ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P4.07 / 617–636
https://doi.org/10.26493/1855-3974.2740.7ab
(Also available at http://amc-journal.eu)
S2 coverings by isosceles and scalene
triangles – adjacency case II*
Catarina P. Avelino †
Center of Mathematics of the University of Minho – UTAD Pole (CMAT-UTAD),
University of Trás-os-Montes e Alto Douro, Vila Real, Portugal and
Center for Computational and Stochastic Mathematics (CEMAT),
University of Lisbon (IST-UL), Portugal
Altino F. Santos
Center of Mathematics of the University of Minho – UTAD Pole (CMAT-UTAD),
University of Trás-os-Montes e Alto Douro, Vila Real, Portugal
Received 4 April 2019, accepted 6 January 2022, published online 11 August 2022
Abstract
The aim of this paper is to complete the study and classification of spherical f-tilings by
scalene triangles T and isosceles triangles T ′ within a subclass defined by the adjacency of
the lower side of T and the longest side of T ′. It consists of eight families of f-tilings (two
families with one continuous parameter, one family with one discrete parameter and one
continuous parameter, and five families with one discrete parameter). We also analyze the
combinatorial structure of all these families of f-tilings, as well as the group of symmetries
of each tiling; the transitivity classes of isogonality are included.
Keywords: Dihedral f-tilings, combinatorial properties, spherical trigonometry.
Math. Subj. Class. (2020): 52C20, 52B05, 20B35
*This research was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tec-
nologia) within the projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UTAD, Center of Mathematics
of University of Minho, and projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID, Center for
Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon.
†Corresponding author.
E-mail addresses: cavelino@utad.pt (Catarina P. Avelino), afolgado@utad.pt (Altino F. Santos)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
618 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
1 Introduction
A folding tessellation or folding tiling (f-tiling, for short) of the sphere S2 is an edge-to-
edge finite polygonal tiling τ of S2 such that all vertices of τ satisfy the angle-folding
relation, i.e., each vertex is of even valency and the sums of alternating angles around each
vertex are equal to π.
F-tilings are intrinsically related to the theory of isometric foldings of Riemannian man-
ifolds, introduced by Robertson [10] in 1977. In several 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 [2], with a complete classification
of all spherical monohedral (triangular) f-tilings. Afterwards, in 2002, Ueno and Agaoka
[11] 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-tilings by two noncongruent classes
of isosceles triangles have recently been obtained [6, 7].
The study of dihedral triangular f-tilings involving scalene triangles is clearly more
unwieldy and was initiated in [1]. In this paper we complete the classification of spherical
f-tilings by scalene triangles T and isosceles triangles T ′ resulting from the adjacency of
the lower side of T and the longest side of T ′.
From now on,
(i) T denotes a spherical scalene triangle with internal angles α > β > γ and side
lengths a > b > c;
(ii) T ′ denotes a spherical isosceles triangle with internal angles (δ, δ, ε), δ ̸= ε, and side
lengths (d, d, e),
as illustrated in Figure 1.
d
e
e
T
g
b
c a
b
T
‘
a
d
d
d
Figure 1: A spherical scalene triangle, T , and a spherical isosceles triangle, T ′.
We shall denote by Ω(T, T ′) the set, up to isomorphism, of all dihedral folding tilings
of S2 whose prototiles are T and T ′ in which the lower side of T is equal to the longest
side T ′.
Taking into account the area of the prototiles T and T ′, we have
α+ β + γ > π and 2δ + ε > π.
As α > β > γ, we also have α > π3 . In [8] it was established that any τ ∈ Ω (T, T
′) has
necessarily vertices of valency four.
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 619
We begin by pointing out that any element of Ω (T, 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.
1
T
b
g
a
I.
e
d
dT
‘
2
T
b
g
a
I.
e
d
dT
‘d c= d c=
T
b
g
a
II
e
d
d
T
‘e c=
Case :I d > e Case :II d < e
Figure 2: Distinct cases of adjacency.
In this paper we will consider the second case of adjacency. The next section con-
tains the main result of this paper. In Section 2 we describe the eight families of spherical
f-tilings that we may obtain from the second case of adjacency (Figure 2-II). The combi-
natorial structure of these tilings, the classification of the group of symmetries and also the
transitivity classes of isogonality are presented. The proof of the main result consists of a
long and exhaustive method and it is presented in Section 3.
2 Main result - Elements of Ω (T, T ′) in the case of Adjacency II
Theorem 2.1. Let T and T ′ be a spherical scalene triangle and a spherical isosceles
triangle, respectively, such that they are in adjacent positions as illustrated in Figure 2-II.
