Cutting up the icosidodecahedron

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Cutting up the icosidodecahedron

Postby quickfur » Wed Jun 12, 2013 9:31 pm

Yesterday, an idea occurred to me w.r.t. finding CRFs via cut-n-paste operations. So far, the only cut-n-paste operations that we've really explored in depth is only augmentation/diminishing of uniform polychora (and that's still far from complete), and stacks of segmentochora.

So I thought, what about a parallel to the gluing of the 600-cell lunes to make a 600-cell? But let's say we relax the requirement that the lunes be CRF, as long as the result of gluing them is CRF (and closes up properly). That got me thinking about 3D shapes that could potentially be formed this way, so I thought of cutting up the icosidodecahedron (o5x3o):

Image

We already know that this can be cut into two pentagonal rotunda. Now, look at the vertex-first projection:

Image

This shows that we can actually cut o5x3o into four pieces, which are lune-like shapes containing bisected decagons, that come in two opposite pairs. Let's call the two kinds of lunes piece A and piece B. So in clockwise order, the icosidodecahedron can be reconstituted by gluing together A-B-A-B. But the interfaces between them are bisected decagons, so that means if we rearrange the order of gluing to A-A-B-B, the shape should still close up! And since all of the bisected decagons will be internal, shouldn't that mean the result will be CRF?

So my question is, why isn't this shape in Johnson's list (or is it)? Is there some unexpected non-convexity going on here?
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Re: Cutting up the icosidodecahedron

Postby quickfur » Wed Jun 12, 2013 9:41 pm

Huh, weird. I just studied it again, and it turns out that A-A-B-B makes a pentagonal orthogyrobicupola (J34)! It's kind of a different construction than the usual method of rotating half of the o5x3o by 36°, but it seems to produce the same result!

In any case, I was thinking about whether this alternate construction might lead to new CRFs in 4D. If a polychoron can be cut into lunes of different shapes but compatible interfacing cells, we might be able to reassemble them in a permuted order to make new CRFs, perhaps?
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Re: Cutting up the icosidodecahedron

Postby quickfur » Wed Jun 12, 2013 9:45 pm

quickfur wrote:Huh, weird. I just studied it again, and it turns out that A-A-B-B makes a pentagonal orthogyrobicupola (J34)! It's kind of a different construction than the usual method of rotating half of the o5x3o by 36°, but it seems to produce the same result!
[...]

Hahaha I'm making a fool of myself. It turns out that the permutation A-B => B-A of two lunes that make up a pentagonal rotunda is equivalent to rotating the original by 36°. So A-A-B-B is actually the exact same construction as before, just analyzed slightly differently. :oops:

Nevertheless, the point stands... if we can find a 4D polychoron that can be cut into permutable lunes (not necessarily CRF, as long as the surface cells are CRF), I think there should be some new CRFs to be discovered there!
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Re: Cutting up the icosidodecahedron

Postby Klitzing » Thu Jun 13, 2013 10:35 am

quickfur wrote:... and it turns out that A-A-B-B makes a pentagonal orthogyrobicupola (J34)! ...


"Ortho-gyro-" truely is nonsense. Either ortho (upright mirror image) or gyro (rotated object). :P
Further, pentagonal cupola is the cap of the small rhombicosidodecahedron (srid, x3o5x), while the pentagonal rotunda is the one you meant, i.e. the hemisphere of the icosidodecahedron (id, o3x5o).

J34 = pentagonal orthobirotunda.

--- rk
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Re: Cutting up the icosidodecahedron

Postby quickfur » Thu Jun 13, 2013 4:18 pm

Haha you're right, I should have checked what I typed before I posted it. It's pentagonal orthobirotunda, not the nonsense that I typed. :oops: :oops:
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