ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 419-443 https://doi.org/10.26493/1855-3974.1401.7d9 (Also available at http://amc-journal.eu) S2 coverings by isosceles and scalene triangles - adjacency case I Catarina P. Avelino * Centre of Mathematics of the University ofMinho - UTAD Pole (CMAT-UTAD), University ofTras-os-Montes e Alto Douro, Vila Real, Portugal affiliated also with: Center for Computational and Stochastic Mathematics (CEMAT), University ofLisboa (IST-UL), Portugal Altino F. Santos Centre of Mathematics of the University ofMinho - UTAD Pole (CMAT-UTAD), University ofTras-os-Montes e Alto Douro, Vila Real, Portugal Received 9 May 2017, accepted 16 October 2017, published online 9 February 2019 The aim of this paper is the study and classification of spherical f-tilings by scalene triangles T and isosceles triangles T'. Due to the complexity of this wide class of tilings, we consider a subclass performed by the adjacency of the shortest side of T and the longest side of T'. It consists of seven families of f-tilings (four families with one discrete parameter and one continuous parameter, two families with one discrete parameter and one sporadic f-tiling). We also analyze the combinatorial structure of all these families of f-tilings, as well as the group of symmetries of each tiling and the transitivity classes of isohedrality and isogonality. Keywords: Dihedral f-tilings, combinatorial properties, spherical trigonometry, symmetry groups. Math. Subj. Class.: 52C20, 52B05, 20B35 1 Introduction A folding tessellation or folding tiling (f-tiling, for short) of the sphere S2 is an edge-to-edge finite polygonal tiling t of S2 such that all vertices of t satisfy the angle-folding relation, i.e., each vertex is of even valency and the sums of alternate angles around each vertex are equal to n. *This research was partially supported by Funda^ao para a Ciencia e a Tecnologia (FCT) through projects UID/MAT/00013/2013 and UID/Multi/04621/2013. E-mail addresses: cavelino@utad.pt (Catarina P. Avelino), afolgado@utad.pt (Altino F. Santos) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 420 Ars Math. Contemp. 16(2019)445-463 F-tilings are intrinsically related to the theory of isometric foldings of Riemannian manifolds, introduced by Robertson [8] in 1977. In some situations (beyond the scope of this paper), the edge-complex associated to a spherical f-tiling is the set of singularities of some spherical isometric folding. The classification of f-tilings was initiated by Breda [1], with a complete classification of all spherical monohedral (triangular) f-tilings. Afterwards, in 2002, Ueno and Agaoka [9] have established the complete classification of all triangular monohedral tilings of the sphere (without any restrictions on angles). Curiously, the triangular tilings of even valency at any vertex are necessarily f-tilings. Dawson has also been interested in special classes of spherical tilings, see [3, 4, 5], for instance. Spherical f-filings by two noncongruent classes of isosceles triangles have recently obtained [2, 7]. From now on, (i) T denotes a spherical scalene triangle with internal angles a > ft > y and side lengths a > b > c; (ii) T' denotes a spherical isosceles triangle with internal angles (S, S, e), S = e, and side lengths (d,d,e), as illustrated in Figure 1. Figure 1: A spherical scalene triangle, T, and a spherical isosceles triangle, T'. Taking into account