Within this case, the f-tilings of Ω (T, T ′) are
Lβ , Dkε (k ≥ 4), Mγ , N k (k ≥ 6), Pk (k ≥ 3), Qk (k ≥ 4), Rk (k ≥ 6)
and Sk (k ≥ 7),
that satisfy, respectively:
(i) α+ δ + β = π, ε = π2 , γ =
π
3 , where α and β satisfy
sin2(α+ β) (1 + 2 cos(α− β)) = 2 sinα sinβ and β ∈
(
π
3
, arccos
√
6
6
)
;
(ii) α+ δ = π, δ + β + ε = π, kγ = π, δ = δ1k(ε), ε ∈
(
εmin,
(k−1)π
k
)
,k ≥ 4,
where δ1k(ε) = arctan
2 sin ε cos2 ε2
cos πk − cos2 ε
and εmin = arccos
√
1 + 8 cos πk − 1
4
;
(iii) α+ δ = π, ε = π2 , β + δ + γ = π, δ = γ and γ ∈
(
π
4 ,
π
3
)
;
620 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
(iv) α+ δ = π, ε = π2 , β + 3δ = π, kγ = π and δ = δ
2
k = arccos
√
1
2 cos
π
k , k ≥ 6;
(v) α+ δ = π, ε = π2 , 2β + 2δ = π, β + δ + kγ = π and δ = δ
3
k = arctan
(
sec π2k
)
,
k ≥ 3;
(vi) α+ δ = π, ε = π2 , 2β + 2δ + γ = π, β + δ + kγ = π, δ = δ
4
k, k ≥ 4, where
δ4k = arctan
(
sin
(k − 1)π
2k − 1
sec
π
2k − 1
)
;
(vii) α + δ = π, ε = π2 , 2β + 2δ = π, kγ = π and α = α
2
k = 2arctan(cos
π
k +√
1 + cos2 πk ), k ≥ 6;
(viii) α+ δ = π, ε = π2 , 2β + 2δ + γ = π, kγ = π and δ = δ
5
k, k ≥ 7, where
δ5k = arctan
(
sin
(k − 1)π
2k
sec
π
k
)
.
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).
Particularizing suitable values for the parameters involved in each case, the correspond-
ing 3D representations of these families of f-tilings are given in Figure 4. In each case, we
present two perspectives in order to provide a more effective visualization of each f-tiling’s
combinatorial structure. Regarding the f-tiling P k, k ≥ 3, it can be observed that, if we
consider the great circle that contains the four vertices surrounded by (β, δ, δ, β, γ, γ, ..., γ)
(marked in red) as the equator line and rotating the southern hemisphere 90 degrees (around
the “vertical” axis) we obtain the f-tiling R2k. Also, it is interesting to relate the mono-
hedral edge-to-edge tilings TI16n+8 and I8n described by Ueno and Agaoka in [11] with
the families of f-tilings Qk, k ≥ 4, and Sk, k ≥ 7, obtained by subdividing the prototypes
in the monohedral tilings into two triangles satisfying the conditions of Figure 2-II. Seeing
from another perspective, we obtain TI16n+8 and I8n eliminating the vertices surrounded
by (α, α, δ, δ) (marked in green) and two suitable edges emanating from those vertices of
Qk, k ≥ 4, and Sk, k ≥ 7, respectively.
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 621
g
g
ee
d
a
g
g
g
g
g
e
e
g
g
g
b
g
g
b
6 8
2
2
D
k
e
M
P
k
Q
k
R
k
e e
aa
d d
d
a
d
d
b
b
d
b
g
d
g
b
g
g
24
2k 2k
2
dd
b
db
d
d d
4
2
2k
k
Lb
g
N
k
ee
e e
ee
e e
ee
e e
ee
e e
ee
e e
aa
d d
4
aa
d d
2k
aa
d d
4k
aa
d d
2k
aa
d d
g
g g g
g
g
g
g
g
g
g
g
g
g g g
b
d d
g
g
b
b
d
b
bb
d
d d
4
b
g g g
d
d
b
g
g
g
g
g
d d b
b
d d
g
g
b
b
g
g g g
g
g
g
g
g
g
g
gb
d
b
bb
d
d d
4
2k 2k
2( 1)-k
k
2(2 1)-k 4( 1)-k4(2 1)-k
2S
k ee
e e
aa
d d
4k
g
g g g
g
g
g
g
g
g
g
g
2k 2k
d d b
b
d d
g
g
b
b
Figure 3: Distinct classes of congruent vertices.
622 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
a Lβ b Mγ
c D4ε d D5ε
e N 6 f N 7
g P3 h P4
Figure 4: Elements of Ω(T, T ′) in the case of adjacency II.
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 623
i Q4 j Q5
k R6 l R7
m S7 n S8
Figure 4: Elements of Ω(T, T ′) in the case of adjacency II.
The combinatorial structure of the classes of spherical f-tilings mentioned in Theo-
rem 2.1, including the symmetry groups, is summarized in Table 1 (the analysis of the
symmetry groups is similar to that applied in previous articles, e.g. [9]). Our notation is as
follows:
• |V | is the number of distinct classes of congruent vertices;
• N1 and N2 are, respectively, the number of triangles congruent to T and T ′, respec-
tively;
• G(τ) is the symmetry group of each tiling τ ∈ Ω (T, T ′) and the index of isogonality
for the symmetry group is denoted by #isog.;
• Cn is the cyclic group of order n;
• V ≃ C2 × C2 is the Klein group;
• Dn is the nth dihedral group (it consists of n rotations and n reflections);
• O is the chiral group with 24 elements;
624 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
• i(k) =
{
3k
2 + 1 if k even
3k+1
2 + 1 if k odd.
f-tiling α β γ δ ε |V | N1 N2 G(τ) #isog.
