A forum for discussing current researches into polyhedra seems sorely lacking these days. So I thought I'd see if one here gathers any interest.

Several people are working actively on abstract polytope theory, in which the abstraction is a set-theoretic construct which captures the connectivity or ...

There are many more stellations of the icosahedron, being the ones which do not come under "Miller's rules".
Some are also quite beautiful.
A selection, which various people have pointed out to me over the years, may be found here: https://www.steelpillow.com/polyhedra/icosa/lost/lost.html

These are fascinating to read about, and to examine the one thumbnail image which has been posted. They sound beautiful. But there is no polyhedron software which installs cleanly on my system and can view .off files equally cleanly. And Ulrich's image server is either broken or hates my system.
The ...

Looks good. Sorry to have fed you a red herring.

metachirality asked about regions suddenly flipping to a different density. I was trying to explain that using the conventional definition of a dense interior region, this cannot happen; as the polyhedron morphs, one region progressively gives way to another one of different density. A sudden flip ...

I think there's only an issue when the centre of reciprocation lies in a facial plane (ie hemi faces). You'd get this issue even with convex polyhedra if you decide to put the centre of reciprocation on one of the faces. Nonconvex polyhedra work fine as long as they don't have hemi-faces.

Isn't ...

Thank you.
There seems to be a more or less "unlimited" option in there, excellent. Now I just need to get to grips with WINE.

Yes, "tidy" is a term I introduced, but the whole point of it is that it has no rigorous definition - it is a rag-bag of intuitive notions, which sometimes lead to ...

I recall discussing this many years ago.
The fundamental issue is that the "interior" of an infinite face is not defined. Polyhedral reciprocation is only equivalent to projective reciprocity for convex solids. Non-convexity introduces anomalies as to the filled-in bits of the face plane, and the ...

https://i.postimg.cc/9FMSRjrM/image.png


The issue here is, how do you define a "region"? When you pull apart the vertical and horizontal ends of the 4, are you pulling the horizontal away from the vertical, or are you pulling the vertical away from the horizontal? In each case, where does the ...

I have the 1989 reprint. It is definitely a mistake.
And it's not on my list at https://www.steelpillow.com//polyhedra/ ... inger.html
I'd better do a quick update.
Thank you for posting it here.

Hi Rob,
Long time no see. Off and on I have been working on the underlying theory of polytopes, with a view to applying it to stellations and facetings. Now that I have at last got there, I am interested in enumerating the facetings of the regular dodecahedon in particular. (I published its faceting ...

Thanks for the heads-up, Rob.

It is basically a non-uniform morph of the rhombicosahedron. One way of making it would be to create a compound of the regular icosahedron and the rhombic triacontahedron, as a new compound polyhedron (I assume Great Stella can do that), and then explore its stellations.

Hi Rob,

Just been playing with MoStella Free, so I thought I'd drop you a line. First time in many years that I have had a box to run one of your apps on natively, this time a Planet Gemini PDA running Android. Thoughts follow in random order:

In faces+wireframe view, the frame tends to disappear ...

Map colouring is a complex and difficult topic. The four-colour theorem, that to always avoid even edges meeting you need four colours, was first proved by a computer exhausting all the possibilities and, I think, more recently proved analytically. The problem you pose is way more complex.

I think ...