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

Posted: **Fri Oct 09, 2020 6:10 am**

by **senkoquartz**

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.

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.

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

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

Posted: **Fri Oct 09, 2020 7:41 pm**

by **senkoquartz**

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

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

Posted: **Sun Oct 11, 2020 6:36 am**

by **senkoquartz**

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)

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.

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

Posted: **Tue Nov 17, 2020 2:42 am**

by **senkoquartz**

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".

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.

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

Posted: **Wed Nov 18, 2020 1:58 pm**

by **Ulrich**

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

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

Posted: **Fri Nov 20, 2020 10:48 pm**

by **senkoquartz**

@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.

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

Posted: **Sat Nov 21, 2020 8:50 am**

by **Ulrich**

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

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

Posted: **Mon Nov 23, 2020 8:59 am**

by **Ulrich**

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:

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

Ulrich

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

Posted: **Tue Nov 24, 2020 7:36 am**

by **senkoquartz**

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.

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

Posted: **Mon Nov 30, 2020 6:19 am**

by **senkoquartz**

@Ulrich

I did the same thing with the excavated icosahedron, and found 2 more pairs. One pair looks almost identical to W115_02 and W115_02_d in your .pdf table.

.off files

https://1drv.ms/u/s!AgVMslM5Ti8XlD8vbF0 ... t?e=jbM027

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

Posted: **Mon Nov 30, 2020 7:34 am**

by **Ulrich**

Great!

I tried starting from geodesic spheres and many different bodies created with the morphing tools. Thus we get trillions of isogonal polyhedra but all isohedral facetings from these were crowns and disphenoids. All this is poking in the fog.

Ulrich

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

Posted: **Tue Dec 01, 2020 1:51 am**

by **senkoquartz**

Turns out the convex hull and general structure those last 4 share actually has a more unique and fascinating specimen I missed the first time - a special 30-disphenoid compound pair

30-disphenoid compounds have 120 faces and 120 vertices. It's very easy to double up the faces OR double up the vertices to make either the faceplane-count or vertex-count only 60 - no unique proportions are required to do so, it's a continuum. This dual pair IS unique in that both of them have both doubled-up faces AND vertices.

There's only one other known pair of disphenoids like this, known to Brückner in the 1900s.

(Plus a similar special case of 12 disphenoids with cubic symmetry and fissary vertices.)

I failed to notice this the first time I was looking through the facetings of this hull. I came across it anyway when I added the new chiral noble to its mirror image and noticed the result had doubled-up faces, which prompted me to try faceting it into triangles.

.OFF files

https://1drv.ms/u/s!AgVMslM5Ti8XlETrBW1 ... a?e=eCEB8r

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

Posted: **Tue Dec 01, 2020 2:01 am**

by **senkoquartz**

The chiral pair @Ulrich discovered yields another special disphenoid compound in the same way.

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

Posted: **Tue Dec 01, 2020 8:54 am**

by **senkoquartz**

.off files for these two compound pairs, plus two more pairs found from two nobles' hulls

https://1drv.ms/u/s!AgVMslM5Ti8XlEhCNb_ ... g?e=CgAQIi

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

Posted: **Wed Dec 02, 2020 8:16 am**

by **Ulrich**

I struggled with the cubic ones some time ago. Trying to reproduce Bückners 24,3 I faceted a rhombic cuboctahedron in a way Branko Grünbaum described years ago. In the preview window everything looks allright but if I hit shift ctrl F the face is splitted to coplanar triangles and I get 12 disphenoids. And even Robert cannot explain how he managed to create 24,3 for the Brückner library.