Lβ α(β)
(
π
3 , arccos
√
6
6
)
π
3 π − α − β
π
2 3 48 24 O 3
Dkε , k ≥ 4 π − δ π − δ − ε πk δ
1
k(ε) (εmin, εmax) 3 4k 4k D2k 3
Mγ π − γ π − 2γ
(
π
4 ,
π
3
)
γ π2 3 8 8 V 3
Nk, k ≥ 6 π − δ π − 3δ πk δ
2
k
π
2 4 4k 8k D2k 4
Pk, k ≥ 3 π − δ π2 − δ
π
2k δ
3
k
π
2 4 8k 8k C2 × C2 × C2 i(k)
Qk, k ≥ 4 π − δ (k−1)π2k−1 − δ
π
2k−1 δ
4
k
π
2 4 14k 14k V 4k − 2
Rk, k ≥ 6 α2k π2 − δ
π
k π − α
π
2 4 4k 4k C2 × Dk 4
Sk, k ≥ 7 π − δ (k−1)π2k − δ
π
k δ
5
k
π
2 4 8k 8k D2k 4
Table 1: Combinatorial structure of the dihedral f-tilings of S2 by scalene triangles T and
isosceles triangles T ′ performed by the lower side of T and the longest side of T ′ in the
case of adjacency II.
3 Proof of Theorem 2.1
In order to better understand the structure of each tiling and due to the complexity of a
global planar representation, in the following proof some f-tilings τ are illustrated only by
a fundamental region F that generates τ by successive reflections and rotations of F . Com-
paring the fundamental region F with its associated f-tiling τ (in Figure 4), it becomes clear
how it is generated. In two of the situations (tilings Qk and Sk), instead of a fundamental
region, we illustrate planar representations that correspond to a half of the f-tilings.
In the case of adjacency II, any element of Ω (T, 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 tiling, except where
otherwise indicated.
Observe that we have ε > π3 (since we are considering the case of adjacency II). Also,
as e = c and using spherical trigonometric formulas, we get
cos γ + cosα cosβ
sinα sinβ
=
cos ε+ cos2 δ
sin2 δ
. (3.1)
Proof of Theorem 2.1. Suppose that any element of Ω (T, T ′) has at least two cells con-
gruent, respectively, to T and T ′, such that they are in adjacent positions as illustrated in
Figure 2-II.
With the labeling of Figure 5a, we have θ1 ∈ {ε, δ}. It is easy to verify that θ1 must
be δ. In fact, if θ1 = ε, v1 cannot have valency four (see side lengths), α + ε + ρ > π,
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 625
b
g
a
e1 2
d
d
v
1
q1
a
b
g
a
e1 2
d
d
v
1
q2
d
e
3dq1
b
Figure 5: Local configurations.
∀ρ ∈ {α, β, δ, ε}, and if α + ε+ kγ = π, k ≥ 1, an incompatibility between sides cannot
be avoided.
Now, at vertex v1 (see Figure 5b) we must have
α+ δ < π or α+ δ = π.
1. Suppose firstly that α + δ < π. If θ2 = δ and ε + δ = π (Figure 6a), we reach a
contradiction at vertex v2, as ε+ β + ρ > π, for all ρ ∈ {α, β, γ, δ, ε}. In fact, taking into
account the side lengths, v2 cannot have valency four and also observe that ε + β + ρ1 ≥
α+ β + γ, ρ1 ∈ {α, β, γ}, and ε+ β + ρ2 > ε+ δ = π, ρ2 ∈ {δ, ε}.
b
g
a
e1 2
d
d
v
1
d
e
3
dq2
e
d
4d
d
5
d
e
v
2
a
b
g
a
e1 2
d
d
v
1
d
e
3
q2
e
d
4d
d
5
d
v
2
e
d
g
a
10
9
6
g
b
v
3
a
8
7
a g
a
b
g
b
g
b
a
b
b
Figure 6: Local configurations.
On the other hand, if θ2 = δ and ε + δ < π, we must have ε + δ + ρ ≤ π, for some
ρ ∈ {α, β, γ}. If ρ = α, we get ε > δ > α > β > γ; but then ε+ δ+α > α+β+ γ > π,
which is not possible. If ρ = β, we obtain δ > β and α > ε, which implies α+ δ+ ρ̄ > π,
∀ρ̄, which is a contradiction. Finally, due to an incompatibility between sides, it is not
possible to have ε+ δ + kγ = π, k ≥ 1.
Therefore, θ2 = ε and, due to the side lengths, we must have ε+ ε = π, and obviously
α > δ, with δ ∈
(
π
4 ,
π
2
)
.
1.1 If α ≥ ε, at vertex v1 (Figure 5b) we must have α + δ + kγ = π, with k ≥ 1, and
α > β > δ > γ. The last configuration extends to the one illustrated in Figure 6b.
