Yes, both the rhombohedra I describe have this property - the one with the golden rhombs lengthways, the other with them sideways. Are you any good at geometry and basic algebra? It is reasonably easy to develop formulas for the stretch/shrink ratios. Then feed the formula into a high-precision calc...

There are two kinds of rhombohedron, depending on whether you stretch or squash the cube along a diagonal. Once the two rhombohedra are scaled to the same edge length, all your figures can be assembled from copies of just these two, and in this respect they bear a close parallel to the original rhom...

Accurate two-sided printing requires specialist printers able to align more accurately than usual, and it is even more critical that software does not introduce even minor scaling or distortion issues. If you are using colour, thin printed lines can often be useful. In some models they help even out...

On the hyperbolic honeycomb shown: We can tell from its Schläfli symbol {3, 7, 3} that it is self-dual because the symbol is symmetrical. I love the way it brings to life what one might call hyperbolic perspective. Traditionally a hyperbolic plane is represented as a disc, with objects of the same (...

It was adrian who wrote: consider that reciprocating in the ellipsoid will produce perpendicular dual edges at the same tangency points I do not think this is true here. It is generally true that dual edges will be at right angles for reciprocation about any sphere: the symmetry of the sphere forces...

I see what adrian means, I had forgotten that possibility. :( If we take the polyhedron with its mid-ellipsoid and squash it down to make the ellipsoid spherical, the construction will still be projective but will the polyhedron necessarily be canonical? Given that there are many morphs with edge-ta...

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 star...

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 dua...

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 surf...

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 lea...

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 duplicate...

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 w...

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?