the area of the prototiles T and T', we have a + ft + y > n and 25 + e > n. As a > ft > y, we also have a > 3. In [6] it was established that any f-tiling by T and T' has necessarily vertices of valency four. We begin by pointing out that any f-tiling by T and T', in which the shortest side of T is equal to the longest side of T', has at least two cells congruent to T and T', respectively, such that they are in adjacent positions and in one and only one of the situations illustrated in Figure 2. Our aim in this paper is to classify f-tilings in the first case of adjacency (Figure 2-Case I). Next section contains the main results of this paper. In Subsection 2.1 we describe six families of spherical f-tilings and one single f-tiling that we may obtain in this case of adjacency. The combinatorial structure of these f-tilings and the classification of the group of symmetries and also the transitivity classes of isogonality and isohedrality are presented in Subsection 2.2. The proof of the main result consists in a long and exhaustive methodology and it is presented in Section 3. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 421 J3 I.1 I.2 II Case I: & > e Case II: & < e Figure 2: Distinct cases of adjacency. 2 Main result 2.1 f-tilings in the adjacency case I Theorem 2.1. Let T and T' be a spherical scalene triangle and a spherical isosceles triangle, respectively, such that they are in one of the adjacent positions illustrated in Figure 2-Case I. Then, from this we obtain six families of spherical f-tilings and one isolated f-tiling, Gk (k > 4), Gk (k > 4), H, D (k > 3), F (k > 4), that satisfy, respectively: I (k > 3), Jf (k > 4), Sk n ■^min 2 (i) a + S = n, S + fi + e = n, kY = n, e = ek (S), S G ek (S) = 2 arccot ^2 cos ^ csc 2S — cot S^ and •/1 + 8 cos f — 1 2), k > 3, where Smk in = arccos 4 (ii) a + S = n, a + fi + e = n, S + fi + y = n, kY = n, S = Sk, k > 4, where Sf=arccot tan 2k (2—sec2 2k )); (iii) a + S = n, a + fi + e = n, S + fi + y = n, 2fi + y + e = n, kY = n, S = Sk, k > 4; (iv) a + S = n, a + y + Y = n, 3fi + e = n, 5y = n, where fi = fi0 = 4 arcta^ yÇ+4V5—2^22+6^5; (v) a + S = n, 2fi + y + e = n, kY = n, a = a1(fi), fi G (firkin, fiO), k > 4, where / n ni n \ \ af (fi) = arccos ( — cos — sec — cos (fi +--) ) k k 2k 2k fimin = maX { p arccos Q sec 2k) — 2k ^ and n lk = (k — 1)n ; fimax 2k ; 422 Ars Math. Contemp. 16(2019)445-463 (vi) a + S = n, 2fi + e = n, kj = n, a = a|(fi), fi G (fim|n, 2 ), k > 3> where a|(fi) = arccos cos ^ cosfij and o2 k I n \/cos2 I +8 - cos | 1 fi^n = maW k, arccos -^-1 ^ ; (vii) a + e = n, fi + 2S = n, ky = n, a = a| (fi), fi G (|, fi^L), k > 4, where fi n (fi ) = arccos ( 2 sin2--cos — and „, ,/1 + 8 cos | - 1 fimax = 2 arcsin 1 4 For each family of f-tilings we present the distinct classes of congruent vertices in Figure 3 (including the respective number of vertices in each tiling). 2k 2k Gk 2k 2k 2k Gk 3k y y P P k H 4k P E P P 2k y Py 2k JP 2k 2k Figure 3: Distinct classes of congruent vertices. 3 a fc k k k k k Particularizing suitable values for the parameters involved in each case, the corresponding 3D representations of these families of f-tilings are given in Figures 4-10. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 423 Figure 4: f-tilings in the adjacency case I; the family. Figure 5: f-tilings in the adjacency case I; the Qk family. Figure 6: f-tilings in the adjacency case I; the Gk family. 424 Ars Math. Contemp. 