Dihedral f-tilings of the sphere by rhombi and triangles

We classify, up to an isomorphism, the class of all dihedral f-tilings of S 2 , whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classiﬁed in Breda and Santos (2004). Here we complete the classiﬁcation considering the case of non-equiangular rhombi, see Tab. 1 and Fig. 24.


Introduction
An isometric folding is a non-expansive locally isometry that sends piecewise geodesic segments into piecewise geodesic segments of the same length.An isometric folding is a continuous map that need not to be differentiable.The points where it is not differentiable are called singular points.The foundations of isometric foldings of Riemannian manifolds are introduced by Robertson (1977).
For surfaces, the set of singular points gives rise to two colored tilings (called f-tilings) with the property that each vertex has even valency and obey the angle folding relation, i.e., the sums of alternating angles around each vertex is π.
The classification of f-tilings started out by Breda (1992), where a complete classification of all monohedral f-tilings of the sphere by triangles was done.
The complete classification of monohedral tilings of the sphere by triangles (which, obviously, includes the monohedral triangular f-tilings) was made clear by Ueno and Agaoka (2002).This classification was partially done by Sommerville (1922), and an outline of the proof was provided by Davies (1967).
For additional information on tilings, see Grünbaum and Shephard (1986).
Our interest focuses in dihedral spherical f-tilings.

Preliminaries
Definition 2.1 Let S 2 be the euclidean 2-sphere of radius 1.A spherical moon L is said well centered if its vertices belong to the great circle S 2 ∩ {(x, y, z) ∈ R 3 | x = 0} and the semi-great circle bisecting L contains the point (1, 0, 0).If L 1 and L 2 are two well centered spherical moons with orthogonal vertices, then L 1 and L 2 are said to be orthogonal.
Definition 2.2 By a well centered spherical quadrangle (WCSQ) we mean a spherical quadrangle which is the intersection of two well centered spherical moons with distinct vertices.
The properties of WCSQ was described by Breda and Santos (2003), where was established that any spherical quadrangle with congruent opposite internal angles is congruent to a WCSQ.Definition 2.3 Let Q and T be, respectively, a spherical triangle and a spherical quadrangle.A dihedral tiling of the sphere S 2 with prototiles Q and T is an edge-to-edge polygonal subdivision τ of S 2 , such that each tile (face) of τ is isometric either to Q or T .
In addition, if the number of geodesic rays (edges) emanating from each vertex of τ is even, making alternated angles then τ is called a dihedral folding tiling or dihedral f-tiling for short.
In short, dihedral means that every tile of τ is congruent to one of the fixed sets Q and T , called the prototiles of τ .Definition 2.4 Let v be a vertex of a (dihedral) f-tiling.The number of edges emanating from v is called the valency of v.
In Fig. 1 is illustrated a vertex v obeying the angle folding relation (α In Breda and Santos (2004) the classification of dihedral f-tilings of S 2 by spherical triangles and equiangular spherical quadrangles (i.e., spherical quadrangles that are congruent to a WCSQ which is the intersection of two orthogonal well centered spherical moons) was presented.
Here we extend the study of f-tilings presenting the classification of all dihedral f-tilings by spherical triangles and non-equiangular spherical rhombi, in other words, spherical quadrangles whose edges have the same length and with distinct pairs of opposite internal angles.
In Breda and Santos (2003) it was shown that any non-equiangular spherical rhombus Q is congruent to a WCSQ which is the intersection of two non-orthogonal well centered spherical moons L 1 and L 2 with the same angle measure θ (Fig. 2).
Notation: We shall denote by Ω(Q, T ) the set, up to an isomorphism, of all dihedral f-tilings of S 2 , whose prototiles are a non-equiangular spherical rhombus Q and a spherical triangle T .Proposition 2.1 (Breda and Santos (2004)) Let τ ∈ Ω(Q, T ).If M > 0 and N > 0 denote, respectively, the number of spherical quadrangles congruent to Q and the number of spherical triangles congruent to T of τ , and E and V denote, respectively, the number of edges and vertices of τ , then: ii) there are at least 6 + M vertices of valency four; iii) π < β + γ + δ < 3π 2 and α 1 + α 2 > π, where α 1 and α 2 are the pairs of opposite internal angles of Q, and β, γ and δ are the internal angles of T .
We shall describe Ω(Q, T ), considering separately different cases depending on the nature of Q and T .In order to get the dihedral f-tilings of Ω(Q, T ) we find useful to start by considering one of its planar representation (PR), beginning with a common vertex to a spherical quadrangle and a spherical triangle in adjacent positions (it must be pointed out that there is always a vertex satisfying this condition).
