I wrote my Thesis at Valencia  University Spain (1982) ,its title was:

Finite type Riemann Surfaces associatted to Schwarz Functions Triangles.

Superficies de Riemann de tipo finito asociadas a las Funciones Triangulares de Schwarz

Alfonso García Marcos

alfonso@systembit.es

In my papier I have asociatted a Riemann Surface to each Schwarz function triangle.

After that ,I´ve got genus and geometric density of spherical tesselation expresions in a new method.

Then I  see Their Poincare Groups ( of each above Riemann Surfaces) as index normal finite subgroup of  Г(2)  (Thanks to  Modular Function Lambda).

Then I  calculate signature of these fuchisian Groups.

Finally I  see there are only nine ( of  above Riemann Surfaces) more Dihedrical cases.

I think my idea  is a new interpretation of Schwarz  triangles ,  different  one to  the Famous  Schwarz Classification based on 14 Schwarz  triangles +Dihedrical cases.

Klein in its The Icosahedron (Dover 1956,p.56) give us a  Albebraic Curve

x=x(n)

for the Octahedro  (432) also for Tethaedro and Icosahedro..

In order to obtain it , He works with different Invariants Forms.

And  in its  Vorlesungen Uber die Hipergeometriche funktion (1933, p.229) make this  for (5/2,3,2).

In my work I tried to classify all (Platonic Solid,Kepler-Poinsot Solids and in General Schwarz Triangles) of these Algebraic Curves.

In my notation

z=z(w).

I define a Riemann Surface associate to each Regular Tesselation. And after that for Schwarz Triangles.

.

I know Puiseux series in Branch Points.of

1) Proyection   z    -------->     w

2)     "               w     -------->     z

and Total Order of Branch.(thanks to Hypergeometric Functions)

Mixing around this concept I can obtain Algebraic Genus and Geometric Density (for overlapping Tesselation) formulas. For cases of Finite Regular Spherical Tesselation are the same than Coxeter Formulas. (Page.32 ,3.1.1,1 and Page 38, 3.1.2,2)

For Schwarz Triangles  genus´s  is new ( Page 68, 4.3,1) (  Page 69,4.5,1)

Also I , and for Finite Regular Spherical Tesselation I work with regular nets on a Riemann Surface and I obtain with this methode genus same formula.

Klein (1933,p.294 ) also  speak about The Transformation Invariant  Group  of its x=x(n).

In my notation this group is The Automorfism Group of  the Covering  You can see in my p.34.

I can see (Thanks to Modular Function Landa) Poincare Group of all Riemann Surfaces associatted to Schwarz Triangles as   Normal Finite index  Subgroups of   Γ( 2) ,

Cocients Groups are Geometric Rotation Groups of Tesselations.

I calculate signature of all of these Fuchsian Groups.(Page 71, 4.6)

Then I can say

There are only nine Different Riemann Surface associatted to Schwarz Triangles. More dihedrical cases.