A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
C. P. AVELINO and A. F. SANTOS
Abstract. In this paper we present some spherical f-tilings by two (distinct) right triangles. We
classify the group of symmetries of the presented tilings and the transitivity classes of isohedrality are
also determined. The combinatorial structure is given in Table 1.
1. Introduction
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Let S 2 be the Riemannian sphere of radius 1. By a dihedral folding tiling (f-tiling for short) of the
sphere S 2 whose prototiles are spherical right triangles, T1 and T2 , we mean a polygonal subdivision
τ of S 2 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 this paper we shall discuss dihedral f-tilings by two spherical right
triangles.
F-tilings are intrinsically related to the theory of isometric foldings of Riemannian manifolds
introduced by S. A. Robertson [4] in 1977. In fact, the set of singularities of any spherical isometric
folding corresponds to a folding tiling of the sphere.
We shall denote by Ω (T1 , T2 ) the set, up to an isomorphism, of all dihedral f-tilings of S 2 whose
prototiles are T1 and T2 . From now T1 is a spherical right triangle of internal angles π2 , α and β
Received February 24, 2012.
2010 Mathematics Subject Classification. Primary 52C20, 05B45, 52B05, 20B35.
Key words and phrases. dihedral f-tilings; spherical trigonometry; combinatorial properties; symmetry groups.
Research funded by the Portuguese Government through the FCT – Fundação para a Ciência e a Tecnologia – under
the project PEst-OE/MAT/UI4080/2011.
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) 6= (γ, δ) and (α, β) 6= (δ, γ). The case α = β or γ = δ was analyzed in
[2], and so we will assume further that α 6= β and γ 6= δ.
a
g
a
c
f
d
T1
T2
p
2
d
p
b
2
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b
Figure 1. Prototiles: spherical right triangles T1 and T2 .
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It follows straightway that
π
π
and γ + δ > .
2
2
A spherical isometry σ is a symmetry of a spherical tiling τ if σ maps every tile of τ into a tile
of τ . The set of all symmetries of τ is a group under composition of maps denoted by G(τ ). In
this paper the group of symmetries of each f-tiling τ ∈ Ω (T1 , T2 ) will also be presented.
(1)
α+β >
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We say that the tiles T and T 0 of τ are in the same transitivity class, if the symmetry group
G(τ ) contains a transformation that maps T into T 0 . If all the tiles of τ form one transitivity class
we say that τ is tile-transitive or isohedral. If there are k transitivity classes of tiles, then τ is
k-isohedral or k-tile-transitive. Dihedral f-tilings are k-isohedral for k ≥ 2. In this paper we also
determine the transitivity classes of isohedrality of each presented tiling.
A fundamental region of τ is a part of S 2 as small or irredundant as possible which determines τ
based on its symmetries. More precisely, the image of a point in S 2 under the symmetry group of
τ forms an orbit of the action. A fundamental region of τ is a subset of S 2 which contains exactly
one point from each orbit, therefore if |G(τ )| = n, then the area of a fundamental region of τ is
4π
n .
It is well known that any spherical isometry is either a reflection, a rotation or a glide-reflection
which consists of reflecting through some spherical great circle, and then rotating around the line
orthogonal to the great circle and containing the origin.
Let v and v 0 be vertices of a spherical f-tiling τ and let σ be a symmetry of τ such that σ(v) = v 0 .
Then every symmetry of τ that sends v into v 0 is composition of σ with a symmetry of τ fixing v 0 .
On the other hand, the isometries that fix v 0 are exactly the rotations around the line containing
±v 0 and the reflections through the great circles by ±v 0 .
In what follows, Rθx , Rθy and Rθz denote the rotations through an angle θ around the xx axis,
yy axis and zz axis, respectively. The reflections on the coordinate planes xy, xz and yz are
denoted, respectively, by ρxy , ρxz and ρyz with the notation used in [1] . For instance, it follows
x
x
that Rθx ρxy = ρxy R−θ
, Rθx Rπy = Rπy R−θ
, ρxy Rθz = Rθz ρxy and ρxy ρyz = ρyz ρxy = Rπy . Besides, 2k
is the smallest positive integer such that (ρxy Rzπ )2k = id.
k
The nth dihedral group Dn (group of symmetries of the planar regular n-gon) consists of n
rotations and n symmetries (reflections). If a is a rotation of order n and b is a symmetry, then
h a, b : an = 1, b2 = 1, ba = an−1 b i is a group presentation for Dn . Moreover, the elements 1, a,
. . . , an−1 , b, ab, . . . , an−1 b are pairwise disjoints.
