When in a cell diagram there are two independent pairs of enantiomorphous cells (so any combination can be added)
There are 9 combinations to consider.

So for example we have an polyhedron B and we have the cells c1L, c1R, c2Land c2R (and maybe other c cells) that could be added we should ...

Abstractly, all isotoxal polyhedra come in dual pairs (some identical).
Most people assume a polar reciprocity about a concentric sphere in order to construct the dual. However the construction is not rigorous in Euclidean space. It is rigorous in projective space, and applying a homogeneous metric ...

Astro Logix have a Facebook page, but their last post was over five years ago.
You could try sending them a message on Facebook, but I wouldn't hold out much hope.
https://www.facebook.com/astrologix/

Rule (v) remarks that "But we allow the combination of an enantiomorphous pair having no common part (which actually occurs in just one case)". Any interpretation which allows more than one such solution to the full set is therefore wrong.
There are 12 cell sets (distinguishing enantiomorphs), so 2 ...

Thank you. Remarkable stuff. Have you tried submitting your paper to a peer-reviewed publication?
You remark that "there are several dual pairs among them". Surely, the standard dual of an isotoxal figure will also be isotoxal. Thus, they must all either exist as a member of a dual pair, or be self ...

Is Wine any easier to set up than it used to be? I could never get past the default display settings, if it ever worked at all.

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 ...