If, at vertices v2 and v3, we have
(i) β + δ + δ = π, we reach a vertex surrounded by six angles δ, implying δ = π3 = β,
which is not possible as β > δ;
626 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
(ii) β + δ + β = π, we obtain the configuration illustrated in Figure 7a. Taking into
account the edge lengths and the fact that β > δ > π4 , at vertex v4 we reach a
contradiction.
Note that it is easy to conclude that is not possible to include angles γ in the previous sums.
b
g
a
e1 2
d
d
v
1
d
e
3
q2
e
d
4d
d
5
d
e
v
2
e
d
g
a
10
d
9
6
g
b
v
3
a
8
7
a g
a
b
g
b
g
b
a
b
b
b b
b
b
11
12
13
14
a
a
g
g
a
a
g g
15
16
v
4g
g
a
a
b
b
17
18 a
g
b
g
a d
19
20
21
22d d
d
d
d
d
e
e
ee
23
24
d
d
d
d
e
a
b
g
a
e1 2
d
d
v
1
q3
d
e
3dq1
q2
e
d
4
d
5
de
d
v
2
b
Figure 7: Local configurations.
1.2 Suppose now that α < ε.
1.2.1 If δ ≥ γ, with the labeling of Figure 7b, we have θ3 ∈ {δ, γ}. Additionally is
important to note that ε = π2 > α > β > δ ≥ γ, δ >
π
4 and β + γ >
π
2 .
If θ3 = δ, we obtain the configuration of Figure 8a. Observe that θ4 cannot be δ, as
δ + δ + δ < δ + δ + α = π and δ + δ + δ + ρ > π, with ρ ∈ {α, β, δ, ε}; ρ2 cannot be γ
due to an incompatibility between sides. Moreover, θ4 cannot be β, as δ + δ + β < π and
δ + δ + β + γ > π.
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
4
d
5
de
d
v
1
g
a
9
6
b
8
7
a
g
a
g
b
b
dq3
e
d
d
d
e
10
ee
d
d
d
d
11
12
q4
v
2
a
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
4
d
5
de
d
v
1
g
a
9
6
b
8
7
a
g
a
g
b
b
dq3
e
d
d
d
e
10
ee
d
d
d
d
11
12
q4
v
2
15
13
14
b
b b
g
a
a
g g
a
16
g
b
a
v
3
b
Figure 8: Local configurations.
Now, at vertex v2 we have necessarily β + δ + β = π or β + δ + kγ = π, with k ≥ 2.
These cases lead to the configurations illustrated in Figure 8b and Figure 9a, respectively.
In both cases, at vertex v3 we reach a contradiction. In fact, due to the edge and angles
lengths there is no way to satisfy the angle-folding relation around this vertex.
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 627
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
4
d
5
de
d
v
1
g
a
9
6
b
8
7
a
g
a
g
b
b
dq3
e
d
d
d
e
10
ee
d
d
d
d
11
12
q4
v
2
1513
14
b
a
a
g
a
18
g
b
a
v
3
g
b
g
g
g
b
b
b
a a
17
16
a
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
4
d
5
de
d
v
1 g
12
6 b
8
7
a
g
a
g
b
b
q3
16
e
d
d
14
13
15
g
g
b
g
a
a g
b b
b
a
a
a
v
2
10
9
b
b b
g
a
a
g g
11
a
a
a
b
b
g
17g
18
d
19 a
g
b
20
e
d
v
3
b
Figure 9: Local configurations.
If θ3 = γ (Figure 7b), at vertex v2 we have necessarily β+δ+β = π or β+δ+δ = π.
These cases lead to the configurations illustrated in Figure 9b and Figure 10a, respectively.
In the first case, at vertex v3 we have β + δ + δ < π and β + δ + δ + ρ > π, for all
ρ ∈ {α, β, γ, δ, ε}. In the last case, at vertex v3 we also reach a contradiction, as δ = π3
implies β = π3 and, due to the edge and angles lengths, it is not possible that this vertex
has valency greater than three.
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
4
d
5
de
d
v
1 g
12
6
8
7
a
b
q3
d
14
13g
g
a
a g
b b
b
a
v
2
9
10 b
g
a
11
e
dd
d
d
d
ee
e
d
d
v
3
a
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
4
d
5
de
d
v
1
g
6
8
7
a
b
a
g
b b
v
2
a
g
b
Figure 10: Local configurations.
1.2.2 If δ < γ, then ε = π2 > α > β > γ > δ >
π
4 . At vertex v1 (Figure 7b) we must
have one of the following situations:
(i) α+ δ+α = π; in this case (Figure 10b), there is no way to satisfy the angle-folding
relation around vertex v2.
(ii) α+ δ+ δ = π; as we can observe in Figure 11a, an incompatibility between sides at
vertex v3 cannot be avoided.
(iii) α+ δ+ γ = π; in this case (Figure 11b), there is no way to satisfy the angle-folding
relation around vertex v4.
