Altino F. Santos

A class of spherical dihedral f-tilings

The classification of all dihedral triangular f-tilings of the Riemannian sphere S^2 whose prototiles are an equilateral triangle and an isosceles triangle and the identification of their symmetry groups is given. We also determine their classes of isogonality and isohedrality.

Spherical f-tilings by scalene triangles and isosceles trapezoids, I

The study of the dihedral f-tilings of the sphere S^2 whose prototiles are an equilateral or isosceles triangle and an isosceles trapezoid was described in [C.P. Avelino, A.F. Santos, Spherical f-tilings by (equilateral and isosceles) triangles and isosceles trapezoids, 2008 (submitted for publication)]. In this paper we generalize this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.

DEFORMATION OF F-TILINGS VERSUS DEFORMATION OF ISOMETRIC FOLDINGS

We present some relations between deformation of spherical isometric foldings and deformation of spherical f-tilings. The natural way to deform f-tilings is based on the Hausdorff metric on compact sets. It is conjectured that any f-tiling is (continuously) deformable in the standard f-tiling τs = {(x, y, z) ∈ S2 : z = 0} and it is shown that the deformation of f-tilings does not induce a continuous deformation on its associated isometric foldings.

Dihedral f-tilings of the sphere by equilateral and scalene triangles-I

The study of dihedral f-tilings of the Euclidean sphere S 2 by triangles and rsided regular polygons was initiated in 2004 where the case r = 4 was considered [5]. 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 classication of all f-tilings of S 2 whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical f-tilings by equilateral triangles and scalene triangles of angles , and ( > > ) whose edge adjacency is performed by the side opposite to was done in [4]. In this paper we extend these results considering the edge adjacency performed by the side opposite to .

Geometric and combinatorial structure of a class of spherical folding tessellations — I

A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed. The combinatorial structure of each tiling is also analysed.

Spherical folding tessellations by kites and isosceles triangles IV

The classification of the dihedral folding tessellations of the sphere and the plane whose prototiles are a kite and an equilateral triangle were obtained in [C. Avelino and A. Santos, Spherical and planar folding tessellations by kites and equilateral triangles, Australasian Journal of Combinatorics, 53 (2012), 109–125.]. Recently, this classification was extended to isosceles triangles so that the classification of spherical folding tesselations by kites and isosceles triangles in three cases of adjacency was presented in [C. Avelino and A. Santos, Spherical Folding Tessellations by Kites and Isosceles Triangles: a case of adjacency, Mathematical Communications, 19 (2014), 1–28.; C. Avelino and A. Santos, Spherical Folding Tessellations by Kites and Isosceles Triangles II, International Journal of Pure and Applied Mathematics, 85 (2013), 45–67.; C.Avelino and A.Santos, Spherical Folding Tessellations by Kites and Isosceles Triangles III, submitted.]. In this paper we finalize this classification presenting all the dihedral folding tessellations of the sphere by kites and isosceles triangles in the remaining three cases of adjacency, that consists of five sporadic isolated tilings. A list containing these tilings including its combinatorial structure is presented at the end of this paper.

A New Metric on Spherical f-Tilings

Abstract The continuous deformation of any spherical isometric folding into the standard spherical folding fs, defined by fs(x, y, z) = (x, y, |z|), has been an open problem since 1989. Some relations between the deformation of spherical isometric foldings and the deformation of spherical f-tilings are analyzed, as they are closely related. The natural way to deform f-tilings is based on the Hausdorff metric on compact sets. However, this metric does not induce a continuous deformation on its associated isometric foldings. A new metric on spherical f-tilings will be introduced and a new contribution to the deformation of isometric foldings (via deformation of f-tilings) will be given.

Dihedral f‐Tilings of the Sphere by Isosceles Triangles and Isosceles Trapezoids

We present the study of dihedral f‐tilings of the sphere, with prototiles a spherical isosceles triangle and a spherical isosceles trapezoid. The combinatorial structure, including the symmetry group of each tiling, is given in Table 1.