265
Acta Math. Univ. Comenianae
Vol. LXXXI, 2 (2012), pp. 265–281
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
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 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.
It follows straightway that
(1)
α+β >
π
π
and γ + δ > .
2
2
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.
266
C. P. AVELINO and A. F. SANTOS
a
g
a
c
f
d
T1
T2
p
2
d
p
b
b
2
e
Figure 1. Prototiles: spherical right triangles T1 and T2 .
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.
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
x
notation used in [1] . For instance, it follows that Rθx ρxy = ρxy R−θ
, Rθx Rπy =
y x
xy z
z xy
xy yz
yz xy
y
Rπ R−θ , ρ Rθ = Rθ ρ and ρ ρ = ρ ρ = Rπ . 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
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
267
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)).
a
a
a
a
(a)
(b)
Figure 2. Prototiles subdivisions.
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
p
g
d
b
2
2
v
b
Figure 3. Case of adjacency.
As a = e, using spherical trigonometric formulas, we obtain
cos β
cos γ
(2)
=
.
sin α
sin δ
268
C. P. AVELINO and A. F. SANTOS
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 π2 + π2 + 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 =
π
2 + α + γ + (k1 − 1)β + (k2 − 1)δ which is not possible.
Therefore, vertex v has valency four as illustrated in Figure 4(b).
v4
d a
2
d a
2
g
p
2
2
...
v
b
p
p
p
b
b
b
2
2
2
2
4
1
p
g
g
v2 g
1
p
v3
3
d a
v1
(a)
(b)
Figure 4. Local configurations.
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.
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
269
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
(k+2)π
with α ∈ (k−2)π
. Planar and 3D representations are given in Figure 6.
2k ,
2k
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).
(k−2)π (k+2)π
2k ,
2k
We denote this family of f–tilings by Rkα with α ∈
3D representation of
Rkα
and k ≥ 3. A
for k = 3, is given in Figure 6(b).
d a
2
g
g
g
g
1
p
p
p
p
2
2
2
2
4
6
3
d a
d a
p
p
p
p
2
7
2
2
2
b
b
b
b
5
8
da
...
Figure 5. Local configuration.
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
3
3
ρyz , ρ1 = ρyz ◦Rz2π , ρ2 = ρyz ◦Rz4π , and so the symmetry group of R3α is isomorphic
3
3
to D3 generated by Rz2π and ρyz . Similarly, we prove that G Rkα is isomorphic
3
270
C. P. AVELINO and A. F. SANTOS
d
d a
g g
g
S
g
g
b
2
1
p
p
p
p
2
2
2
6
b
b
d a
d a
p
p
p
p
2
2
2
a
10 d
d a
b
N
b
3
2
7
a
2
4
g
a
b
b
5
N
a
8
9
b
p
2
p
p
2
2
p
S
2
(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.
to Dk . It follows immediately that Rkα is 2-isohedral with respect to the symmetry
group.
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
p
2
p
2
2
d g
d g
6
1
p
p
p
p
b
b
2
2
2
2
4
3
g d a
v1
Figure 7. Local configuration.
271
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
g
g
p
g
g
g
p
2
p
2
2
N
b
b b
p
p
2
p
2
p
b
d
a
a
2
p
2
b
b
b b
b
2
p
2
a
a
p
2
g
g
p
2
p
2
p
2
p
2
a
p
2
d
p
p
2
n
g
g
g g
m
p
p
p
p
2
a
p
2
b
b
b b
b
j
d
2
a d
p
d
p
p
i
b
g
2
p
k
2
2
p
2
2
p
2
p
p
2
2
d
p
2
p
2
p
2
p
2
d
p
p
2
2
p
d a a d
d
a a
b b
b
b b
p
2
g
p
p
p
2
(a) Planar representation of Da
p
o
h
g
c
m
b
L2
n
l
j
a
i
p
2
d a
a
a d
a
p
p
d
2
f
k
x
e
L1
S
(b) 3D representation of Da
Figure 8. Sf-tiling Da .
