Right Triangular Spherical Dihedral f–Tilings with Two Pairs of Congruent Sides

By a dihedral folding tiling (f-tiling, for short) of the sphere S whose prototiles are spherical right triangles, T1 and T2, we mean a polygonal subdivision τ of S such that each cell (tile) of τ is congruent to T1 or T2 and the vertices of τ satisfy the angle-folding relation, i.e., each vertex of τ is of even valency and the sums of alternating angles around each vertex are π. In fact, the crease pattern associated to the subjacent graph of a spherical f-tiling satisfy the Kawasaki’s condition at any vertex v. In this paper we shall discuss dihedral f-tilings by spherical right triangles, considering two distinct cases of adjacency. We assume that from all the sides of the prototiles involved there are two pairs of congruent sides. The 3-dimensional representations of the obtained f-tilings are presented as well as the combinatorial structure. f-tilings are intrinsically related to the theory of isometric foldings of Riemannian manifolds, introduced by S. A. Robertson [6] in 1977. The classification of f-tilings was initiated by Ana Breda [1], with a complete classification of all spherical monohedral f-tilings. Later on, in 2002, Y. Ueno and Y. Agaoka [10] have established the complete classification of all triangular monohedral tilings (without any restrictions on angles). The study of dihedral f-tilings by spherical right triangles is a very extensive and exhaustive work and some particular cases were recently obtained in papers [4, 5]. A list of all dihedral f-tilings of the sphere by triangles and parallelograms including the combinatorial structure of each tiling can be found in [2]. Robert Dawson has also been interested in special classes of spherical tilings, see [7–9] for instance. We shall denote by Ω (T1, T2) the set, up to an isomorphism, of all dihedral f-tilings of S whose prototiles are T1 and T2. From now on T1 is a spherical right triangle of internal angles π2 , α and β, with edge lengths a (opposite to β), b (opposite to α) and c (opposite to π2 ), and T2 is a spherical right triangle of internal angles π2 , γ and δ, with edge lengths d (opposite to δ), e (opposite to γ) and f (opposite to π2 ) (see Figure 1). We will assume throughout the text that T1 and T2 are distinct triangles, i.e., (α, β) ̸= (γ, δ) and (α, β) ̸= (δ, γ). The case α = β or γ = δ was analyzed in [4], and so we will assume further that α ̸= β and γ ̸= δ.