628 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
4
d
5
de
d
v
1
g
6
8
7
a
b
v
2
dd
d d
ee
13
b
g
a
10 e e
d
d d
d
9
a
a
b
b
g
g
11
12
v
3
a
b
g
a
e1 2
d
d
v
1
d
e
3
q2
e
d
4d
d
5
d
v
2
e
d
g
a
10
d
15
6
g
b
v
3
a
8
7
a g
a
b
g
b
g
b
a
b
b
b b
b
b
9
11
14
16
a
a
g
g
a
a
g g
12
13
v
4g
g
a
a
b
b
21
18 a
g
b
g
a d
20
22
17
19d d
d
d
d
d
e
e
ee
b
Figure 11: Local configurations.
(iv) α+δ+β = π; in this situation, the last configuration extends to the one illustrated in
Figure 12a. Now, at vertex v4 we have necessarily γ+γ+ρ = π, with ρ ∈ {α, β, γ}.
It is easy to verify that the two first cases lead to impossibilities. The last case
(ρ = γ) leads to a continuous family of f-tilings formed by 72 tiles. Due to the large
dimension of the corresponding planar representation, we only illustrate its eighth
fundamental region in Figure 12b.
b
g
a
e1 2
d
d
v
1
d
e
3
q2
e
d
4d
d
5
d
v
2
e
d
g
a
10
15
6
g
b
v
3
a
8
7 a
g
a
b
g
b
g
b
a
b
b
b
b
b
9
11
14
16
a
a
g
g
a a
g g
12
13
v
4
g
g
a
a
b
b
a
b
g
e
d
de
d
d
d
d
a
ab
b
b
b
g g
a
a
a
a
g
gg
e
b
b A eighth fundamental region of
Lβ
Figure 12: Local configurations.
We denote this continuous family of f-tilings by Lβ , where
α+ δ + β = π, 2ε = π and 3γ = π.
Using Equation (3.1), we get
sin2(α+ β) (1 + 2 cos(α− β)) = 2 sinα sinβ, with π
3
< β < arccos
√
6
6
.
3D representations of Lβ are illustrated in Figure 4a.
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 629
2. Suppose now that α + δ = π. We have α > δ and α > π2 . In fact, if α ≤ δ, we would
have ε > δ ≥ α > β > γ, with δ ≥ π2 , and consequently ε+ θ2 > π, θ2 ∈ {ε, δ}.
2.1 If θ2 = δ (Figure 5b), then it is a straightforward exercise to prove that γ = πk ,
for some k ≥ 4, and the complete planar representation derives uniquely as illustrated in
Figure 13.
b
g
a
e
1 2
d
d
d
d
e
3
a
b
g
4
d
e
5
d
d
e
6
v
2
a
g
7
8
g
a
d d
d
d e
9
a
g
b 10
13
d
11
a
g
a
g
b
12
v
3
d
d
14
g
a
b
15
d
e
a
16
g
b
17
q2 d
d
d
b
18
20
d
d
g
b
a
21
22
g
19
g
a
g
ba
q1 d
eb
b
e
25
e
26
e
ed d
d
d
e
b
g
a
g
b
a
b
g
a
a
b
b
28
29
23
27
d
e
d
d
d
e
d
e
e
d
d
24
30
31
Figure 13: Planar representation of Dkε , k ≥ 4.
This family of f-tilings is denoted by Dkε , where α+ δ = π, δ+β+ε = π and kγ = π,
with k ≥ 4. Using (3.1), we get
δ = δk(ε) = arctan
2 sin ε cos2 ε2
cos πk − cos2 ε
, k ≥ 4,
with ε ∈
(
εmin,
(k−1)π
k
)
, where εmin = arccos
√
1+8 cos πk −1
4 . 3D representations of D
4
ε
and D5ε are given in Figures 4c – 4d.
2.2 If θ2 = ε, we have β ≥ δ or β < δ.
2.2.1 If β ≥ δ, we have α > π2 = ε > δ >
π
4 and the last configuration extends to the one
illustrated in Figure 14a. Now, we have θ3 ∈ {β, δ, γ}.
2.2.1.1 If θ3 = β, at vertex v2 we must have δ + β + β + kγ = π, with k ≥ 0. It is easy
to verify that k has to be zero, giving rise to the configuration of Figure 14b. At vertex v3
we obtain α + k̄γ = π, with k̄ ≥ 2. Taking into account Equation (3.1) and the relations
between angles, we get 2 cos δk = cos δ csc
δ
2 . Consequently, we obtain sin δ ≤ cos δ and
δ ≤ π4 , which is not possible.
2.2.1.2 If θ3 = δ, at vertex v2 we have δ + β + δ + kγ = π, with k ≥ 0. It is a
straightforward exercise to show that (i) if k = 0, although a complete configuration is
achieved, it leads to β = γ = π3 , which is not possible; (ii) if k = 1, again a complete
configuration is achieved, with δ = π3 = β + γ, which is a contradiction; (iii) the case
k > 1 leads to an incompatibility between sides.
2.2.1.3 If θ3 = γ, at vertex v2 we must have one of the following situations:
630 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
5
d
6
de
d
v
1
g
4 a
b
q3
v
2
a
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
5
d
6
de
d
v
1
g
4 a
b
q3
7
b
g
g
a
b
b
a
a
g
9
8
11
v
2
g a
b
10
a
b
g
v
3
b
Figure 14: Local configurations.