16(2019)445-463 Figure 7: f-tilings in the adjacency case I; the isolated f-tiling. (m) F Figure 8: f-tilings in the adjacency case I; the Fk family. (n)15 0o)Xi (p) I5 Figure 9: f-tilings in the adjacency case I; the Ik family. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 425 (q )Jâ Figure 10: f-tilings in the adjacency case I; the Jk family. (r )J$ (s) JS 2.2 Symmetry groups and combinatorial structure In this subsection we present the group of symmetries of each spherical f-tiling mentioned in Theorem 2.1. The number of transitivity classes of tiles and vertices of each tiling is indicated in Table 1. Any symmetry of Dk, k > 3, fixes the north pole N = (0,0,1) (and consequently the south pole S = -N) or maps N into S (and consequently S into N). The symmetries that fix N are generated, for instance, by the rotation Rfn (of an angle tt around the z axis) k k and the reflection pyz (on the coordinate plane y o z) giving rise to a subgroup of G(Dk) isomorphic to Dk, the dihedral group of order 2k. Now, the map $ = R| o pxy = pxy o R| is a symmetry of D^ that changes N and S. One has 4>2k-i o pyz = pyz o ^ and ^ has order 2k. It follows that ^ and pyz generate G(Dk), and so it is isomorphic to D2k. Moreover, Dk is 2-tile-transitive and 3-vertex-transitive with respect to this group. The analysis considered to the combinatorial structure of Dk also applies to the family of f-tilings Gk, k > 4. And so G(Gk) = D2k. Gk is 3-isohedral and 4-isogonal. Concerning the family of f-tilings Gk, k > 4, we have that G(Gk) = Dk, since in this case there is no symmetry sending the north pole into the south pole. Moreover, Gk has 6 transitivity classes of tiles, and so it is 6-isohedral. The vertices of Gk form 8 transitivity classes. Regarding the symmetry group of H, the symmetries that fix N are generated by the rotation Rzfn and the reflection pyz on the plane x = 0. On the other hand, 5 ^ = Rzn ◦ p _ ◦ pxy 5 is also a symmetry of H that sends N into S. Thus, we conclude that G(H) is isomorphic to Dio, the dihedral group of order 20. H is 4-tile-transitive and 5-vertex-transitive. Any symmetry of I^, k > 3, fixes N or maps N into S. The symmetries that fix N are generated, for instance, by the rotation Rzfn of order k and the reflection pyz, giving rise k to a subgroup S of G(I^) isomorphic to Dk. To obtain the symmetries that send N into 426 Ars Math. Contemp. 16 (2019) 445-463 S it is enough to compose each element of S with pxy. Since pxy commutes with RZ and k pyz, we may conclude that G (Ik) is isomorphic to C2 x Dk. Ik has 2 transitivity classes of tiles with respect to the group of symmetries and 3 transitivity classes of vertices. Similarly to previous cases, we have G(F|) = G( ) = D2k. is 3-isohedral and 4-isogonal and is 2-isohedral and 3-isogonal. The combinatorial structure of the class of spherical f-tilings described in the previous subsection, including the symmetry groups, is summarized in Table 1. Our notation is as follows: • |V | is the number of distinct classes of congruent vertices; • Ni and N2 are, respectively, the number of triangles congruent to T and T', respectively; • G(t) is the symmetry group of each tiling t and the indices of isohedrality and isogonality for the symmetry group are denoted, respectively, by #isoh. and #isog. 3 Proof of Theorem 2.1 In the case of adjacency I, any f-tiling by T and T' has at least two cells congruent to T and T', respectively, such