In the next section we present some examples of f-tilings by right-triangles on a case of adjacency.
The complete classification of all f-tilings by the considered prototiles is far from being achieved.
We believe that this very hard study leads to infinite families of f-tilings (with discrete or continuous
parameters) without no patterns, precisely due to the fact that both prototiles are right triangles.
In contrast with other prototiles (consider, for instance, the rectangle illustrated in Figure 2(a)
with α > π2 that cannot be subdivided in two tiles of the same family), the right triangles are
special prototiles since they can be subdivided into two new right triangles (therefore within the
same family of prototiles), and so on (see Figure 2(b)).
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a
a
a
a
(a)
(b)
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Figure 2. Prototiles subdivisions.
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2. Some examples of f–tilings by right–triangles
on a case of adjacency
We will suppose that any element of Ω (T1 , T2 ) has at least two cells congruent, to T1 and T2 ,
respectively, such that they are in adjacent positions as illustrated in Figure 3.
d a
c
a =e
f
T2
T1
p
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g
d
b
2
2
v
b
Figure 3. Case of adjacency.
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As a = e, using spherical trigonometric formulas, we obtain
cos β
cos γ
(2)
=
.
sin α
sin δ
We will assume that all the edges of T1 and T2 are pairwise distinct (except a and e). The study
of right triangular spherical dihedral f–tilings on this case of adjacency, where the prototiles have
two pairs of congruent sides, was already presented in [3].
In order to pursue any dihedral f-tiling τ ∈ Ω (T1 , T2 ), we start by considering one of its local
configurations, beginning with a common vertex to two tiles of τ in adjacent positions, and then
enumerating the following tiles according to the angles and edges relations until a complete f-tiling
or an impossibility is achieved.
Lemma 2.1. With the previous assumptions vertex v (Figure 3) has valency four.
Proof. Suppose that vertex v has valency greater than four (Figure 4(a)). Then, if
• β > α, we must have π2 + β + kγ = π or π2 + β + kδ = π for some k ≥ 1. In the first
case an incompatibility between sides cannot be avoided around vertex v. In the last case
we obtain π2 + β + kδ = π = π2 + α + γ + (k − 1)δ which is not possible (observe that
π
π
2 + 2 + α + β + γ + δ > 2π).
• β < α, one gets π2 + k1 β + k2 γ = π or π2 + k1 β + k2 δ = π for some k1 , k2 ≥ 1. Analogously,
in the first case an incompatibility between sides cannot be avoided around vertex v. In the
last case we obtain π2 + k1 β + k2 δ = π = π2 + α + γ + (k1 − 1)β + (k2 − 1)δ which is not
possible.
Therefore, vertex v has valency four as illustrated in Figure 4(b).
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Now, we analyze the cases when the valency of the vertices v1 , v2 , v3 and v4 is four. As there
was not imposed any strict order relation between the angles, it is enough to consider vertices v1
and v2 , for instance. As previously mentioned, when the valency of these vertices is greater than
four, the study is not complete and we only present some examples of f–tilings.
Proposition 2.2. Let T1 and T2 be spherical right triangles such that they are in adjacent
positions as illustrated in Figure 4(b). If at least one of the vertices v1 and v2 has valency four,
then Ω(Q, T ) 6= ∅ iff α + δ = π and β = γ = πk for
some k ≥ 3. In this case, for each k ≥ 3, there
is a family of f-tilings denoted by Rkα with α ∈
are given in Figure 6.
(k−2)π (k+2)π
2k ,
2k
. Planar and 3D representations
Proof. Suppose that any element of Ω (T1 , T2 ), has at least two cells congruent, respectively, to
T1 and T2 , such that they are in adjacent positions as illustrated in Figure 4(b).
1. If v1 has valency four, taking into account the edge lengths, we must have α + δ = π and the
last configuration is extended to the one illustrated in Figure 5. Repeating the same argument, we
get β = γ = πk with k ≥ 3. The extension of the last configuration gives rise to the“closed” planar
representation, see Figure 6(a). We denote this family of f–tilings by Rkα with α ∈
(k−2)π (k+2)π
2k ,
2k
and k ≥ 3. A 3D representation of Rkα for k = 3, is given in Figure 6(b).
v4
d a
2
d a
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Figure 4. Local configurations.