Whenever a tiling is drown in one hemisphere, it means that the plane x = 0, say, is a plane of symmetry.The whole tiling is then obtained by reflecting the picture in this plane.
Spherical trigonometry is based on the following cosine rules: The internal angles α, β and γ of a spherical triangle T obey: where a, b and c are the lengths of the edges opposite to α, β and γ, respectively.This formulas are of crucial interest to prove the results that follow.For a detailed discussion on spherical trigonometry, see Berger (1996).
3 Prototiles: Non-equiangular rhombus and equilateral triangle Proposition 3.1 If Q and T are, respectively, a non-equiangular spherical rhombus and an equilateral spherical triangle, then Ω(Q, T ) is the empty set.
Proof: Let Q be a non-equiangular spherical rhombus with pairs of opposite internal angles α 1 and α 2 (α 1 > α 2 ), and let T be an equilateral spherical triangle with internal angle β.
2 ) and β > π 3 , then v has valency four, and the cells surrounding v have, in cyclic order, angles measure Hence in order to have the angle folding relation satisfied we must have α 2 + α 2 = π, i.e., α 2 = π 2 .This information allows us to adjust some more cells to the initial PR leading to the PR illustrated in Fig. 3-II.In order to go on with the PR of τ we must analyze the angles behavior around v 1 .We already know that the cyclic sequence of their positions contains the sequence (. . ., β, α 2 , β, . . .).As 2β < π and 2β + ρ > π for all ρ ∈ {β, α 1 , α 2 }, then there is no way to position another cells around v 1 in order to have the angle folding relation full filled.And so Ω(Q, T ) = ∅.✷ The argumentation used to prove the Propositions stated in the following Sections are somehow similar to the one used on proof of Prop.3.1.Thus, we decided to present only the proofs we consider more representative.A detailed study containing the proofs of those results can be found in a technical report placed in the web site http://www.mat.ua.pt/ambreda/rhombi.pdf 4 Prototiles: Non-equiangular rhombus and isosceles triangle Through this section Q is a non-equiangular spherical rhombus with pairs of opposite internal angles α 1 and α 2 (α 1 > α 2 ), and T is an isosceles spherical triangle with internal angles β, γ, γ (β = γ).
Any element of Ω(Q, T ) has, at least, two cells congruent, respectively, to Q and T , such that they are in adjacent positions in one of the situations illustrated in Fig. 4.
Next, by "unique f-tiling" we mean unique up to an isomorphism.
) and γ = π k , for some k ≥ 4. In this situation for each k ≥ 4, Ω(Q, T ) contains a unique dihedral f-tiling, denoted by R k , such that the angles around vertices are the ones represented at the head of the Fig. 7 , where R 3 and U i , 1 ≤ i ≤ 4 are non-isomorphic dihedral f-tilings, such that the angles around vertices are the ones represented in Fig. 7.
Proof: We only present the proof of the i) case.
In order to obtain a complete PR of an element τ ∈ Ω(Q, T ), the PR illustrated in Fig. 4-C is extended in a unique way to get the PR illustrated in Fig. 5, where α 1 + γ = π, α 2 + β = π and kγ = π, for some k ≥ 4. (Note that the case k = 3 implies α The side length of T opposite to γ and the side length of Q are the same.Using the formulas (2.1) we may show that 3D representations for k = 4 and k = 5 are illustrated in Fig. 6.
a 1 g Fig. 6: 3D representations.Fig. 7 exhibits the vertices in the dihedral f-tilings corresponding to the i) and ii) cases.In Fig. 8 a 5 Prototiles: Non-equiangular rhombus and scalene triangle Here Q is a non-equiangular spherical rhombus with pairs of opposite internal angles α 1 and α 2 (α 1 > α 2 ), and T is a scalene spherical triangle with internal angles β, γ and δ, such that β > γ > δ.
If τ ∈ Ω(Q, T ), then there are necessarily two cells of τ congruent, respectively, to Q and T , such that they are in adjacent positions in one of the situations illustrated in Fig. 9. Suppose that the PR illustrated in Fig. 10 is contained in a complete PR of any element τ ∈ Ω(Q, T ).It follows that α 2 + γ ≤ π.We shall consider separately the cases α 2 + γ < π and α 2 + γ = π.
where the internal angles of the prototiles present in the dihedral f-tilings are: The angles around vertices are positioned as illustrated in Fig. 11.Proof: We only present the proof of the i) case.