(i) β+δ+kγ = π, k ≥ 1; the case k > 1 leads to δ = β = π3 , which is not possible, and
so k = 1. In this case we obtain the planar representation of Figure 15. We denote
this family of f-tilings by Mγ , where α + δ = π, β + δ + γ = π and γ ∈
(
π
4 ,
π
3
)
.
Using Equation (3.1), we get δ = γ.
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
5
d
6
de
d
v
1
g
4 a
b
q3
7
g
g
a
b
b
a
a
g
9
811
v
2
g
a
b
10
a
b
gv
3
b
d
d
e
12 d d
e
13
e
d
d
14
e
d
d
15
g
b
a
Figure 15: Planar representation of Mγ .
3D representations of Mγ , γ ∈
(
π
4 ,
π
3
)
, are illustrated in Figure 4b.
(ii) β+δ+β+kγ = π, k ≥ 1; as we can observe in Figure 16a, we reach an impossibility
as there is no way to complete the sum of alternate angles around vertex v3.
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
5
d
6
de
d
v
1
g
4 a
b
q3
7
g
g
a b b
a
a
g
9
8
11
v
2
a
b 10
a
b
g
v
3
b
12
13
b
g
g
a a
g
b
a
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
5
d
6
de
d
v
1
g
4 a
b q3
b
Figure 16: Local configurations.
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 631
(iii) β + δ + δ + kγ = π, k ≥ 1; in this case there is no way to satisfy the angle-folding
relation around vertex v2.
2.2.2 If β < δ (Figure 5b), it is easy to conclude that α > π2 = ε > δ > β > γ. Now,
with the labeling of Figure 16b we have θ3 ∈ {δ, β}.
2.2.2.1 If θ3 = δ, we obtain the configuration illustrated in Figure 17a. Note that θ4
cannot be ε, otherwise there is no way to satisfy the angle-folding relation around vertex
v2. Also, θ5 cannot be ε, as it implies θ6 = α. Now, at vertex v3, we have necessarily
3δ+β = π. In fact, if 3δ = π, at vertex v2 we obtain δ+δ+β < π and δ+δ+β+ρ > π,
ρ ∈ {β, γ}, as β + γ > δ. Then, the last configuration extends uniquely to a complete
b
g
a
e1 2
d
d
d
e
3dq1
q2
e
d
5
d
6
de
d
v
1
g
4 a
b dq3
d
e
7
dq4 e
d
8
dq5
e
d9
e
11
dq6
d
e
10
d
d
v
3
v
2
a
g
g
g
aa
bb
d
d
e
a
d
d
d
a
ee
b
e
e
g
b
e
e
d
d
d
e
d
d
d
d
d
d
d
d
b A kth fundamental region of N k , k ≥ 6.
Figure 17: Local configurations.
planar representation formed by 12k tiles. Due to its large dimension, we only illustrate
the kth fundamental region in Figure 17b. As β > γ, using Equation (3.1), we must have
kγ = π, with k ≥ 6. We denote this family of f-tilings by N k, where α+δ = π, 3δ+β = π
and kγ = π, k ≥ 6. Moreover,
δ = δk = arccos
√
1
2
cos
π
k
, k ≥ 6.
3D representations of N k, for k = 6, 7, are illustrated in Figures 4e – 4f.
2.2.2.2 If θ3 = β, we obtain the configuration of Figure 18a.
It is a straightforward exercise to prove that if vertex v2 has valency six, we obtain
α+ kγ = π, k ≥ 2, or β + δ + γ = π, and in either cases Equation (3.1) has no solution.
Moreover, this equation also has no solution if there is a vertex with a sum of alternate
angles of the form β + δ + δ = π. Now, we consider separately the cases θ4 = θ5 = γ,
θ4 = θ5 = β, and θ4 = β and θ5 = γ.
2.2.2.2.1 If θ4 = θ5 = γ, vertex v2 must have valency greater than eight. In fact, valency
eight implies the existence of a vertex with a sum of alternate angles of the form β+δ+δ =
π.
Now, with the labeling of Figure 18b, if v2 has valency greater or equal to ten and
• there is an additional angle β in the sum of alternate angles (note that is not possible
to have an additional angle δ, as 2δ + 2(β + γ) > π), the last configuration extends
632 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
b
g
a
e
1
2
d
d
de
3
dq1
q2
e
d
5
d
6
d e d
v
1
g
4
a
b
bq3
78 aa
b
g g
q4
q5
v
2
a
b
g
a
e
1
2
d
d
de
3
dq1
q2
e
d
5
d
6
d e d
v
1
g
4
a
b
bq3
78 aa
b
g g
gq4
g
b
b
a
a
9
11
12
10
gq5
g
b
b
a
a
v
2
b
Figure 18: Local configurations.
to the one illustrated in Figure 19a and there is no way to satisfy the angle-folding
relation around vertex v3.
b
g
a
e
1
2
d
d
de
3
dq1
q2
e
d
5
d
6
d e d
v
1
g
4
a
b
bq3
78 aa
b
g g
gq4
g
b
b
a
a
9
11
12
10
13
14
d
d
d
e
e
15
d
gq5
g
b
b
a
a
v
2
a
b
g
a
16
g
v
3
b
a
b
g
a
e
1
2
d
d
de
3
dq1
q2
e
d
5
d
6
d e d
v
1
g
4
a
b
bq3
78 aa
b
g g
gq4
g
b
b
a
a
9
11
12
10
13
14
d
d
d
d
d
e
e
15
e
e
d
d
gq5
g
b
b
a
a
v
2
a
b
b
g a
16
17
18
d
20
19
d
d
d
d
e
e
g
v
3
v
4
b
Figure 19: Local configurations.
