Hurwitz group

[ericr]

Construction
source : wikipedia

Hurwitz groups and surfaces are constructed based on the tiling of the hyperbolic plane by the (2,3,7) Schwarz triangle.
To obtain an example of a Hurwitz group, let us start with a (2,3,7)-tiling of the hyperbolic plane. Its full symmetry group is the full (2,3,7) triangle group generated by the reflections across the sides of a single fundamental triangle with the angles π/2, π/3 and π/7. Since a reflection flips the triangle and changes the orientation, we can join the triangles in pairs and obtain an orientation-preserving tiling polygon. A Hurwitz surface is obtained by 'closing up' a part of this infinite tiling of the hyperbolic plane to a compact Riemann surface of genus g. This will necessarily involve exactly 84(g − 1) double triangle tiles.

The following two regular tilings have the desired symmetry group; the rotational group corresponds to rotation about an edge, a vertex, and a face, while the full symmetry group would also include a reflection. Note that the polygons in the tiling are not fundamental domains – the tiling by (2,3,7) triangles refines both of these and is not regular.


order-3 heptagonal tiling   
order-7 triangular tiling
Wythoff constructions yields further uniform tilings, yielding eight uniform tilings, including the two regular ones given here. These all descend to Hurwitz surfaces, yielding tilings of the surfaces (triangulation, tiling by heptagons, etc.).

From the arguments above it can be inferred that a Hurwitz group G is characterized by the property that it is a finite quotient of the group with two generators a and b and three relations


thus G is a finite group generated by two elements of orders two and three, whose product is of order seven. More precisely, any Hurwitz surface, that is, a hyperbolic surface that realizes the maximum order of the automorphism group for the surfaces of a given genus, can be obtained by the construction given. This is the last part of the theorem of Hurirz

i ask no more new formulas but it is  possible to make of something like this

I see  subime fractal on   http://www.josleys.com/show_gallery.php?galid=342 in 3D
I want too know if M3d can (and I) do the same
Tank you in advence

ericr

[DarkBeam]

With fragmentarium, I think there is a script

[ericr]

yes I know that I use it on Frangarmtarium
but I have in mind Mobius tp on MB3

[ericr]

I founds 2 prog  that do tesselation in                2d circle   of Poincaré
                                                                        3d sphere of Poincaré + Appolonius in a lot of case ( I mean a lot of Polyedrons to start)
 
Free prog very easy to use no formuas
ERICR
                                                                        
jenn3d_win_2008_03_13.zip  in   http://www.math.cmu.edu/~fho/jenn/       3d sphere+Appolonius
 
sorry don t remember the other but look google

[blob]

This one perhaps: http://dmitrybrant.com/2007/01/24/hyperbolic-tessellations

[ericr]

yes yes it"s the good on tanks