  In this paper, following Grothendieck {\it Esquisse d'un programme}, which
was motivated by Belyi's work, we study some properties of surfaces $X$ which
are triangulated by (possibly ideal) isometric equilateral triangles of one of
the spherical, euclidean or hyperbolic geometries. These surfaces have a
natural Riemannian metric with conic singularities. In the euclidean case we
analyze the closed geodesics and their lengths. Such surfaces can be given the
structure of a Riemann surface which, considered as algebraic curves, are
defined over $\bar{\mathbb{Q}}$ by a theorem of Belyi. They have been studied
by many authors of course. Here we define the notion of connected sum of two
Belyi functions and give some concrete examples. In the particular case when
$X$ is a torus, the triangulation leads to an elliptic curve and we define the
notion of a "peel" obtained from the triangulation (which is a metaphor of an
orange peel) and relate this peel with the modulus $\tau$ of the elliptic
curve. Many fascinating questions arise regarding the modularity of the
elliptic curve and the geometric aspects of the Taniyama-Shimura-Weil theory.
