User talk:Tomruen/hyperbolic honeycombs
Non-Wythoffian Polytopes and Tilings. edit
The distinction between "Wythoffian" vs non-wythoffian, is one of my terms.
Wythoff's paper deals with the construction of Mrs Stott's various uniforms in C_600, by using mirrors of that group. The construction will work in any symmetry generated by reflections, but leads to uniforms only in the case where the group is a simplex. In this case, one can always find points where it is (0 or 1) from every face. Wythoffian groups lead to Dynkin symbols.
The symmetry group o---o-6-o o-U-o has a group in the shape of a triangle-prism. The group leads to eight uniforms, but in every case, the height of the prism is different. No particular prism gives more than one uniform, and many give none.
In hyperbolic H3, there is a laminate-truncate group, that leads to a pentahedral group. The group is of the form P,Q,R, where PQ is a platonic solid (eg 5,3), and Q,R is a hyperbolic tiling (eg 3,7). The laminate on 5,3,7 is formed by cutting off the corner for (3,7), and replacing it by a mirror perpendicular to the three incident mirrors. One can, for example, construct figures of the nature of x5x3o7o, which have faces x5x3o (truncated dodecahedra), and x3o7o (this is a polytope, not a tiling). You can vary the edges of it so that the faces of x3o7o fall in the same plane, and this plane can be used as a mirror. This makes the edges unequal, and therefore not uniform.
The sole example of uniform here is for x4x3o8o, where the equal-edge case already has x3o8o's faces falling in the same plane. The duals ought give a uniform tiling of prisms, but this never gives uniform prisms. The example here has the octagonal prism formed by the top and bottom faces of the truncated-cube of the 4,3,8.
Using mirror-edge constructions with non-simplex groups nearly never gives rise to uniforms.
--Wendy.krieger (talk) 09:53, 15 March 2009 (UTC)
- Thanks Wendy. If you want to add any descriptions to my test article, especially nonwythoffian forms, I'd be glad. I don't feel qualified myself. I felt comfortable with the vertex figures and cells since they are the most clear aspects. SockPuppetForTomruen (talk) 18:08, 15 March 2009 (UTC)
Snubs x3o:s3sPs. edit
The cases in #2(a) etc, of the non-wythoffian figures needs some comment.
There is only one kind of tetrahedron here, not two as indicated.
It does well to suppose that the core figure (octahedron, icosahedron) were some alternate colour (like yellow), and that the tetrahedra have three red faces and one yellow face. The yellow faces hook up to the O, I, while the red faces hook up to an other red face.
The case for a3e:s2sPs is derived from a3e3o4o (a, e = x,o freely). Eight triangles (that correspond to the edges of complex polygon 3{3}3), become linked to pairs of tetrahedra. —Preceding unsigned comment added by Wendy.krieger (talk • contribs) 09:48, 15 March 2009 (UTC)
--Wendy.krieger (talk) 09:53, 15 March 2009 (UTC)
- Thanks! I changed the table to show one type of tetrahedron, with vertex figure counts as a sum. SockPuppetForTomruen (talk) 17:56, 15 March 2009 (UTC)
source edit
Who found the non-Wythoff tilings of H3?
- Wendy Krieger of course! I drew the figures from her hand-sketches. She had at least one published, so I included it the on the main list.
I wonder how inefficient it would be to search for them by building polyhedra out of all the polyhedron (and plane tiling) vertex figures: if the thing can be circumscribed, it's a VF, and its circumradius tells you which space it belongs to. —Tamfang (talk) 04:45, 4 April 2011 (UTC)
- Hyperbolic constructions are all outside of my searches for now, but have fun! You're a master of 2D hyperbolic tilings. It would be great to add perspective views of the 3D hyperbolic tilings! :)