that they are in adjacent positions and in one and only one of the situations illustrated in Figure 2. After certain initial assumptions are made, it is usually possible to deduce sequentially the nature and orientation of most of the other tiles. Eventually, either a complete tiling or an impossible configuration proving that the hypothetical tiling fails to exist is reached. In the diagrams that follow, the order in which these deductions can be made is indicated by the numbering of the tiles. For j > 2, the location of tiling j can be deduced directly from the configurations of tiles (1,2,... ,j - 1) and from the hypothesis that the configuration is part of a complete f-tiling, except where otherwise indicated. Observe that we have S > n. Also, as d = c and using spherical trigonometric formulas, we get cos y + cos a cos 8 e -—-— = cot S cot —. (3.1) sin a sin p 2 Proof of Theorem 2.1. We consider separately the subcases illustrated in Figure 2-Case I. Case I.1: With the labeling of Figure 11(a), at vertex vi we must have a + S < n or a + S = n. Case I.1.1: Suppose firstly that a + S < n. If a < S, we must have a + S + ke = n, with k > 1. Due to the existence of vertices of valency four, it follows that S = 2, and consequently, by Equation (3.1), cos y + cos a cos 8 = 0. Nevertheless, this is not possible, since cos y > cos 8 > cos a > 0. Therefore, a > S. It follows that a>p>S>e>Y and a + S + kY = n, with k > 1; see Figure 11(b). Note that 6\ = y, otherwise at vertex v2 we get a + 8 = n = y + e, which is an impossibility. Now, we have 02 = y, ®2 = S or 02 = e. Case I.1.1.1: If 02 = y, we obtain the configuration illustrated in Figure 12(a). Due to the edge lengths, at vertex v3 we must have d3 + 8 + P < n, with p > e, which implies d3 = e. At vertex v4 we reach a contradiction, as a + S + p > n, for all p G {a, 8, S,e}. Table 1: Combinatorial structure of the dihedral f-tilings of S2 by scalene triangles T and isosceles triangles T' performed by the shortest side of T and the longest side of T' in the case of adjacency I. f-tiling a /3 7 5 e M ATi ^2 G(r) #isoh. #isog. X>£, k > 3 7T — Ô 7t — S — e 7T fc V mm' 2 / £k(ô) 3 4k 4k D2k 2 3 gk, k > 4 7T — Ô (*-!)* s k 7T fc Sk 2<5 (fc-.1),r k 4 8k 4k D2k 3 4 gk, k > 4 7T — Ô (fc-i)W s k 7T fc Sk 2(5 k 5 8k 4k Dk 6 8 U 3tt 5 /3° 7T 5 2ir 5 7T - 3/3° 4 60 20 Dw 4 5 k> 4 (oik oik \ \rmin' rmax/ 7T fc 7T — a (fc-l)* 2/3 3 8k 4k D2k 3 4 k > 3 (o2k tt\ ^mim 2 / 7T fc 7T — a 7T -2/3 3 4k 2k C2 x Dfc 2 3 Jp, k > 4 ( — «3fc ) V k ' ' max/ 7T fc 7T-/3 2 7T — a 3 4k 4k D2k 2 3 428 Ars Math. Contemp. 16(2019)445-463 Figure 11: Local configurations. Figure 12: Local configurations. Case I.1.1.2: If 02 = S (Figure 12(b)), we reach an impossibility at vertex v4, since S + S + p > n, for all p G {a, /, S, e}. Note that d3 cannot be 7 (tile 11), as it implies a sum of alternate angles at vertex v3 including the angles /, pi and p2, with pi g {a, /} and p2 G {a, /, S, e}, which is not possible due to the dimensions of the involved angles. Case I.1.1.3: Finally we consider 02 = e (Figure 13(a)). At vertex v3 we must have S + / + = n, k > k. Nevertheless, an incompatibility between sides at this vertex cannot be avoided. Case I.1.2: Suppose now that a + S = n (consequently / + 7 > S > |). If a = S = |, we also get 7 = f, which is not possible. On the other hand, if S > | > a (> / > 7), we obtain cot S < 0, thereby making Equation (3.1) infeasible. Thus, a > | > S. With the C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 429 Figure 13: Local configurations. labeling of Figure 13(b), we have 01 = 6, = e, = $ or 0i = a. Case I.1.2.1: If 01 = 6, we get the configuration illustrated in Figure 14(a). Note that, at vertex v2, it is not possible to have 6 + 6 + ky = n, with k > 1, and 6 + 6 + $ + y > n. At vertex v3 we must have a + $ + ke = n, with k > 1. Nevertheless, at this vertex we 430 Ars Math. Contemp. 