## Equilateral convex polyhedra (new class after 400 years?)

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robertw
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Certainly some polyhedra can't be given equal edge lengths unless they lose their convexity, like many of the duals of the Archimedean solids.

Here's some more convex equilaterals though.

From Stella's near miss library, these two don't need any further variation. I even named them "Equal-edged Near Miss", so that's more equilaterals that were known before that paper. These were ones I found, as convex hulls of new Stewart Toroids.

Other near misses in Stella's library can be adjusted for equal edge lengths, such as these near misses discovered by Robert Austin:

guy
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robertw wrote:Certainly some polyhedra can't be given equal edge lengths unless they lose their convexity, like many of the duals of the Archimedean solids.
Yes of course, all those funny groupings of triangles round a vertex for a start.

Something like a heptagonal pyramid requires its equilateral morph to be wound up to give it a star base. That can be done with either a {7/2} or {7/3} base, giving two distinct geometric solutions.

I suspect that those Archimedean duals would also yield two solutions, one with augmentations and the other with excavations.

What could Stella's spring algorithm make of all that?

polymorphic
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I haven't been able to make one with the rhombic enneacontahedron truncated on the 5-valance vertices. It works on the 3 and 6 valance vertices, but on the 5 it collapses into non-convex.

Roger

robertw
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polymorphic wrote:I haven't been able to make one with the rhombic enneacontahedron truncated on the 5-valance vertices. It works on the 3 and 6 valance vertices, but on the 5 it collapses into non-convex.
I agree, I don't think that one works.

Here's a really simple one I just found:

polymorphic
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There are many that can't work. For example if you augment a cube with cubes it will be non-convex and is already equilateral.

That models seems familiar. How did you create it?

robertw
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polymorphic wrote:There are many that can't work. For example if you augment a cube with cubes it will be non-convex and is already equilateral.
Or even if you put a pyramid on each side of a cube.
That models seems familiar. How did you create it?
Weirdly, I started with "6J91 (P4)" which is in Stella under "Stella Library->More Stella Toroids->6J91 (P4)" (it's not a toroid but the shape of a hole in one of the toroids), and took the convex hull, then adjusted for equal edge lengths. But I'm sure there's a more straight forward way.

robertw
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robertw wrote:But I'm sure there's a more straight forward way.
Actually, it's a rectified truncated octahedron I suppose.

robertw
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Here's a bunch more. These are all convex hulls of new Stewart toroids that were discovered by Alex Doskey and myself, and which are all available in Stella. No adjustment required as the edges are already the same length due to the nature of Stewart toroids.

polymorphic
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robertw wrote:
robertw wrote:But I'm sure there's a more straight forward way.
Actually, it's a rectified truncated octahedron I suppose.
It can be found by truncating the on 3-valance vertices of the truncated octahedron by a factor of 1/2.

It can also be found by beveling only on the 3-valance vertices in conway notation which *would* translate to this operation string. (Beveling itself would not allows the three 'b3').

dk3datO

Conway Notation specification does not specify that beveling can only be done by vertex valances. I think this may have been an oversight! Beveling is actually truncation by 1/2. Therefore since Conway Notation allows for truncation by vertex valance, so should it be with Beveling. I am going to propose this change.

polymorphic
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polymorphic wrote: Conway Notation specification does not specify that beveling can only be done by vertex valances. I think this may have been an oversight! Beveling is actually truncation by 1/2. Therefore since Conway Notation allows for truncation by vertex valance, so should it be with Beveling. I am going to propose this change.
I experimented with this and the only difference is how much is truncated from the polyhedron before planarization occurs. Normal truncation is only 1/3 of the way. Beveling is 1/2 of the way. So the results of tN and bN are ultimately the same.

The interesting polyhedron can actually already be generate by ambo of the truncated Octahedron.

conway atO | antiview

Roger

robertw
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polymorphic wrote:Conway Notation specification does not specify that beveling can only be done by vertex valances. I think this may have been an oversight! Beveling is actually truncation by 1/2. Therefore since Conway Notation allows for truncation by vertex valance, so should it be with Beveling. I am going to propose this change.
Are you confusing bevelling with rectification (ambo)? Rectification ("a" operator) is truncating half way so that the neighbouring truncations meet. Bevelling is not just a deeper truncation. It is truncating both vertices and edges. If you limited the set of vertices to be truncated, you'd then also have to decide which edges to truncate.

Rob.

Gerry in Quebec
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### Goldberg polyhedron

Rob W. wrote:
Finally, here's a couple of Goldberg polyhedra I made using this new feature, to give them equal edge lengths.

Hi Rob & others:
Great discussion about equilateral Goldberg polyhedra. I've experimented with these on and off in recent years. Here's a SketchUp model (by Taff Goch... from 2010 as I recall), showing the I {3,0} polyhedron.

- Gerry in Quebec

Gerry in Quebec
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### Re: Goldberg polyhedron

Let me try posting this again, this time with the link to the 3D warehouse:

https://3dwarehouse.sketchup.com/model. ... 32e8ebf2b1

Hi Rob & others:
Great discussion about equilateral Goldberg polyhedra. I've experimented with these on and off in recent years. Here's a SketchUp model (by Taff Goch... from 2010 as I recall), showing the I {3,0} polyhedron.

- Gerry in Quebec[/quote]