ericr


« on: August 08, 2012, 12:32:12 PM » 

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 orientationpreserving 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.
order3 heptagonal tiling order7 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
