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.

## "Branko Grünbaum" Vertically Transitive Polyhedra

- robertw
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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.

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.

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.

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.

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.