16(2019)445-463 reach a contradiction, since (S + S + P) + (a + P + e) > (S + S + e) + (a + P + 7) > 2n. Case I.1.2.2: If ^ = e, we obtain the configuration of Figure 14(b). Note that if 02 = 7, we would get the angles (S, e, 7, A ...) in one of the sum of alternate angles at vertex v2; but (S + e + 7 + P) + (a + S) = (S + S + e) + (a + P + 7) > 2n, which is not possible; at tile 11, it is easy to observe that 03 = a, 7, S; on the other hand, 03 cannot be e, otherwise, at vertex V3, we get S + S + P = n, but (a + S) + (S + S + P) + (e + S + P + e +----) > 2(S + S + e) + (a + P + 7) > 3n, which is a contradiction; a similar reasoning applies to the choice of and the fact that A; = 1 in the sum S + P + fee = n, at vertex v2. We denote the continuous family of f-tilings illustrated in Figure 14(b) by Df, where a + S = n, S + P + e = n and ^7 = n, with k > 3. As 0 ■ cos — = cos S sin S cot —+ cos2 S k 2 e n -<=>■ cot — = 2 cos — csc 2S — cot S. 2 k Therefore, e = em(S) = 2 arccot ^2 cos ^ csc 2S — cot S^ , k > 3, with S G (Skkin, 2), where , Vl + Scos f — 1 n Smin = arccot —-- > 3 is obtained when e = S. The graph of this function for Sf in < S < | is outlined in Figure 15, for different values of k. 3D representations of D3, D4 and D| are given in Figures 4(a)-4(c). Case I.1.2.3: Consider = P (Figure 16(a)). At vertex v1 we cannot have a + P = n = e + 7, as a>S>e and P > 7. Thus, a> | >S>P>Y>e and a + P + ke = n, k > 1. It is easy to observe that k = 1, as k > 1 lead to a vertex with a sum of alternate angles including the angles S, S and p, with p G {a, P, S, e}, which is not possible due to the dimensions of the involved angles. The last configuration extends to the one illustrated in Figure 16(b). At vertex v2 we have necessarily one of the following situations: (i) S + P + P = n; (ii) S + P + 7 = n. Note that S + P + ke = n, k> 1 lead to a vertex with a sum of alternate angles including the angles S, S and p, with p g {a, P, S, e}. (i) If S + P + P = n, we obtain the configuration illustrated in Figure 17(a). Note that, at vertex v3, we cannot have a + 7 + 7 + kp = n, with p G {7, e} and k > 1, otherwise we get (a + 7 + 7 + kp) + (a + S) + (S + P + P) > (a+P + 7) +(a + P+y) + (S + S + e) > 3n, which is not possible. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 431 Figure 15: e = ek (J), with S^in < S < f, and for k = 3,4,5,..., to. P 8 Y 4 a 3 5 5 Y a 1 5 01= P P * a Y V1 (a) 10 e 4 3 Y a S S Y Y a S S 1 6i p 7 a v, 1 p p ea a Y S S e 6 a 8 Y 9 e S S p (b) Figure 16: Local configurations. v At vertex v4 we must have k7 = n, with k > 4. As S = 27 and n < S + S + e = 47 + e, we conclude that k = 4, which is not possible as S < f. (ii) If S + fi + y = n, the last configuration gives rise to the one illustrated in Figure 17(b), where 02 can be e or S. According to the selection for 02, we obtain the planar representations illustrated in Figures 18(a) and 18(b), respectively. In the first case we have a + S = n, a + fi + e = n, S + fi + 7 = n, kY = n, with k > 4, and 1 n ( 2 n ) " — tan - I 2 — sec2 2 S=Sk=arcco^1tan2k (2 - sec2 2k )) 432 Ars Math. Contemp. 