g
g
g
p
2
d
2
d
d
2
N
p
a
d
2
2
p
b
S
b
p
p
2
p
b
b
2
2
p
p
2
d
b
a
a
p
2
2
p
2
2
a
p
2
2
2
b b
g g
g
g
g
g
g
2
ad
a d
a
d
a
d
d a
2
2
p
2
p
2
p
p
2
p
p
p
2
d
d
a
a d
p
p
a
p
p
2
b b
b
b
b b
a
d d a
a
e
d
a
b
b b
b
d a
d a
2
2
2
p
2
2
b
p
2
g
g
g
g g
a dd
a
p
b
b
p
g
g
g
g ...
g g
2
l
2
p
d
b
p
2
p
g g
p p
2
d a
2
p
2
2
b
p
2
b
b
2
2
b
2
p
g
p
a
a
d
g
2
2
p
2
d
d
p
p
2
2
2
g
2
p
p
a
p
2
h
d
a
c
g
g
p
d
b
2
p
o
a
d
a
p
b
b
d a
d
a
d
b
f
p
p
d
2
2
d
a
d
a d
a
2
p
p
a
2
d
2
a
p
p
p
d
2
g
g
g
2
p
g
g
g
g
g
2
p
2
272
C. P. AVELINO and A. F. SANTOS
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 =
π
2k
(iii) δ + α + γ = π, β =
(iv) α + 3δ = π, γ =
π
4
and β =
π
3
π
3,
k = 2, 3,
and α ∈ π6 , π3 ,
and β =
π
3.
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
π
6 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
3
symmetries that send N into S are Rπx , RπL1 = Rz2π ◦ Rπx and RπL2 = Rz4π ◦ Rπx .
3
3
2
= id, i = 1, 2, and the symmetry group of Da is G(Da ) =
Note that RπLi
hRz2π , Rπx i ' D3 ; Da is 16-isohedral (8 transitivity classes of triangles T1 and 8
3
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
2
and ρyz with 8 symmetries (for the other vertices the analysis is 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 other hand, the reflection ρxy
3
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 .
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 rise to a subgroup G of G(L) isomorphic to
3
D3 . To obtain the symmetries that send N into S, it is enough to compose each
273
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
g
g
p
b
b b
p
d
2
p
2
p
a
d
a
a
a
d
p
p
2
2
p
2
p
2
p
p
2
2
a
a d
d
d
d
d
a
a
p
a
2
d
2
2
g
g
2
2
b
b
b b
b
p
p
p
b
d
2
d a
a
d
a d
2
p
p
a
g
g
g
2
p
2
p
p
2
p
2
p
b
b
2
2
d a
d
a
p
2
2
p
p
2
p
2
p
b bb
b b
2
p
2
p
2
p
2
g
2
p p
2
b
2
p
2
2
2
2
p
2
2
b
p
p
g
g
p
a
a
g
p
2
2
2
g
b
p
p
b
g
g g
g
b
b
b
b
a
a
p
2
p
d
b
2
g g
g ...
g g
ad
a d
a
d
a
d
p
d
p
b b
p
a
a d
d
p
b
d
d
2
2
2
2
p
d
a
p
p
a
p
2
d
a
d
a
p
b
b
d a
2
g g
p
b
2
g
g
g
g
2
2
2
g
2
p
p
p
g
g
p
2
a
d
2
g
p
p
2
p
2
2
p
p
p
p
2
b b
p
b b
p
b b
g
2
p
p
p
2
2
g
g
p
d
pa
a
d d
2
a
b
b
b b
b
b
p
p
d
2
2
p
g
2
g
p
2
g
g
g
g
g
g
2
2
p
2
(a) Planar representation of Db
N
a
b
L
W
C
E
S
(b) 3D representation of Db
Figure 9. Sf-tiling Db .
element of G with a = Rzπ ρxy . Now, one has
3
5 yz
a ρ
z
xy yz
= R 5π ρ ρ
3
= Rz5π Rπy = Rπy Rzπ3 = ρyz ρxy Rzπ3 = ρyz a.
