Projective geometry is a funny thing. Despite its most pure form having no concept of angle or distance (i.e. no concept of coordinates), it is most often taught using a Euclidean metric with yet another coordinate bolted on top. let me know if you get baffled.

Also, be warned - projective geometry ...

And, as noted in last reply, I'm curious to see what might come from using it as the surface of reciprocation.
Polar reciprocation is a construction in pure projective geometry. This geometry has no idea of metric, i.e. of distance or angle. To a projective geometer a sphere, ellipsoid, hyperbolic ...

Would it be possible for Stella to explore plane tilings as well as polyhedra?

There seem to be two approaches to tiling:

Symmetries (e.g. kaleidoscopes) in the plane can generate tilings, much as spherical ones generate polyhedra. There are even some "dense" or overlapping tilings analogous to ...

First, please could you shrink that huge screenshot? It is vastly bigger than my poor screen and makes all the text shoot off to the right.

Duality of polyhedra exists at several different levels. Sometimes, a polyhedron will have a dual at one level but not at another.

Combinatorial or abstract ...

I do not think there is any serious mathematical approach to them, they are just symmetrical shapes which look a bit like polyhedra and have the same symmetries.

Technically they are finite bounded manifolds whose boundary is disjoint, but that applies to anything with holes punched through its ...

Hi Rob,

'Fraid I am still firmly wedded to Linux and can't be ***ed with WINE so my copy of Stella is equally firmly parked for the foreseeable.

I would think that all the basic operations one encounters - truncation, bevelling, runcination, etc. - would be worth implementing, if only to help us ...

Of course, a 20-sided pyramid with something like a {20/9} star icosagon base can have equilateral sides.

In general, for an {n/m}star base, n/m < 6 will allow a uniform star pyramid.
Also, n and m must be co-prime (no common factor) or you get a compound, and m < n/2 or you get a backward ...

Certainly some polyhedra can't be given equal edge lengths unless they lose their convexity, like many of the duals of the Archimedean solids.
Yes of course, all those funny groupings of triangles round a vertex for a start.

Something like a heptagonal pyramid requires its equilateral morph to be ...

robertw wrote:I've also just realised that pretty much every near miss included with Stella can be adjusted to an equilateral version.

H'mm. Setting symmetry aside, are there any convex polyhedra that can not be adjusted to equilateral form? Whole families? Non-convex? What are the rules?

By all, that's wonderful. Why solve equations when you can jiggle about?

I also need a way to ... maybe even to generate Goldbergs with equal edge lengths as per the paper.
I think that will require the solution of multiple simultaneous equations, there are no shortcuts for such arbitrary metric equalities in projective reciprocation. Can one even assume that all the ...

Well Goldberg was interested in cages, and probably not interested in flat faces, because he was looking at molecular arrangements. I think these guys just realised, while doing the same thing, that they could expand this to say something interesting about polyhedra, as an aside to their molecular ...

Hi Robert,

Yes, I think your remarks about the other types of equilateral polyhedron are spot on.

I don't think I have yet found two equivalent definitions of a Goldberg polyhedron. One would have to go back and read Goldberg's original papers and see how rigorous he was - probably no more so than ...

C'est peut-etre une bulle appelé la Langue Anglaise?

Congratulations.

When can we see the 50-arm star with interior lighting?