Isohedral Stellations

For general discussion of polyhedra, not necessarily Stella-specific.
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ndl
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Isohedral Stellations

Post by ndl » Wed Feb 15, 2017 5:17 am

I'm interested in finding all the convex isohedral stellations of various polyhedra. I found (including stellation core):
Dodecahedron: 4
Icosahedron: 10
Rhombic Dodecahedron: 3
Rhombic Triacontahedron: 12

I'm wondering if anyone ever researched this before.

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robertw
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Post by robertw » Fri Feb 17, 2017 3:37 am

Not sure what you mean by "convex". None of the stellations are convex other than the original core.

People are certainly interested in isohedra, and even more in isogonal isohedra (aka noble polyhedra), but I don't know whether anyone's enumerated all those to be found among stellations of the Archimedean duals, for example. I presume not, as I've found a few myself that seem to be novel, eg this stellation of the dual of the snub cube: http://www.software3d.com/NobleSnub.php

ndl
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Post by ndl » Fri Feb 17, 2017 2:33 pm

Hi Robert, thanks for your reply.
I was referring to partial stellations of different sub-symmetries that use only some of the face planes of the stellation core. For example using 12 planes of the icosahedron you can make the pyritohedron and with 8 you can make the octahedron, etc.

Here's a picture of my Zometool models for Dodecahedron and Icosahedron:

https://drive.google.com/file/d/0B8ESz- ... sp=sharing
(The image didn't load in so I just made it a link)

P.S. Thanks for your amazing program, Great Stella!

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robertw
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Post by robertw » Mon Feb 20, 2017 1:19 am

Oh OK, I see what you mean. I haven't seen that investigated before. For the dodecahedron I think you're right that there's 4: the original dod, one with 5-fold dihedral symmetry, and two with 3-fold dihedral symmetry. It's just about finding subsets of faces within any subsymmetry group which are of the same type within that group.

Actually this is quite easy to investigate in Stella. Start with whichever polyhedron, say icosahedron. In turn, try each subsymmetry group (except for pyramidal and no symmetry). See how many face types there are in the info window, ignoring types with less than 4 non-parallel faces and types which include all faces. Add them all up and that should be your answer.

For icosahedron:
Full symmetry: 1
Tetrahedral: 3
5-fold dihedral: 2
3-fold dihedral: 3
2-fold dihedral: 0. This one's a bit tricky. There are 5 types, but 2 are same as for tetrahedral symmetry, and 3 don't enclose space.

So I get a total of 9, but 10 I think if you include both mirror images of chiral polyhedra.

ndl
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Post by ndl » Mon Feb 20, 2017 2:14 am

Thanks for your idea, that is an easy way to see possibilities, but you have to watch out because some are combinations of face types. I found 4 distinct polyhedra with D3 symmetry from the icosahedron, one 12 faced, two 6 faced reflexible (using the faces around the poles for one and alternating faces around the middle for the second like in S6 symmetry), and one 6 faced chiral.

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