3
2
p
p
2
p
p
2
2
2
2
p
p
a
p
p
2
2
p
2
d a
a
d
a d
2
b
p
p
g g
g
g
p
2
b b
2
2
2
2
2
d a a d
d
d
a a
b
p
g
p
p
2
2
2
b
2
p
p
d
a
a
d
a d
2
2
b
p
2
p
2
p
d
d a a
p
g
p
a d
a
d
2
274
C. P. AVELINO and A. F. SANTOS
g
g
g
p
N
b
b b
p
d
2
p
2
p
p
2
d
d
2
p
2
a
a
a
d
2
p
a
2
p
2
2
2
2
a d
p
p
g
g
p
2
b
b
b
b b
a
d
a a d
d
p
2
b bb
b b
2
p
2
2
ad
a d
a
d
a
d
p
2
2
p p
b
p
2
p
2
2
g
g
2
e
p
p
p
2
2
p
b
p
p
p
2
2
2
p
2
2
c
b
b
2
2
b
2
a
d
d d a a
2
p
2
p
p
p
2
b b
b b
p
d
d a a d
d
a a
b b
b
b b
2
2
b
b
p
p
g
2
d
2
g
a
d
g
p
2
p
2
d a
a
d
a d
a
2
p
a
d p a pa
b
b
b b
2
bb
p
p
d
2
p
2
2
2
p
2
p
2
2
a d
a
d
a
d
a
d
p
2
p
2
p
p
2
p
S
g
2
b
g
p
2
p
p
p
2
g
p
2
2
p
g g
g
g
2
2
p
2
p
p
p
p
b
b
d
b
2
d
2
g
g
g g
a d
a a d d
2
2
p
p
p
2
2
2
2
p
b
p
b
p
b b
2
p
2
g
b
2
p
b
a
a
d a
b
2
g
p
2
p
p
g
p
a
a
d
g
p
p
2
d
d
2
p
2
2
2
p
d
a
p
p
a
p
2
d
p
g
h
p
2
d a
2
g g
2
2
f
g
g
p
d
d
a
p
b
g g
g
g
g
g ...
g g
p p
2
p
2
p
p
b
d
p
d a
a
d
a d
2
p
2
b
da a
p
p
a
g
g
2
2
2
g
2
p
p
p
g
g
p
2
g
g
g
g
g
g
2
2
p
2
(a) Planar representation of Dc
N
d
e
f
g
h
c
b
a
S
(b) 3D representation of Dc
Figure 10.S f-tiling Dc .
Moreover, | < a > | = 6 and ρyz ∈<
/ a >. Therefore, < a, ρyz >= G(L) ' D6 and
x
L is 2-isohedral. Observe that Rπ ∈ G(L); in fact, Rπx = ρyz Rz2π ◦ Rzπ ρxy .
3
3
275
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
p
2
d
d
d a
a
p
d
p
2
2
p
p
2
g
g
g
g
p
p
p
p
N
2
2
2
2
b
b
b
b
2
2
2
2
a
b
b
g
b
p
2
g
g
g
p
2
d
d a
d
p
a
d
g
p
2
g
g
2
g
g
g
a d
a d
d
d
p
2
p
2
2
p
2
g
g
d
p
d
2
p
g g
2
2
2
2
g
2
g
d
a
p
d
2
2
p
a
a
2
N
a
p
p
p
2
d d
g
p
p
2
d
p
p
a
g
p
p
d
p
g
g
p
p
p
p
2
2
p
b
2
b
b b
b
b
b
S
2
a
d
d
d
a
d
p
2
S
(a) Planar representation of L
(b) 3D representation of L
Figure 11. f-tiling L.
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
2
the other hand, a = ρxy Rzπ is also a symmetry of M that maps N into S. Since a
4
has order 4, then G(M) is 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
π
π
Figure 14, respectively. One has β = π3 , γ = 2k
, δ = arcsec 4 cos 2k
and
α = π − 2δ, k = 2, 3.
