Search found 75 matches

by guy
Sat Sep 24, 2016 9:03 am
Forum: Polyhedra
Topic: Coloring of Snub Dodecahedron
Replies: 3
Views: 10855

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 y...
by guy
Wed Aug 10, 2016 8:49 am
Forum: Polyhedron Models
Topic: Quasicrystals
Replies: 11
Views: 17964

Very nice, congratulations and thank you for sharing. How about removing a few building blocks so that a central cavity connects to the outside? Besides looking cool, such "holey" crystals have found various applications as catalysts, molecular or atomic filters, etc. but I have never seen a quasicr...
by guy
Tue Aug 09, 2016 9:01 am
Forum: Polyhedron Models
Topic: Quasicrystals
Replies: 11
Views: 17964

I understand what you were talking about, but for whatever reason the numbers were still not right. By the way, the solid in the middle between the two pyramids would be a triangular antiprism instead of square antiprism :) Would the length of the space diagonal always be equally divided by the 3 s...
by guy
Mon Aug 08, 2016 9:04 am
Forum: Polyhedron Models
Topic: Quasicrystals
Replies: 11
Views: 17964

The trick is to choose the right slices through the 3D object. It is easiest to explain for the cube. Pick one vertex and identify the three adjacent ones. Cut round these three to remove a triangular pyramid. Do the same on the opposite side so you now have two pyramids and a square antiprism. Thes...
by guy
Sun Aug 07, 2016 7:03 pm
Forum: Polyhedron Models
Topic: Quasicrystals
Replies: 11
Views: 17964

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...
by guy
Sat Aug 06, 2016 6:47 pm
Forum: Polyhedron Models
Topic: Quasicrystals
Replies: 11
Views: 17964

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...
by guy
Sat Jun 11, 2016 6:03 am
Forum: Polyhedron Models
Topic: Printing Nets
Replies: 2
Views: 8300

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...
by guy
Mon Aug 24, 2015 9:11 am
Forum: Stella Forum
Topic: How does Stella4D determine the dual?
Replies: 30
Views: 51817

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 (...
by guy
Sun Aug 23, 2015 7:28 pm
Forum: Stella Forum
Topic: How does Stella4D determine the dual?
Replies: 30
Views: 51817

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...
by guy
Sun Aug 23, 2015 4:27 pm
Forum: Stella Forum
Topic: How does Stella4D determine the dual?
Replies: 30
Views: 51817

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...
by guy
Sun Aug 23, 2015 1:49 pm
Forum: Stella Forum
Topic: How does Stella4D determine the dual?
Replies: 30
Views: 51817

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 ...
by guy
Sun Aug 23, 2015 8:32 am
Forum: Stella Forum
Topic: How does Stella4D determine the dual?
Replies: 30
Views: 51817

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 ...
by guy
Fri Aug 21, 2015 9:32 am
Forum: Stella Feature Requests
Topic: Plane tilings
Replies: 0
Views: 14178

Plane tilings

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...
by guy
Fri Aug 21, 2015 8:48 am
Forum: Stella Forum
Topic: How does Stella4D determine the dual?
Replies: 30
Views: 51817

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...
by guy
Thu Aug 13, 2015 7:16 pm
Forum: Polyhedra
Topic: Archimedean polyhedra with missing faces
Replies: 1
Views: 11374

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...