"Branko Grünbaum" Vertically Transitive Polyhedra

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oxenholme
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"Branko Grünbaum" Vertically Transitive Polyhedra

Post by oxenholme » Wed Jan 16, 2008 6:15 am

I would dearly love to see Stella provide the facial planes for the Vertically Transitive polyhedra described by Branko Grünbaum.

Their faces are non-convex pentagons, there are examples of each type for all symmetry groups (tetrahedral, octahedral and icosahedral), and they are chiral - i.e. symmetrical about a point.

Indubitably were they to be made from coloured 160gsm they would look superb.

I am sure that the equations in Branko's paper are sufficient for anyone with co-ordinate geometry, but my mathematics are limited.

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Post by robertw » Wed Jan 16, 2008 7:42 am

I'm not familiar with these polyhedra. How does "Vertical" have meaning with regard to polyhedra? Do you have a link to a site describing these polyhedra in more detail?

Probably no new feature is required in Stella for this. Just need someone to figure out the face normals and put them into "Stellation->Stellation Planes", followed by some work finding the faces within the stellation diagrams :)

Rob.

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oxenholme
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Post by oxenholme » Wed Jan 16, 2008 8:22 am

Vertically is with respect to vertices...

http://www.math.washington.edu/~grunbaum/

The polyhedra are not necessarily stellations, though very similar solids can sometimes be arrived at by stellating.

I obtained the paper on them from Branko himself, having read about them in a general book on polyhedra. Unfortunately I find the mathematics beyond me.

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Post by robertw » Wed Jan 16, 2008 2:22 pm

I don't see any mention of "vertical" on that link you sent.

Did he send you a physical copy of the paper? If he just emailed it, maybe you could forward it to me. I'd be interested to have a look, whether or not I'm able to make any sense of it myself :?

Rob.

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Post by oxenholme » Wed Jan 16, 2008 4:03 pm

I've emailed Branko with a hyperlink for this thread.

I have the paper in hardcopy only. If all else fails I will see whether I can get someone to scan it for me.

One of the stellations of the pentagonal hexecontahedron (dual of snub dodecahedron) looks very similar to one of the solids concerned - I made a model using 5 colours and it looks superb. I have a .jpeg of it on my computer at home - I intend to make an avatar from it.

Image

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guy
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Post by guy » Mon Feb 11, 2008 11:25 am

Branko does not list any title "Vertically transitive polyhedra" on the pages linked. Is that the title of the hardcopy you have, or is it called something else?

I think it is safe to assume that "vertically transitive" means that all vertices lie within the same symmetry orbit, i.e. they are all equivalent (transitive) under the symmetries of the polyhedron. Nowadays we say that such a polyhedron is isogonal (same-cornered). So I am guessing that is quite an old paper.

BTW, the most interesting isogonal polyhedra are those which are also facially-transitive or isohedral. Such polyhedra are called noble.

BTW2, the polyhedron you have illustrated is not isogonal, so maybe I have gone off in the wrong direction.

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