The great circle x = 0 depicted in the planar representation of Q2 has 6 segments
of length π3 . Any symmetry of Q2 fixes C or maps C into L. As before, the
symmetries that fix C are generated, for instance, by the rotation Rx2π and the
3
reflection ρxz , and so G(Q2 ) contains a 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.
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
276
C. P. AVELINO and A. F. SANTOS
d
g
d
d
p
2
p
2
d
p
d
a
a
p
p
2
2
p
a
d
p
d
2
p
p
2
N
g
g
g
g
p
2
p
2
b
g
g
a
p
2
2
p
d
p
b
b
2
2
2
2
c
h
d
p
d a
d
p
p
2
g
g
d
p
2
p
p
2
2
2
d
a
g
p
2
p
g g g
2
g
2
g g g
2
2
2
g
d
da
a
a
d a
2
d
d d
d aa
p
2
p
2
p
p
2
2
p
2
p
2
p p
2
2
b b
b
b
b b
p
b
p
2
d
b b
b
b b
2
p
p
p
p
p
p
2
2
2
p
2
2
p
d 2
d d
a
a d
2
p
g
g
2
2
g
2
p
g
g
2
g g
g
p
p 2 p
2
p 2
2
p
2
p
p
2
g g
g g
g
g
g
g
S
g g
g g
g
2
p
p
2
a
2
2
p
b bb
b
b b
p
2
a d
p
b b
b
b
2
2
2
d d
d
a
2
d
d
d
d
a a
p
2
p
a
d
d
d d
2
2
2
aad
d
d d
2
p
b
b
p
2 p
p 2
p
2
p
g
p
p
2
p
2
2
p
d a
a
dp
2
p
p
p
2
d
g
a
a
d
p
2 p 2
2 d
p p
g
p
2
2
2
a d
d d
p
2
g
g
g
d d
d
p
g
a
d d pd
p
p
p
p
2
2
g g gg
g
g p2
p p
g
p
d
g
g
p
p
d
2
2
p
2
p
2
2
g
b ad
2
p
d a2
p
dd 2
p
2
2
g
g
g
2
2
2
p
2
p
d d
d
d
a
a
p
p
p
g g
2
p
d
e
p
p
p
b
a
g
g
a d
a d
d
d
b
b
2
2
2
f
g
b
p
p
g
b
p
b
a
d
a d
b
b
b
p
p
2
2
d
p
g
g
p p
gg gg g
p
g2
d 2
p
dp
p
2
2
2 g
g
g g g g
da
g
a
d
p
g
p
2
d 2
d
g
p
g 2
2
2
i
p
p
a
b
g
g g
g
g
2
2
d a
a
2 p
2
2
p
p
d
2
2
2
b
b
a
a
d d
d
p
d
g g
g
g
g
2
p
d
b
b
p
p
p
b
2
d
2
d
d
a
2
2
p
g
g
g
a
d
a d
d
d p
p 2
2
p
p p
2
2
d d d
da a
g
p p
2
p
2
2
2
p
p
2
b b b
p
2
p
2
2
b
b
b
p
p
2
2
p
2
p
2
(a) Planar representation of M
N
h
g
i
c
b
a
f
e
d
S
(b) 3D representation of M
Figure 12. f-tiling M.
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.
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
d
d a
p
d
d
a
p
2
g
g
g
g
g g
p
2
p
2
p
2
g
g
p
2
p
p
p
p
2
2
b
c
p
2
p
2
2
b
b
2
2
a
b
b
C
b
b
p
d a
p
a
d d
p
g
g
2
g
g
g
2
2
2
g
g
g
a d
a d
d
d
g
p
2
g
g g
g
p
p
2
2
p
2
2
p
p
p
p
2
2
2
2
g
p
p
2
p
d
277
g
g
d
d
p
2
p
d
d
a
2
2
p
2
b
b b
b
b b
p
p
2
2
d
d
d
d
L
a
p
p
a a
d
2
p
d d
a
d
a
b
a
p
2
c
L
p
C
2
2
(a) Planar representation of Q2
(b) 3D representation of
Q2
Figure 13. f-tiling Q2 .
