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  Topic: Coloring of Snub Dodecahedron
guy

Replies: 3
Views: 8936

PostForum: Polyhedra   Posted: Sat Sep 24, 2016 9:03 am   Subject: Coloring of Snub Dodecahedron
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 ...
  Topic: Quasicrystals
guy

Replies: 11
Views: 13649

PostForum: Polyhedron Models   Posted: Wed Aug 10, 2016 8:49 am   Subject: Quasicrystals
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" crystal ...
  Topic: Quasicrystals
guy

Replies: 11
Views: 13649

PostForum: Polyhedron Models   Posted: Tue Aug 09, 2016 9:01 am   Subject: Quasicrystals
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 instea ...
  Topic: Quasicrystals
guy

Replies: 11
Views: 13649

PostForum: Polyhedron Models   Posted: Mon Aug 08, 2016 9:04 am   Subject: Quasicrystals
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 triangu ...
  Topic: Quasicrystals
guy

Replies: 11
Views: 13649

PostForum: Polyhedron Models   Posted: Sun Aug 07, 2016 7:03 pm   Subject: Quasicrystals
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 e ...
  Topic: Quasicrystals
guy

Replies: 11
Views: 13649

PostForum: Polyhedron Models   Posted: Sat Aug 06, 2016 6:47 pm   Subject: Quasicrystals
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 assemble ...
  Topic: Printing Nets
guy

Replies: 2
Views: 6600

PostForum: Polyhedron Models   Posted: Sat Jun 11, 2016 6:03 am   Subject: Printing Nets
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 ...
  Topic: How does Stella4D determine the dual?
guy

Replies: 30
Views: 41767

PostForum: Stella Forum   Posted: Mon Aug 24, 2015 9:11 am   Subject: How does Stella4D determine the dual?
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 hyp ...
  Topic: How does Stella4D determine the dual?
guy

Replies: 30
Views: 41767

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 7:28 pm   Subject: How does Stella4D determine the dual?
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 du ...
  Topic: How does Stella4D determine the dual?
guy

Replies: 30
Views: 41767

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 4:27 pm   Subject: How does Stella4D determine the dual?
I see what adrian means, I had forgotten that possibility. Sad
If we take the polyhedron with its mid-ellipsoid and squash it down to make the ellipsoid spherical, the construction will still be proje ...
  Topic: How does Stella4D determine the dual?
guy

Replies: 30
Views: 41767

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 1:49 pm   Subject: How does Stella4D determine the dual?
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 anot ...
  Topic: How does Stella4D determine the dual?
guy

Replies: 30
Views: 41767

PostForum: Stella Forum   Posted: Sun Aug 23, 2015 8:32 am   Subject: How does Stella4D determine the dual?
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 ...
  Topic: Plane tilings
guy

Replies: 0
Views: 12873

PostForum: Stella Feature Requests   Posted: Fri Aug 21, 2015 9:32 am   Subject: 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 ...
  Topic: How does Stella4D determine the dual?
guy

Replies: 30
Views: 41767

PostForum: Stella Forum   Posted: Fri Aug 21, 2015 8:48 am   Subject: How does Stella4D determine the dual?
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. ...
  Topic: Archimedean polyhedra with missing faces
guy

Replies: 1
Views: 10085

PostForum: Polyhedra   Posted: Thu Aug 13, 2015 7:16 pm   Subject: Archimedean polyhedra with missing faces
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 ...
 
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