# Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles-I

### Ana M. d'Azevedo Breda

Universidade de Aveiro, Portugal### Patrícia S. Ribeiro

E.S.T. Setúbal, Setubal, Portugal### Altino F. Santos

Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal

## Abstract

The study of dihedral f-tilings of the Euclidean sphere $S_{2}$ by triangles and $r$-sided regular polygons was initiated in 2004 where the case $r=4$ was considered [6]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and $r$-sided regular polygons, for any $r≥5$, was described. Later on, in [3] the classfication of all f-tilings of $S_{2}$ whose prototiles are an equilateral triangle (a 3-sided regular polygon) and an isosceles triangle are obtained. In this paper we extend these results presenting the study of all spherical f-tilings by equilateral triangles and scalene triangles in a particular way of adjacency. For this particular case of adjacency, we get two continuous families of f-tilings denoted by $F_{1}$ and $F_{2}$. The combinatorial structure including the transitive properties under the action of their symmetry groups is given in Table 1.

## Cite this article

Ana M. d'Azevedo Breda, Patrícia S. Ribeiro, Altino F. Santos, Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles-I. Rend. Sem. Mat. Univ. Padova 124 (2010), pp. 1–23

DOI 10.4171/RSMUP/124-1