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
3
the other hand, a = ρxy Rzπ is a symmetry of Gα that maps N into S. Similarly
3
to some previous cases, we conclude 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 presented in Figure 16. One has β = π3 , γ = π4 , δ = 12 arccos 2−2
≈
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
2
reflection ρyz giving rise to a subgroup G of G(C) isomorphic 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.
278
C. P. AVELINO and A. F. SANTOS
p
2
p
p
2
p
2
p
p
2
p
2
d
d a
g g
g
v1
g
g
g g
a
p
d
p
d
p
2
a
b
2
2
c
g
2
2
p
p
p
p
2
b
b
b
2
2
2
b
g
a d
a d
d
d
b
b
p
d
d
d
d
a
p
2
p
2
2
d
2
a
b
p
p
d a
d
p
a
d
p
g
g
2
b
2 p
2
d
d
a
a d
b b
b
p
2
b
2
g
2 p
2
a
p
2
g
v2 g
p
p
2
g
g
g g
g
d
d
d
a
2
2
d
a
b b
b
b b
b
p
p
2
2
2
2
p
2
p
p
p
p
p
a a
d
d
g g
g
g
g
p
v4
g
g g g
2
2
d
d
d
g
g
g
d
p
p
2
2
2
a d d
d
d d
2
p
p
g
aa
p
2
2
g
g
2 p 2
2
g
p
p
p
p
2
2
g
g
g
2
2
d
g
p
p
p 2
v3
g
2
2
2
2
2
p
p
p
p
2
p
2
d
2
p
2
p
p
g
p
p
p
p
g
g
g
g
g
g
g
2
2
p
p
2
d a
dd a
d
p
2
p
p
2
2
d d
a
a d
b
b
b
b
b b
d
2
d
d d
v3
a
d
a
v4
p
p
2
2
c
p
2
b
p
2
p
p
a
2
2
v1
v2
(a) Planar representation of Q3
(b) 3D representation of Q3
Figure 14. f-tiling Q3 .
a
a
p
2
d
g
d g
p
2
p
2
p
p
2
2
b
N
b
b
b
b
b
a
a
a
2
p
d
d
b
g
g
p
2
p
p
p
p
2
p
2
2
a
p
p
2
g
b
b
2
2
p
2
d a
2
2
p
p
2
2
p
2
g d
g
p
p
2
d a
g d
b
b
b
a
a
N
d
a
g
d
S
b
p
p
a
a d g g
a
b
g
p
2
2
2
p
2
d
(a) Planar representation of Gα , α ∈
S
`π
,
6
π
3
´
Figure 15. f-tiling Gα , α ∈
(b) 3D
of Gα ,
` representation
´
α ∈ π6 , π3
`π
6
,
π
3
´
.
