Mrrl wrote:[...]

What if you take two n,3 skew duoprisms and glue the together by the prismatic cells? Is there a chance to get convex body? Also we may try to glue them by antiprismatic cells (and get two new bodies).

OK, so i've been doing a little thinking along this line of gluing our bipolyhedral rings to make new CRFs.

I think it's possible with the triangular biantiprismic ring; I'll calculate coordinates and try it later.

For the square biantiprismatic ring, I think gluing by the cubical facets won't give you a convex shape -- even in cube-centered projection you can see the square pyramids protrude outwards, which means the dihedral angle is > 90°, so the result of gluing will be non-convex.

However, I think it might be possible to glue two of them by their antiprismic cells; if i'm not mistaken, it should still be convex even though the angle between some of the cells will be very shallow. Unfortunately this is the harder case to investigate. For gluing by the cube face we can just mirror the coordinates, but gluing by the antiprisms is harder to calculate.

EDIT: Actually, I just checked: attaching two copies of square biantiprismic rings by their square antiprism cells is also non-convex.

The tetrahedral cells protrude from the square antiprismic envelope of the parallel projection centered on the square antiprism, which means their dihedral angle > 90°. And actually, in general, I think the shape of an n-gonal antiprism will always cause some of the dihedral angles to be > 90°, so the only convex possibilities are the biantiprismic ring family itself. (Not 100% sure on this one, but I suspect this is true.

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