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

*Finite
type **Riemann
Surfaces associatted to Schwarz **F**unctions
**T**riangles.*

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

__ __**Alfonso
García Marcos**

__Madrid 14.03.06__

**In
my papier I have asociatted a Rieman****n****
****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
Dihed****r****ical
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

**T****here**
**are only nine Different Riemann
Surface associatted to Schwarz Triangles. ****More
dihedrical cases.**