• β + δ + kγ = π, with k ≥ 3, we obtain the configuration illustrated in Figure 19b.
At vertex v3 we must have one of the following situations:
(i) β + δ + ε = π; this condition leads to a sum of alternate angles at vertex v4
containing ε+ δ + β + γ > π, which is not possible;
(ii) β + δ + δ + β = π; in this case we obtain a complete planar representation
formed by 16k tiles. Due to its dimension, we only illustrate one octant of the
sphere (fundamental region) in Figure 20. Observe that one of the hemispheres
is obtained from the other through a 90 degree rotation. Note that if θ6 = δ,
we would obtain β + 3δ = π and consequently no solution would exist for
Equation (3.1). We have δ = arctan
(
sec π2k
)
, β = π2 − δ, γ =
π
2k , α =
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 633
π − δ, ε = π2 and k ≥ 3. We denote this f-tiling by P
k, k ≥ 3, whose 3D
representations, for k = 3, 4, are presented in Figures 4g – 4h.
b
d
e
e
d
d
d
d
d
a
a
a
b
g
g
eg
b
Figure 20: A eighth fundamental region of Pk, k ≥ 3.
(iii) β + δ + δ + β + β = π; in this case we have necessarily k ≥ 4 and it gives
rise to a sum of alternate angles of the form α + k̄γ = π, with k̄ ≥ 2. Due
to the angles relations, we have k̄ = 2. Nevertheless, under these conditions,
Equation (3.1) has no solution.
(iv) β + δ + δ + β + γ = π; in this case we also have k ≥ 4 and we obtain a com-
plete planar representation formed by 28k tiles. Due to its large dimension,
we only illustrate one hemisphere in Figure 21. The other hemisphere is ob-
tained through a 180 degree rotation along the x axis and a reflection. We have
δ = arctan
(
sin (k−1)π2k−1 sec
π
2k−1
)
, β = (k−1)π2k−1 − δ, γ =
π
2k−1 , α = π − δ
and ε = π2 . We denote this f-tiling by Q
k, k ≥ 4, whose 3D representations,
for k = 4, 5, are presented in Figures 4i – 4j.
g
a
g
b
g
e
d
g g
g
g
e
e e
e e
e e
e e
e e
e
e
e e
e e
e e
e e
e e
ee
e
e
d
d
d
d
d
d d
dd
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d d
d d
d
d d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
a a
a
a
a
a
a
a a
a a a a a
a a a a
a
a
b b b
b b
b b
g
b b
b
b
b
b
g
a a
a a
a
a
a
bb g g
g g
g
g
b
b
g
b
b
g g
b b
g g
g
b
g g
b
bg
g
b
d
d
b
g
g
g
b
Figure 21: One hemisphere of Qk, k ≥ 4.
(v) β + δ + δ + δ = π; under this condition, it is easy to verify that we achieve at
vertex v4 (see Figure 19b) a sum of alternate angles containing δ + δ + β + γ,
but δ + δ + β + γ < δ + δ + β + δ = π and δ + δ + β + γ + ρ > π, for all
ρ ∈ {α, β, δ, ε}.
2.2.2.2.2 If θ4 = θ5 = β (Figure 18a), it is easy to observe that vertex v2 cannot be
surrounded by six consecutive angles β, as we obtain a vertex with a sum of alternate
634 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
angles of the form α+ γ+ ρ, with ρ ∈ {α, β, δ}, which is not possible. Moreover, it is not
possible to have angles γ surrounding v2, as it gives rise to a vertex with a sum of alternate
angles containing α and β. Taking into account these restrictions and analyzing the angles
relations and side lengths, at vertex v2 we must have one of the following cases:
(i) β+ δ+ β+ δ = π; in this case we obtain the configuration illustrated in Figure 22a.
b
g
a
e
1
2
d
d
de
3
dq1
q2
e
d
5
d
6
d e d
v
1
g
4
a
b
bq3
78 aa
b
g g
q4
b
a
a
9
11
12
10
13
14
d
15
d
d
q5
a
v
2
b
g
a
16
d
d
e
e
b
g
g
b
d
d
d
e
e
g
v
3
v
4
a
b
g
a
e
1
2
d
d
de
3
dq1
q2
e
d
5
d
6
d e d
v
1
g
4
a
b
bq3
78 aa
b
g g
q4
b
a
a
9
11
12
10
13
14
d
15
d
d
q5
a
v
2
b
g
a
16
17
18
20
19
d
d
e
e
b
g
g
b
d
d
d
e
e
g
v
3
v
4
b
b
d
d
d
d
e
e
a
a
e
e
b
b
d
d
d
d
a
a
g
g
21
23
22
24
e
e
e
e
d
d
d
d
b
b
a
a
g
g
g
g
g
g
d
d
d
d
a
a
31
29
30
32
27
25
26
28
b
b
v
5
v
6
b
Figure 22: Local configurations.