16(2019)445-463 (a) (b) Figure 17: Local configurations. Note that by Equation (3.1) we have cos k + cos S cos(S + k) cos S sin (S + sin(S + k) = cos (S + 2k) n n \ c n cos — cos 0 +--+ cos 0 cos — =0 k V + 2k J + 2k 2 cos 0 cos3 —-— sin 0 cos — sin — = 0 2k k 2k n 1 n 2 n -<=>■ cot S = tan---tan — sec2 —. 2k 2 2k 2k We denote this family of f-tilings by Gk, k > 4. 3D representations of Gk, k = 4, 5, 6, are presented in Figures 5(d) - 5(f). In the second case we have a + S = n, a + ft + e = n, S + ft + 7 = n, 2ft + 7 + e = n, kY = n, with k > 4, and S = Sk; we denote this family of f-tilings by Gk. 3D representations, for k = 4, 5,6, are presented in Figures 6(g) -6(i). Case I.1.2.4: If 01 = a (Figure 19(a)), we must have ft < S, otherwise there is no way to satisfy the angle-folding relation around vertex v1. Then, a> ) >S>ft>Y and S > e. Now, we have 0) = ft, 0) = y or 0) = e. Note that 02 cannot be S, as S + ft + e + p > n, for all p G {ft, y}. n i-a < ri s' 0 1 Si. I £-5 Co s' Oo a s a. a £ (a)Qk, k > 4 (b) Qk,k> 4 Figure 18: Planar representations. 434 Ars Math. Contemp. 16(2019)445-463 Figure 19: Local configurations. Case I.1.2.4.1: If O2 = 0, we get the configuration illustrated in Figure 19(b), where a + 2y = n and, at vertex vi, 30 + ke = n, k > 1. As k > 1 implies the existence of a vertex with a sum of alternate angles containing S + S + 0, and (30 + ke) + (2S + 0) + (a + S) > (a + 0 + y) + 2(2S + e) > 3n, we conclude that k = 1. Now, O3 G {e, 7}. If O3 = e (Figure 20(a)), at vertex v2 we reach a contradiction, as for p G {0,7}, we get S + 0 + e + p > S + 0 + e + 7 > 2S + e > n. On the other hand, if O3 = 7, the last configuration extends to the one illustrated in Figure 20(b). If O4 = e (Figure 21(a)), at vertex v3 we must have S + 20 = n, as S + 20 + p > n, for all p G {a, 0,7, S, e} (note that a + 0 + e = 30 + e = n, implying 7 > e; consequently a > 2 > S > 0 > 7 > e). As k7 = n, 47 = S + 27 < a + 27 = n and 67 = 3S > n, we that b — ^ Tr\int1w with tV»A rAmainino /^rrnHitirrnc wit* retain rv — _ 0 _ _ 5 ' 0 10 ' conclude that k = 5. Jointly with the remaining conditions, we obtain a = 31, 0 = , 7 = n, S = 2t and e = 10. Nevertheless, under these conditions, Equation (3.1) is impossible. On the other hand, if O4 = S, we obtain the planar representation illustrated in Figure 21(b). We have ^3 + 4V5 - 2^22 + 6V5, 3n „ „ a = —, 0 = 4 arctan 5 ' H n 2n 7 =—, S =—, and e = n — 30. 55 We denote this f-tiling by H, whose 3D representation is presented in Figure 7(j). Case I.1.2.4.2: If O2 = 7, we obtain the configuration illustrated in Figure 22(a). Note that, at vertex v1, all the alternate angle sums containing 0+0+7+p, with p g {a, 0,7, S}, exceed n, and so 0 + 0 + 7 + ke = n, with k = 1 (k > 1 implies the existence of a vertex with alternate sum S + S + 0 = n and e > 0). C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 435 Figure 20: Local configurations. Now, 03 must be ft or 7. In the first case (Figure 22(b)), we observe that at vertex v3 we must have S+S+ft = n, implying at vertex v4 the existence of an alternate angle sum containing a + ft + 7 > n, which is an impossibility. On the other hand, if 63 = 7, the last configuration extends to the one illustrated in Figure 23. We denote this family of f-tilings by , where a + S = n, 2ft + 7 + e = n and k7 = n, with k > 4. As 7 = k < ft < S, ft + y > S, using Equation (3.1) we get cos k + cos a cos ft — cos a sin (ft + 2k) sin ft cos (ft + 2k) n ^^ cos — cos k n n / n \ -<=>■ cos a = — cos —+ sec — cos ft +-- k + 2k Vft + 2k) n ( n \ n cos — cos ft +—- + cos a cos —- = 0 k V 2k J 2k n n ( n t n \ k Therefore, i n ni n \ \ a = ai (ft) = arccos ( — cos — sec —- cos (ft +—- ) ) , k > 4, y V k 2k V 2k// " (a) (b) Planar representation of 7-L. Figure 21: Local configurations. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 437 Figure 23: Planar representation of Fk. 438 Ars Math. Contemp. 16(2019)445-463 with £ e (£lkn,£max), where n\k i n (1 n \ n pzkn = ma« —, arccos - sec —-- £m 1 n ' k, V 2 2k I 2k and £ ifc max (k - 1)7 2F~ are obtained, respectively, when e = y or e = S and a = S. Note that if e = S, we get n n / n — cos — sec — cos £ +-- k 2k V£ + 2k cos ( 2£ + k n ( 2 / „ n \ \ n (n n cos 2k (2 cos2 (£ + 2k) - 0 = - cos k cos (£ + 2k cos +1. \ = - cos f + Vcos2 k + 8cos22k VP + 2k7 4 cos ^ 2k cos f£ + *) = - cos n + (2c°f 2k + 1) V^ 2k J 4 cos 2k / n \ 1 n cos £ +--= - sec —. V£ + 2k7 2 2k The graph of a — a.\(£), for £mkn < £ < £max, is outlined in Figure 24, for different values of k. Note that the condition e < S is equivalent to a < 2£ + n. Figure 24: a = a\(£), with £lkn < £ < £max, and for and for k = 4,5, 6,..., to. 3D representations of F|, F| and F^ are given in Figures 8(k) - 8(m). Case I.1.2.4.3: If 02 = e (Figure 19(a)), at vertex v1 we must have (i) £ + £ + ke = n, k > 1, (ii) £ + £ + £ + ke = n, k > 1 or (iii) £ + £ + y + ke = n, k > 1. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 439 Note that in all these cases k must be one, otherwise we reach a vertex with alternate sum S + S + ft = n and other vertex surrounded in cyclic order by (a, e, ft,...), which is not possible. In case (i), ft + ft + e = n, we obtain the planar representation of Figure 25. We denote Figure 25: Planar representation of Ik. this family of f-tilings by Ig, where a + S = n, 2ft + e = n and kY = n, with k > 3. Using Equation (3.1), we get a = ag (ft) = arccos cos ^ cos ftj , k > 3, with n ^cos2 n + 8 - cos n ma^< —, arccos —-k- k I k' 4 k I 8 — c0s k I r, n k < ft < 2 , where the lower and upper bounds are obtained, respectively, when e = y or e = S and a = S. The graph of this function is outlined in Figure 26, for different values of k. Note that the condition e < S is equivalent to a < 2ft. 3D representations of 1$, for k = 3,4,5, are illustrated in Figures 9(n) -9(p). In case (ii), ft + ft + ft + e = n, using similar arguments applied before, the local configuration extends to the f-tiling H, obtained in Case 1.1.2.4.1. In the last case, by symmetry we obtain the families of f-tilings and Gk, k > 4, of Cases I.1.2.4.2 and I.1.2.3(ii), respectively. Case I.2: With the labeling of Figure 27(a), at vertex v\ we must have ft + S: or ft + S < n. Case I.2.1: Suppose firstly that ft + S = n. As S = ft = | implies y = f, we have S = ft. If S > ft, by Equation (3.1), we conclude that a > |, preventing a feasible assignment for and In turn, if S < ft, we obtain a vertex (v2) surrounded by four angles S. As 2S < ft + S = n, we would have 2S + p < n, with p G {a, ft, y, S, e}, which is not possible. n 440 Ars Math. Contemp. 