Two new noble polyhedra - a new source of convex hulls to search from?

For general discussion of polyhedra, not necessarily Stella-specific.
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senkoquartz
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Two new noble polyhedra - a new source of convex hulls to search from?

Post by senkoquartz » Fri Oct 09, 2020 6:10 am

https://1drv.ms/u/s!AgVMslM5Ti8XkjGq79b ... j?e=p4mPy6 .stel and .off file

By taking uniform polyhedron duals, and isohedrally faceting them, and then taking the duals of those, you get isogonal polyhedra that are stellations of the original uniform polyhedra.

Some of these stellations have the same convex hulls that yield already-known noble polyhedra, so mightn't some have convex hulls that yield undiscovered nobles?

By faceting the dual of the great ditrigonal dodecicosidodecahedron, we get the polyhedron on the left, and its dual (a stellated great ditrigonal dodecicosidodecahedron) on the right.
Image

Taking the convex hull of that polyhedron on the right, and faceting it with the criteria "Isohedral", we get a new noble and its dual.
Image

They are the only nobles so far (with icosahedral symmetry) that are made of 5gons and non-chiral.

senkoquartz
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by senkoquartz » Fri Oct 09, 2020 7:41 pm

I think that .off file maybe doesn't work, but here's a list of working .off files for all 48 known (afaik) nobles with greater than dihedral symmetry that are not just regular or compounds

https://1drv.ms/u/s!AgVMslM5Ti8XkjQb9FM ... o?e=aCl7JJ

senkoquartz
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by senkoquartz » Sun Oct 11, 2020 6:36 am

So I've done this with almost every uniform polyhedron and compound, and haven't found any new nobles - though I found the first noble again from quit gissid and giid, and found the second noble from sidditdid and tid.
I did find several isohedral polyhedra with isogonal faces though, especially this one (appearing in compound form as a faceting of the compound of 5 rhombic dodecahedra)
Image

And as always, the big blind spot with finding nobles with "Isohedral Facets" in Stella is that it can't do it with 120 vertices.

senkoquartz
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by senkoquartz » Tue Nov 17, 2020 2:42 am

Four new ones

I took an isohedral faceting of the dual of the small retrosnub icosicosidodecahedron's (aka sirsid)'s convex hull, and then isohedrally faceted the dual of that faceting, yielding a new noble polyhedron and its dual, and then faceted that noble dual, and got yet another pair.

First pair - to recap, the one on the left is a stellation of an isohedral faceting of the dual of sirsid's convex hull. They look eerily [suspicuously] like the compounds of 5 great dodecahedra and 5 small stellated dodecahedra (aka presipsi and passipsi) - so I call them "notpresipsi" and "notpassipsi".
Image
Image
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The second pair - the one on the left is a faceting of of notpassipsi. These two are very similar to the two new ones in the OP post.
Image

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Ulrich
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by Ulrich » Wed Nov 18, 2020 1:58 pm

I was not able to reproduce this with my equipment because stella always crashes when I try to create an isohedral faceting of the small retrosnub icosicosidodecahedron's dual or its convex hull. Could you describe how you managed to do so?

Thanks
Ulrich

senkoquartz
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by senkoquartz » Fri Nov 20, 2020 10:48 pm

@Ulrich
Ah, no. I'll clarify. I didn't facet sirsid's dual, or the dual's hull. I took the hull of sirsid, and faceted that hull's dual.

I've now searched the large majority of uniform polyhedron hulls and noble polyhedron hulls for isogonal stellations of those hulls that facet into new noble polyhedra, but haven't found any more new ones. I found the above ones again several times - particularly, "tidoid2" has 5 different isogonal stellations that yield one of the above four nobles.

I even tried faceting the dual of "gridoid2" into a pentakis dodecahedron, and got one already known noble from that. So there could be more, particularly working from 120vert hulls.

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Ulrich
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by Ulrich » Sat Nov 21, 2020 8:50 am

Ah, thanks, I got it now. The first one is really great with its pentagonal faces and I'm eager to build it in paper.

Ulrich

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Ulrich
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by Ulrich » Mon Nov 23, 2020 8:59 am

senkoquartz,
your latest discoveries brought me to the following: I augmented the icosahedron with pyramids, faceted it isohedral and faceted the duals of the convex hulls of the results again. Thus I found four more noble polyhedra:

Image
Image
Image
Image

Trying the same with the first stellation of the icosahedron or the great stellated dodecahedron only resulted in compounds of crowns or disphenoids.

Ulrich

senkoquartz
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Re: Two new noble polyhedra - a new source of convex hulls to search from?

Post by senkoquartz » Tue Nov 24, 2020 7:36 am

Amazing! I love the first one.

I had tried searching from an augmented icosahedron, but hadn't succeeded in finding any. I wonder if other isohedral deltahedra would also yield new results.

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