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p
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In the planar representation, the dark region corresponds to a fundamental region of R3α . In
fact, any symmetry of this f-tiling fixes the north vertex N ; the symmetries that fix N are the
rotations id = R0z , Rz2π , Rz4π , and the reflections ρyz , ρ1 = ρyz ◦ Rz2π , ρ2 = ρyz ◦ Rz4π , and so the
3
3
3
3
symmetry group of R3α is isomorphic to D3 generated by Rz2π and ρyz . Similarly, we prove that
3
G Rkα is isomorphic to Dk . It follows immediately that Rkα is 2-isohedral with respect to the
symmetry group.
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da
...
Figure 5. Local configuration.
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2
p
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(a) Planar representation of Rkα
(b) 3D representation of R3α
S
Figure 6. f-tilings Rkα , with α ∈
“
(k−2)π (k+2)π
, 2k
2k
”
and k ≥ 3.
2. Suppose now that v2 has valency four. As γ 6= π2 (otherwise a = f ), it follows that δ + γ = π
(see Figure 7). At vertex v1 we must have γ + kα = π, k ≥ 1. However, an incompatibility between
sides cannot be avoided around this vertex for all k.
g d a
5
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Figure 7. Local configuration.
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Given the reasons referred earlier, when the valency of all the vertices vi , i = 1, 2, 3, 4, is greater
than four, we present examples of f–tilings satisfying:
√
(i) α + δ = π2 , β = π3 , γ = π4 and δ = arctan 2,
(ii) α + 2δ = π, γ =
Close
π
2k
(iii) δ + α + γ = π, β =
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and β =
π
3
π
3,
k = 2, 3,
and α ∈ π6 , π3 ,
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(iv) α + 3δ = π, γ = π4 and β = π3 .
In the first case we consider three f-tilings, say Da , Db and Dc .
In Figure 8 a planar and 3D representations of Da are given. The dark region in the planar
representation corresponds to a fundamental region of Da . In fact, the dark line corresponding
to a great circle composed by 12 segments of length π6 is invariant under any symmetry. Thus,
any symmetry of Da fixes N or maps N into S. The symmetries that fix N are generated by the
rotation Rz2π and the symmetries that send N into S are Rπx , RπL1 = Rz2π ◦Rπx and RπL2 = Rz4π ◦Rπx .
3
3
3
2
Note that RπLi = id, i = 1, 2, and the symmetry group of Da is G(Da ) = hRz2π , Rπx i ' D3 ; Da
3
is 16-isohedral (8 transitivity classes of triangles T1 and 8 transitivity classes of triangles T2 ).
The f-tilings Db and Dc (Figure 9 and Figure 10) are obtained from Da rotating the southern
hemisphere π6 for the left and the right, respectively.
A planar and 3D representations of Db are illustrated in Figure 9. The three great circles x = 0,
y = 0 and z = 0 depicted in the planar representation have 8 segments of length π4 and there
exist exactly six vertices surrounded by 8 angles γ (say N, S, W, E, C, L). The symmetries that fix
vertex N are generated by Rzπ and ρyz with 8 symmetries (for the other vertices the analysis is
2
similar). And so the symmetry group contains 48 symmetries. It follows immediately that G(Db )
is isomorphic to C2 × S4 , the octahedral group, and Db is 2-isohedral.
Concerning to the tiling Dc , the dark line (corresponding to the equator) is invariant under any
symmetry. Thus, any symmetry of Dc fixes N (and S) or maps N into S (and vice-versa). The
symmetries that fix this points are generated by the rotation Rz2π and the reflection ρyz . On the
3
other hand, the reflection ρxy sends N into S and commutes with the previous symmetries. It
follows that G(Dc ) is isomorphic to C2 × D3 and Dc is 8-isohedral. Finally, the dark region in the
planar representation of Dc corresponds to a fundamental region of this tiling.
In the case (ii) we consider four f-tilings denoted by L, M, Q2 and Q3 .
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(a) Planar representation of Da
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Figure 8. Sf-tiling Da .
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(b) 3D representation of Db
Figure 9. f-tiling Db .
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Figure 10.S f-tiling Dc .
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(a) Planar representation of L
Figure 11. f-tiling L.
