More faceting criteria

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 Joined: Fri Dec 20, 2019 10:02 pm
More faceting criteria
A faceting criterion to exclude nontame polyhedra (no three edges which meet at a vertex are coplanar) and a faceting criterion to exclude any polyhedron with nonvertextransitive faces. The only criterion which reduces the amount of polyhedra found by faceting by a significant amount is "Isohedral" which reduces it way too much.
 robertw
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Re: More faceting criteria
I'm not familiar with tame polyhedra. What's the idea behind that? Have you found that three coplanar edges at a vertex comes up a lot?
And vertextransitive faces. Hmm, wouldn't that limit it to regular faces? Maybe you can post an example or two?
You can click the "Add image to post" link under the post to upload screenshots. I use Greenshot too to easily screenshot any area of the screen.
And vertextransitive faces. Hmm, wouldn't that limit it to regular faces? Maybe you can post an example or two?
You can click the "Add image to post" link under the post to upload screenshots. I use Greenshot too to easily screenshot any area of the screen.

 Posts: 12
 Joined: Fri Dec 20, 2019 10:02 pm
Re: More faceting criteria
Here is a good demonstration of the principle of nonregular vertex transitive faces and of having three or more coplanar edges that meet on a vertex.
The yellow faces have vertex transitivity (with a 2fold dihedral symmetry) even though there are two edge lengths. The red faces are also transitive since they are rectangles (again with 2 edge lengths). The blue lines show a trio of coplanar edges that meet on a vertex. Nontame polyhedra happen to be the majority of a faceting of a polyhedron and are generally undesirable (some people even suggest to not even consider them as polyhedra!)
Another faceting criterion would be to only include locally convex/convex vertex figure facetings
The yellow faces have vertex transitivity (with a 2fold dihedral symmetry) even though there are two edge lengths. The red faces are also transitive since they are rectangles (again with 2 edge lengths). The blue lines show a trio of coplanar edges that meet on a vertex. Nontame polyhedra happen to be the majority of a faceting of a polyhedron and are generally undesirable (some people even suggest to not even consider them as polyhedra!)
Another faceting criterion would be to only include locally convex/convex vertex figure facetings
 robertw
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 Location: Melbourne, Australia
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Re: More faceting criteria
But why the term "tame"? I presume not being tame causes problems somehow?
Yep I thought about vertex transitive faces after I posted last time. I realised it must be about reflective symmetry, not just rotational. If it was just rotational the faces would be regular, but with reflective symmetry as well they could have edges alternating between two lengths.
Yep I thought about vertex transitive faces after I posted last time. I realised it must be about reflective symmetry, not just rotational. If it was just rotational the faces would be regular, but with reflective symmetry as well they could have edges alternating between two lengths.

 Posts: 12
 Joined: Fri Dec 20, 2019 10:02 pm
Re: More faceting criteria
Not being tame is weird. Pseudofaces look like exotic polygons (more than 2 edges per vertex) and their shells (locally convex hulls retaining the edges of a polytope) are exotic as well which is why some people think they should be excluded from polyhedron definitions. Only allowing tame polychora greatly reduces the number of isogonal polyhedra (the number of isogonal polyhedra with isogonal faces is reduced from millions to thousands), and most untame polyhedra look similar to each other and are not attractive. Tame polyhedra look less wild than their untame counterparts so that is where I assume the name comes from.