2.1.The first situation leads us immediately to a contradiction as illustrates Fig. 13.In fact, this case gives rise to a vertex w surrounded by the cyclic sequence (α 2 , α 2 , α 2 , α 2 , . . .).As α 2 > γ, then 2α 2 +γ > π = α 2 +2γ > 2α 2 .And so the sums of alternating angles around w must obey 2α 2 +δ = π, as illustrated in the figure.This procedure gives rise to a vertex surrounded by α 1 and γ in adjacent positions, leading us to an incompatibility between angles around this vertex (see side length of the prototiles).2.2.In the second case we may extend without more restrictions the PR illustrated in Fig. 10 to get the one illustrated in Fig. 14.This construction leads to two antipodal vertices surrounded by at least six δ angles.It is a straightforward exercise to show that those vertices are enclosed exclusively by δ angles.Therefore kδ = π, for some k ≥ 4. On the other hand α 2 + 2γ = π with γ < α 2 .Hence γ < π 3 < α 2 and so δ = π 4 or δ = π 5 , as γ + δ > π 2 .Firstly, consider that δ = π 4 .Then α 1 = 3π 4 and the PR illustrated in Fig. 14  as shown in Fig. 15.The sum of alternating angles around vertices verify (5.2) On the other hand b is also the side length of Q, therefore (5.3)By (5.2) and ( 5.3) we conclude that And so To complete the proof we shall consider now that δ = π 5 (Fig. 14).As before the side length of T , b, opposite to β is common to Q, leading to conclude that γ = arccos 1 − cos π 5 = arccos 3 , contradicting (5.1).3D representations of some of these f-tilings are illustrated in Fig. 16.For additional information see also Tab. 1.
Proposition 5.4 If α 2 + γ = π, then Ω(Q, T ) is the family of f-tilings denoted by R 2 δ δ∈]0, π 3 [ , where cos γ + cos δ = 1.(Its 3D representation is illustrated in Fig. 17.) 3 , then γ = π 3 and so α 1 = α 2 = 2π 3 , giving rise to a dihedral f-tiling, which was established in Breda and Santos (2004).Proposition 5.6 If the PR illustrated in Fig. 18 is contained in a complete PR of an element τ ∈ Ω(Q, T ), then: 3 and δ = π 5 .Moreover, the first situation leads to two dihedral f-tilings denoted by M 3 and S 2 , and in the second case one gets three dihedral f-tilings denoted by G 1 , G 2 and G 3 , respectively.In any case Q is decomposed in four triangles congruent to T , bisecting Q twice, by α 1 and by α 2 .The sum of alternating angles around vertices are represented in Fig. 19.For a 3D representation of the mentioned f-tilings, see Fig. 20.
Finally we present the f-tilings obtained from the adjacency illustrated in Fig. 9-E  Any planar representation of R k φ has the configuration of Fig. 5 (replacing one of the γ angles by δ) and the sum of alternating angles around vertices are given by: k , k ≥ 3, when considered the adjacency illustrated in Fig. 9-E In both cases cos β + cos γ + cos δ = 1.
Proof: We shall suppose that τ ∈ Ω(Q, T ) has two cells in adjacent positions as illustrated in Fig. 9-E (the other case is analogous).Using similar argumentation to the one used before we obtain in a unique way a planar representation with the configuration of Fig. 5 (replacing one of the γ angles by δ), where α 1 + δ = π, α 2 + β = π and γ = π k , for some k ≥ 3. Now, as Q has all congruent sides and distinct pairs of opposite internal angles, then Q is congruent to a WCSQ, as seen in Breda and Santos (2003).This means that Q = L 1 ∩ L 2 , for some non-orthogonal well centered spherical moons L 1 and L 2 , with angle measure γ = π k (k ≥ 3), see Fig. 21.
With the labelling of Fig. 21, one gets for some x ∈]0, π 2 [ and k ≥ 3.And so cos π k + cos β + cos δ = 1.The relations (5.4) and (5.5) define, respectively, an increasing continuous bijection between x ∈]0, π 2 [ and β ∈] arccos  In Fig. 24 it is shown a 3D representation of the dihedral f-tilings described on Tab. 1. Observe that for the families M k , E k and S k−1 , k ≥ 3 we have chosen M 5 , E 3 and S 3 as representatives for their 3D representations.
Proposition 4.1 If Ω(Q, T ) = ∅, then there cannot be a pair of tiles in the position of Fig. 4-A or Fig. 4-B.Proposition 4.2 If a f-tiling contains two cells in the position of Fig. 4-C, then