16(2019)445-463 Figure 27: Local configurations. Case I.2.2: Suppose now that ft + 5 < n. As in Case I.2.1, if 5 > f, we obtain a > n and no assignment for 61 and 02 is possible. Thus, 5 < f and, as any tiling has necessarily vertices of valency four, we have a > |. Now, observing Figure 27(a), we have 01 G {5, e, ft}. Case I.2.2.1: If 01 = 5, we obtain the configuration illustrated in Figure 27(b). Vertex v3 must have valency three, but in this case we get a + ft + 5 = n = 5 + e + 7, implying e > a, which is not possible. Case I.2.2.2: If 01 = e, the last configuration extends uniquely to the one illustrated in Figure 28. Note that at vertex v4, 02 must be ft and the vertex must have valency three C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 441 Figure 28: Planar representation of J. and kY = n, with k > 4. Using Equation (3.1) we get cos n + cos a cos / / k — sin — (1 — cos a) 2 sin 4 2 cos — + cos a ^2 cos2 — — 1 J =2^1 — cos2 — ) (1 — cos a) 2 — n cos a = 2 sin2--cos —. 2 k Therefore, with = ak(—) = arccos ( 2 sin2 — — cos — ) , k > 4 — < ¡3 < 2arcsin k •y/1 + 8 cos n — 1 4 where the lower and upper bounds are obtained, respectively, when — = y and e = S. The graph of this function is outlined in Figure 29, for different values of k. 3D representations of J, for k = 4,5,6, are illustrated in Figures 10(q) - 10(s). Case I.2.2.3: Finally, if 01 = —, at vertex v3 (see Figure 27(a)) we have a + — < n. a + — = n = e + y implies e > a > f > S, which is a contradiction. As any tiling has necessarily vertices of valency four, we conclude that a + — + ke = n, k > 1, and a + S = n at vertex v2, as illustrated in Figure 30, configuration coincident with the one presented in Figure 16(b), which leads to the families of f-tilings Gk and Gk (Case I.1). □ a 442 Ars Math. Contemp. 16(2019)445-463 a Figure 29: a = af (£), with n < ^ < ^fL, and for k = 4, 5, 6,..., to. References [1] A. M. R. Azevedo Breda, A class of tilings of S2, Geom. Dedicata 44 (1992), 241-253, doi: 10.1007/bf00181393. [2] A. Breda, R. Dawson and P. Ribeiro, Spherical f -tilings by two noncongruent classes of isosceles triangles - II, Acta Math. Sin. (English Series) 30 (2014), 1435-1464, doi:10.1007/ s10114-014-3302-5. [3] R. J. M. Dawson, Tilings of the sphere with isosceles triangles, Discrete Comput. Geom. 30 (2003), 467-487, doi:10.1007/s00454-003-2846-4. [4] R. J. M. Dawson and B. Doyle, Tilings of the sphere with right triangles I: The asymptotically right families, Electron. J. Combin. 13 (2006), #R48, http://www.combinatorics. org/ojs/index.php/eljc/article/view/v13i1r4 8. C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles ... 443 [5] R. J. M. Dawson and B. Doyle, Tilings of the sphere with right triangles II: The (1,3,2), (0,2 ,n) subfamily, Electron. J. Combin. 13 (2006), #R49, http://www. combinatorics.org/ojs/index.php/eljc/article/view/v13i1r49. [6] A. M. d'Azevedo Breda and A. F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles, Beiträge Algebra Geom. 45 (2004), 447-461, https://www.emis.de/journals/BAG/vol.45/no.2/8.html. [7] A. M. R. d'Azevedo Breda and P. dos Santos Ribeiro, Spherical f-tilings by two non congruent classes of isosceles triangles - I, Math.. Commun. 17 (2012), 127-149, https://hrcak. srce.hr/82991. [8] S. A. Robertson, Isometric folding of Riemannian manifolds, Proc. Roy. Soc. Edinburgh Sect. A 79 (1978), 275-284, doi:10.1017/s0308210500019788. [9] Y. Ueno and Y. Agaoka, Classification of tilings of the 2-dimensional sphere by congruent triangles, Hiroshima Math. J. 32 (2002), 463-540, http://projecteuclid.org/euclid. hmj/1151007492.