(b) 3D representation of L
A planar representation of L is illustrated in Figure 11(a). One has β = π3 , γ = π4 , δ =
√
arccos 42 ≈ 69.3◦ and α = π − 2δ ≈ 41.4◦ . Its 3D representation is given in Figure 11(b).
Similarly to the previous case, any symmetry of L fixes N or maps N into S. The symmetries
that fix N (and S) are generated, for instance, by the rotation Rz2π and the reflection ρyz giving
3
rise to a subgroup G of G(L) isomorphic to D3 . To obtain the symmetries that send N into S, it
is enough to compose each element of G with a = Rzπ ρxy . Now, one has
3
5 yz
a ρ
z
= R 5π ρ ρ
3
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xy yz
=
z
R 5π Rπy
3
= Rπy Rzπ3 = ρyz ρxy Rzπ3 = ρyz a.
Moreover, | < a > | = 6 and ρyz ∈<
/ a >. Therefore, < a, ρyz >= G(L) ' D6 and L is 2-isohedral.
x
Observe that Rπ ∈ G(L); in fact, Rπx = ρyz Rz2π ◦ Rzπ ρxy .
3
3
A planar representation
of M is illustrated in Figure 12(a). One has β = π3 , γ = π8 , δ =
arcsec 4 cos π8 ≈ 74.3◦ and α = π − 2δ ≈ 31.4◦ . Its 3D representation is given in Figure 12(b).
G(M) contains a subgroup S isomorphic to D4 generated by Rzπ and ρyz . On the other hand,
2
a = ρxy Rzπ is also a symmetry of M that maps N into S. Since a has order 4, then G(M) is
4
isomorphic to D8 generated by a and ρyz . Finally, M is 9-isohedral.
We illustrate planar and 3D representations of Q2 and Q3 in Figure 13 and Figure 14, respecπ
π
, δ = arcsec 4 cos 2k
and α = π − 2δ, k = 2, 3.
tively. One has β = π3 , γ = 2k
The great circle x = 0 depicted in the planar representation of Q2 has 6 segments of length
π
2
3 . Any symmetry of Q fixes C or maps C into L. As before, the symmetries that fix C are
generated, for instance, by the rotation Rx2π and the reflection ρxz , and so G(Q2 ) contains a
3
subgroup G isomorphic to D3 . In order to obtain all the symmetries that send C into L, it is
enough to compose each element of G with ρyz which commutes with all elements of G. And so
G(Q2 ) ∼
= C2 × D3 . It follows immediately that Q2 is 3-isohedral.
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Figure 12. f-tiling M.
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(a) Planar representation of Q2
Figure 13. f-tiling Q2 .
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(a) Planar representation of Q3
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Figure 14. f-tiling Q3 .
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(b) 3D representation of Q3
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(a) Planar representation of Gα , α ∈
S
`π
6
,
π
3
´
(b) 3D
of Gα ,
` representation
´
α ∈ π6 , π3
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Figure 15. f-tiling Gα , α ∈
`π
6
,
π
3
´
.
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(a) Planar representation of C
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(b) 3D representation of C
Figure 16. f-tiling C.
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The tiling Q3 has exactly four vertices surrounded by 12 angles γ and denoted by vi , i = 1, 2, 3, 4.
Any symmetry of Q3 that sends vi into vj , i 6= j, consists of a reflection on the great circle
containing the remaining vertices. On the other hand, the symmetries of Q3 fixing one of these
four vertices form a subgroup G isomorphic to D3 . Thus, G(Q3 ) contains exactly 24 symmetries.
Now, we easily conclude that G(Q3 ) is the group of all symmetries of the regular tetrahedron or
the group of all permutations of four objects, S4 . Finally, Q3 is 3-tile-transitive with respect to
this group.
In the case (iii) we consider a family of f-tilings denoted by Gα , α ∈ π6 , π3 . The corresponding
planar and 3D representations are illustrated in Figure 15. Due to the condition (2) we have α = γ,
and so δ = π − 2α.
G(Gα ) contains a subgroup G isomorphic to D3 generated by Rz2π and ρyz . On the other hand,
3
a = ρxy Rzπ is a symmetry of Gα that maps N into S. Similarly to some previous cases, we conclude
3
that G(Gα ) is isomorphic to D6 generated by a and ρyz . Finally, Gα is 2-isohedral.
