Noble polyhedra. Where can I find them?

For general discussion of polyhedra, not necessarily Stella-specific.
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senkoquartz
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Noble polyhedra. Where can I find them?

Post by senkoquartz » Wed May 20, 2020 2:01 pm

I found out about noble polyhedra a few days ago. I was looking at the final icosahedron stellation and realized it's an isohedral isogonal figure made out of ENNEAGRAMS. THAT'S INSANE. Now my mission in life is to learn about all the noble polyhedra I can.

What I'm interested in are the "miscellaneous" ones that are neither disphenoids nor crown polyhedra - the ones with octahedral/icosahedral symmetry.

The problem is, finding them on the internet is extremely hard. They are alluded to, but never listed.

From what I gather, most of the known ones are found in Max Brückner's "Uber die gleichecking-gleichflachigen, diskontinuierlichen und nichtkonvexen Polyheder" (1906). But I can't find a .pdf online, or a place to buy it.

Where can I find them?

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robertw
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Re: Noble polyhedra. Where can I find them?

Post by robertw » Wed May 20, 2020 3:50 pm

If you have Great Stella or Stella4D, then they can be found in the library, in the Bruckner folder:

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And since Stella shows you the dual as well, and nobles are all duals of other nobles, you get two at once:


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You'll even find my own more recent discovery in there. You can see a paper model here:

https://www.software3d.com/NobleSnub.php

senkoquartz
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Re: Noble polyhedra. Where can I find them?

Post by senkoquartz » Sun May 24, 2020 1:36 am

@robertw

I bought it, and I'm enjoying it a lot. In the Bruckner catalogue, after sifting through compounds and dual entries, I count 27 distinct nobles with octahedral or icosahedral symmetry, not including another 8 with degenerate hexagonal faces.

In another thread you mentioned that you found a few novel noble polyhedra among Archimedean dual stellations, the "Noble Snub Cube Faceting" being one of them. What were the others? Were they degenerate or no?

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robertw
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Re: Noble polyhedra. Where can I find them?

Post by robertw » Sun May 24, 2020 2:59 am

Although I added features to Stella to make them easy to find, I unfortunately never got around to publishing my results. But polyart on this forum is getting a paper published about them soon. I'll direct him to this thread, in case he hasn't seen it already.

They can be found as facetings of isogonal polyhedra, ie polyhedra whose vertices are all the same, such as the Archimedean solids or any of the uniform polyhedra (and some others). From the menu, select "Faceting->Faceting Criteria->Isohedral", then go to the Preview Faceting view and hit the up arrow to step through valid facetings. This will include compounds, but won't include polyhedra with faces that visit the same vertex more than once (which may or may not be acceptable to you. Bruckner allowed these). You've just got to sort out which ones are compounds, generally by converting the faceting preview into the base model and seeing if it colours it as a compound.

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Ulrich
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Re: Noble polyhedra. Where can I find them?

Post by Ulrich » Sun May 24, 2020 4:22 pm

In my paper paper for this year‘s Bridges Conference in Helsinki (which has been cancelled due to corona) I describe how I used Stella4D to create noble polyhedra. I did it in the way Robert mentioned above and which is very easy to do. I was only interested in distinct ones, not in componds. I found 61 shapes, 19 of them were previously unpublished. Some of Brückner‘s models could not be reproduced and from the truncated icosidodecahedron no isohedral faceting was available. So there might be some more noble polyhedra to discover.
I‘ll post a link when the Bridges Proceedings 2020 are published

Ulrich

senkoquartz
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Re: Noble polyhedra. Where can I find them?

Post by senkoquartz » Sun May 31, 2020 11:47 pm

@Ulrich
I followed the above method - found 46 total.

There is a weird shape - not a noble - from faceting the 2 snub cubes compound. Face is a 7gon, with 2 types of vertices.

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Ulrich
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Re: Noble polyhedra. Where can I find them?

Post by Ulrich » Tue Jun 02, 2020 8:43 am

I was able to reproduce this, it‘s really nice. Unfortunately it is only isohedral.

I sent you a table with my noble findings the other day in a private message, so that you can compare your list with mine.

Ulrich

senkoquartz
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Re: Noble polyhedra. Where can I find them?

Post by senkoquartz » Thu Jun 04, 2020 2:53 pm

@Ulrich
When I said I found 46, I wasn't counting regular polyhedra and ones with degenerate faces - of which there are 17. So I actually have 3 new ones for you.

The first two are a dual pair with hexagram faces - the first one has the same convex hull as your W115_01_d in the .pdf sent me, and the second has the same convex hull as W115_02_d.

Image

The third is a self-dual, same hull as W110_01_d.

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