Isogonal Polychora Project: n-n duoprismatic compound swirls

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Isogonal Polychora Project: n-n duoprismatic compound swirls

Postby Mercurial, the Spectre » Sun Jun 25, 2017 5:17 pm

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

Postby username5243 » Sun Jun 25, 2017 8:38 pm

Interesting...

Would you mind sharing what all you have for this project so far at some point?
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Re: Isogonal Polychora Project: n-n duoprismatic compound sw

Postby ubersketch » Mon Dec 11, 2017 2:16 pm

username5243 wrote:Interesting...

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

What he said.
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