Given the sums of alternate angles S1 : β+ δ+β+ δ = π and S2 : β+ δ+ kγ = π,
k ≥ 3, it is a straightforward exercise to prove that at vertices v3 and v4 we must have
only S1 or a combination of S1 and S2 (note that we have symmetry, so order does not
matter). If we have a combination of S1 and S2, we obtain a complete representation
of f-tiling Pk, k ≥ 3, previously achieved. On the other hand, if we have only S1,
the last configuration extends to the one illustrated in Figure 22b. At vertices v5 and
v6 we must have only S1 or S2. In the last case, as before we obtain the f-tiling Pk,
with k ≥ 4. If S1 is the sum of alternate angles at vertices v5 and v6, then we obtain
a complete representation formed by 8k tiles. A fundamental region is illustrated
in Figure 23. For each k ≥ 6, we have α = 2arctan
(
cos πk +
√
1 + cos2 πk
)
,
β = π2 − δ, γ =
π
k , δ = π − α and ε =
π
2 . We denote this f-tiling by R
k, k ≥ 6,
whose 3D representations, for k = 6, 7, are presented in Figures 4k – 4l.
b
d
d
a
dee
g
d
a
g
b
Figure 23: A 2kth fundamental region of Rk, k ≥ 6.
(ii) β + δ+ β + δ+ β = π; this case leads to the following additional relations between
angles: α+γ+γ = π and kγ = π, with k ≥ 8. Nevertheless, under these conditions,
(3.1) has no solution.
C. P. Avelino and A. F. Santos: S2 coverings by isosceles and scalene triangles . . . 635
2.2.2.2.3 If θ4 = β and θ5 = γ (Figure 18a), it is easy to observe that vertex v2 cannot
be surrounded by the sequence (. . . , β, β, γ, γ, . . .), as we achieve a vertex with a sum of
alternate angles containing α + β, which is not possible as α + β + ρ > π, for all ρ. As
the sum of alternate angles surrounding v2 must contain at least one angle γ, taking into
account the previous restriction and analyzing angles relations and side lengths, at vertex
v2 we have necessarily β + δ + β + δ + γ = π, as illustrated in Figure 24.
b
g
a
e
1
2
d
d
de
3
dq1
q2
e
d
5
d
6
d e d
v
1
g
4
a
b
bq3
78 aa
b
g g
q4
b
a
a
9
11
12
10
13
14
d
15d
q5
a
v
2
b
g
a
16
d
eb
g
g
g
d
d
d
e
g
v
3
b
b
a
e
e
d
b
17
d
d
18
a
g
d
d
d
e
e
19
20 e
e
d
d
d
d
22
21
bq6
g
a
a
g
b
23
24
Figure 24: Local configuration.
Note that θ6 must be β (tile 23), as θ6 = δ immediately leads to an impossibility. It
is a straightforward exercise to verify that at vertex v3 we must have β + δ + kγ = π,
with k ≥ 4, or β + δ + β + δ + γ = π. In the first case, analyzing the symmetry of the
figure and all possible combinations of angles surrounding specific vertices, we obtain the
f-tiling Qk, k ≥ 4, formerly achieved. In the last case, beside this family of f-tilings, we
also obtain a complete planar representation formed by 16k tiles. Due to its dimension, we
only illustrate one hemisphere in Figure 25. the other hemisphere is obtained through a 180
degree rotation along the x axis and a reflection.
g
a
g
b
g
e
d
g g
g
g
e
e e
e e
e e
e e
e e
e
e
e e
e e
e e
e e
e e
ee
e
e
d
d
d
d
d
d d
dd
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d d
d d
d
d d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
a a
a
a
a
a
a
a a
a a a a a
a a a a
a
a
b b b
b b
b b
g
b b
b
b
b
b
g
a a
a a
a
a
a
bb g g
g g
g
g
b
b
g
b
b
g g
b b
g g
g
b
g g
b
bg
g
b
d
d
b
g
g
g
b
Figure 25: One hemisphere of Sk, k ≥ 7.
We have δ = arctan
(
sin (k−1)π2k sec
π
2k
)
, β = (k−1)π2k − δ, γ =
π
k , α = π − δ and
636 Ars Math. Contemp. 22 (2022) #P4.07 / 617–636
ε = π2 . We denote this f-tiling by S
k, k ≥ 7, whose 3D representations, for k = 7, 8, are
presented in Figures 4m – 4n.
ORCID iDs
Catarina P. Avelino https://orcid.org/ 0000-0003-4335-0185
Altino F. Santos https://orcid.org/ 0000-0002-8638-4644
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[9] A. M. R. d’Azevedo Breda and A. F. Santos, Symmetry groups of a class of spherical foldings
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