In the last case we consider an f-tiling denoted by C whose
planar and 3D representations are
√
2−2
π
π
1
presented in Figure 16. One has β = 3 , γ = 4 , δ = 2 arccos 4 ≈ 49.2◦ and α = π − 3δ ≈ 32.4◦ .
The dark line in the planar representation corresponds to the equator that is invariant under any
symmetry of C. With the labeling used in this figure, the symmetries that fix N are generated, for
instance, by the rotation Rzπ and the reflection ρyz giving rise to a subgroup G of G(C) isomorphic
2
to D4 . On the other hand, a = ρxy Rzπ is also a symmetry of C that maps N into S. The symmetry
4
group of C is then G(C) = < a, ρyz >' D8 , C is 12-isohedral.
3. Summary
In Table 1 is shown a list of the presented spherical dihedral f-tilings whose prototiles are spherical
right triangles, T1 and T2 , of internal angles π2 , α, β, and π2 , γ, δ, respectively, in the case of
adjacency illustrated in Figure 3. Our notation is as follows:
• |V | is the number of distinct classes of congruent vertices,
• N1 and N2 , respectively, are the number of triangles congruent to T1 and T2 , respectively,
used in each tiling.
• G(τ ) is the symmetry group of each tiling τ ∈ Ω (T1 , T2 ); by Cn we mean the cyclic group
of order n; Dn is the dihedral group of order 2n; S4 is the group of all permutations of four
distinct objects; the octahedral group is Oh ∼
= C2 × S4 (the symmetry group of the cube).
• I(τ ) corresponds to the number of isohedrality classes of each tiling
τ ∈ Ω (T1 , T2 ).
The distinct classes of congruent vertices of each f-tiling are illustrated in Figure 17.
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Rka , k b 3
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p
p
d
d
gg gg
p
g
2
ggg p
2
a
d
a
Figure 17. Distinct classes of congruent vertices.
d
d
a
d
a a
d d
a
d
f-tiling
Rkα
Da
Db
Dc
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α
„
«
(k − 2)π (k + 2)π
d
,
2k
2k
π
−δ
2
π
−δ
2
π
−δ
2
π − 2δ
M
π − 2δ
Q2
π − 2δ
Q3
π − 2δ
Gα
“π π”
,
6 3
C
π − 3δ
β
γ
π
k
π
3
π
3
π
3
π
3
π
3
π
3
π
3
π
3
π
3
π
k
π
4
π
4
π
4
π
4
π
8
π
4
π
6
α
π
4
|V |
N1
N2
G(τ )
4
2k
2k
Dk
2
2
5
48
48
D3
16
2
5
48
48
C2 × S4
2
2
6
48
48
C2 × D3
8
2
4
4
12
24
D6
2
5
48
96
D8
9
4
12
24
C2 × D3
3
4
24
48
S4
3
3
12
12
D6
2
5
48
144
D8
12
δ
π−α
arctan
arctan
arctan
√
√
√
I(τ )
√
arccos
“
π”
arcsec 4 cos
8
“
π”
arcsec 4 cos
4
“
π”
arcsec 4 cos
6
π − 2α
√
1
2−2
arccos
2
4
Table 1. Combinatorial structure of some dihedral f–tilings of S 2 by right triangles on the case of adjacency of
Figure 3.
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Acknowledgment. The authors would like to thank the anonymous referee for helpful comments and suggestions.
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1. Breda A. M. and Santos A. F., Symmetry groups of a class of spherical folding tilings, Applied Mathematics &
Information Sciences, 3 (2009), 123–134.
“
“
π”
π”
and γ, γ,
,
2. Avelino C. P. and Santos A. F., Right triangular dihedral f–tilings of the sphere: α, β,
2
2
accepted for publication in Ars Combinatoria.
3.
, Right triangular spherical dihedral f–tilings with two pairs of congruent sides, accepted for publication
in Applied Mathematics & Information Sciences.
4. Robertson S., Isometric folding of Riemannian manifolds, Proceedings of the Royal Society of Edinburgh, 79
(1977), 275–284.
C. P. Avelino, Department of Mathematics, UTAD, 5001 - 801 Vila Real, Portugal, e-mail: cavelino@utad.pt
A. F. Santos, Department of Mathematics, UTAD, 5001 - 801 Vila Real, Portugal, e-mail: afolgado@utad.pt
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