantiprism.

It took me hours to conceptualize this post because finding the exact number of tetrahedra relied on sequences of triangular numbers. Have fun
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Trionian

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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

Interesting...

Would you mind sharing what all you have for this project so far at some point?
Dionian

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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

Would you mind sharing what all you have for this project so far at some point?

What he said.

ubersketch
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

Isn't this just the class of 4D "antiprisms" in the same vein as the grand antiprism? Basically, you have two orthogonal rings of antiprisms and tetrahedra connecting them. The grand antiprism is the only case for which the result is uniform, but basically you can take any n-sided antiprism and have m such antiprisms per ring.

Of course, in the CRF threads what we call an "antiprism" has a different meaning: a bistratic polytope consisting of some base polytope B and its dual B' in a parallel hyperplane, and a bunch of n-pyramids and tetrahedra connecting them together. Each p-gonal face in one of the base cells gives rise to a p-gonal pyramid, and each edge gives rise to a tetrahedron.
quickfur
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

A more interesting direction of research might be to find polytopes (whether CRF or not is up to you) consisting of more than 2 rings of identical cells, that are closed up in a way that's symmetrical (i.e., the rings are transitive).

As an interesting reference point, the 120 dodecahedra of the 120-cell can be divided into 12 rings of 10 dodecahedra each. Of course, in this case, there are no intervening cells, and it so happens that the symmetry is higher than just a 12-fold ring symmetry. But this 12-ring structure (without the higher symmetry) is also present in the so-called "spidrox": a particular truncation of the rectified 600-cell that shows up as 12 rings of alternating pentagonal prisms and antiprisms, with gaps filled in by 30 rings of square pyramids in a twisting formation.

A question of much interest to me is, what numbers of rings are possible? So far, we know n=2 (duocylinder / duoprism and similar polytopes), n=4 (certain decompositions of the 24-cell), n=6 (dual of n=4 case), n=12 (120-cell, spidrox, etc.), n=30 (spidrox). And also n=8, n=16 I think, corresponding with certain decompositions of the tesseract/16-cell. But what about n=3? Is it possible to have 3 transitive cell rings in a 4D polytope? Or n=5? Etc..
quickfur
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

I detect a potential trend here.
Do you have any images? I would highly appreciate them.

ubersketch
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

Depends. Images of what? The various things I referred to that are CRF may already have images posted somewhere on this forum; if you search for spidrox, for example, you'll probably find a few images I rendered. You can see projections of the grand antiprism on my website.

As far as the other (non-uniform) members of the grand antiprism family are concerned, sorry, no, I don't have images of those. But once you understand the grand antiprism itself, it's not hard to imagine the rest of them in your mind's eye.
quickfur
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

quickfur wrote:Depends. Images of what? The various things I referred to that are CRF may already have images posted somewhere on this forum; if you search for spidrox, for example, you'll probably find a few images I rendered. You can see projections of the grand antiprism on my website.

As far as the other (non-uniform) members of the grand antiprism family are concerned, sorry, no, I don't have images of those. But once you understand the grand antiprism itself, it's not hard to imagine the rest of them in your mind's eye.

Oh, I just want to see the swirlchora.

ubersketch
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

For the great antiprism ("gap") itself you might want to have an according look into
At PolyhedronDude's page there also are listed the uniform (non-convex) swirlprisms, e.g. "sisp",
which one can be found on my site likewise, cf. https://bendwavy.org/klitzing/incmats/sisp.htm.

OTOH, PolyhedronDude has also a further page on non-uniform, so still scaliform swirlprisms too,
cf. http://www.polytope.net/hedrondude/scaleswirl.htm.
This is where "spidrox" then gets refered.
That one too can be found on my site at https://bendwavy.org/klitzing/incmats/spidrox.htm
or alternatively on wikipedia at https://en.wikipedia.org/wiki/Rectified_600-cell#Diminished_rectified_600-cell
(Quickfur too has an according link to a special spidrox page within his CRF section,

You also ought have a look to other pages too on all these mentioned websites! These are a rich source of goods!
Or you even could simply use the search functionality within this forum!

--- rk
Klitzing
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

Don't forget this post of Quickfur's.
student91
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

student91 wrote:Don't forget this post of Quickfur's.

Hey student91, good to see you around again.

Ah, yeah, that post was smack in the middle of CRFebruary, when there was a huge explosion of CRF crown jewels we found that eventually led to the EKF polytopes. Awesome trip down memory lane. Good times!
quickfur
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### Re: Isogonal Polychora Project: n-n duoprismatic compound sw

What would the conjugates of these objects be like? I'm thinking something similar to padiap.
Either way, I think you should tell us more about this Isogonal Polychora Project and what it has in store for us.

ubersketch
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