279
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
p
p
2
p
2
g
g
p
a
g
g
p
p
p
2
2
d
d
2
2
p
p
2
p
2
d a
a
d
d
d
2
p
2
d
2
2
d
d
a
d
2
p
2
p
d
d
2
p
2
p
p
b
b
b
d a
d a
d
d
d
2
2
d d
d d
p
2
p
2
g g
g
g
g
p
p
2
p
g
2
2
p
2
g g
g
g
p
p
g
2
p
g
2
2
p
2
g g
g
g
g
p
p
2
p
g
2
2
p
2
g
g
g g
b
2
p
2
d
g g
p
2
2
d d
p
p
p
2
h
p
p
p
f
b
b
b
b
g
g
p
p
p
p
2
2
d d
d
d
2
2
p
d
a
d a
b
p
p
g g
g
g g
g
2
2
g g
g
g
g g
2
2
2
g
g g
p
p
2
2
p
p
b
2
2
g
p
p
2
2
2
p
2
2
a
p
2
b
p
p
d
d a
d d
2
p
2
2
d
d
d
2
g g
g
g
p
g
g
p
p
2
2
2
p
2
g g
g
g
g
p
p
p
p
g
g
g
2
2
2
2
g
p
2
2
p
2
d d
d
d
a
d
a d
p
2
p
2
p
p
p
p
2
2
p
g
2
g g
g
p
2
d
d
2
2
g
g
g
d d
d a
p
2
p
2
d
a
p
p
2
2
p
2
p
p
b
p
2
b b
b
d
a
2
2
b
b
d d
a d
p
p
p
p
2
2
d
d
p
p
p
p
2
2
g
g
2
2
d
d
2
2
g g
g g
g
g
d
2
a a d
d d
p
2
p
2
p p
d
2
p
d
p
2
2
d
p
b
p
b
2
d a a
g
g
2
p
d
b b
b
b
p
p
2
d
2
g
p
2
g
g g
g g
2
p
g
g
g
2
2
d
2
b
b
2
2
2
b
b
p
a
d d d
2
p
2
p
2
g
g
d
N
l
k
c j
b i
a
d
e
h
g
f
S
(b) 3D representation of C
Figure 16. f-tiling C.
p
a d
a d
p
2
2
p
2
p
d
2
2
d
g
p
2
d
g
d
p
2
2
a
p
2
d
(a) Planar representation of C
d
d
b
b
g
g
g
g
2
2
2
g g
p
p
p
p
d a
p
b
g
g
p
2
p
p
b
b
2
p
2
2
a
p
2
p
d
p
p
p
b b
b
d
d
p
d
a
p
2
p
p
d d
d a
2
g
S
d d
d
d
2
2
p
d
g g
a
p
p
2
2
p
2
2
p
p
2
g g
p
d
2
d d
2
2
p
2
2
2
p
p
2
d d
d
d
a
d
ad
p p
d
a
p
2
g
p
d d
d a
2
2
p
g
p
p
p
2
2
g
2
d
p
g
p
2
p
d
d
a
a
p
2
g g
b
b
2
p
2
g
d
d
b
p
a
d
d d
p
g
b
b
2
d a
d
d
p
2
p
p
2
p p
2
p
2
p p
g
g
a
d
2
i
p
g g
p
2
g
g
2
2
d d
g
2
2
2
g g
j
p
p
g
a
d
k
a d
d
a
d
d
d d
g
g
l
d
d
a d
2
e
g
p
p
2
b
a
p
2
c
b
2
g g
g g
b
p
p
g g
2
2
a
p
p
p
d
a
2
p
p
g
2
p
2
p
p
b
b b
d
g g
g
b
b
a
g
g
p
2
p
b
d
d
2
p
2
Ng
d
d
2
2
p
2
g
p
p
d
d
p
g g
2
2
2
d d
p
2
p
p
d d
a
p
g
g g
a
2
2
g
d
a
d
280
C. P. AVELINO and A. F. SANTOS
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 ).
Rka , k b 3
p
p
p
p
Da ,Db
p
p
p
p
Dc
p
p
p
p
L
p
p
p
p
M
p
p
p
p
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Q2
p
p
p
p
Q3
p
p
p
p
Ga
p
p
p
p
C
p
p
p
p
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
g
g
b
b
g
g
b
b
g
g
b
b
g
g
g
g
g
g
b
b
g
g
g
g
g
g g
g g
g g
g g
g
g
g
g
a
d
g
g
a
d
p
2
g g
g
g
g
g
g
g
g ggg
ggg g
b
b
d
g
b
b
g
g
g
g
d
d
a d
d a
a d
a
d
2
p
gggg
g g
a
a
g
g
gggg
b
b
b
b
g
d a
a
d
d
a
a
d
d
a
a
d
a
d
a
d
a
d
d
a
d
p
2
p
2
p
2
p
2
a
d
a a
d d
a
d
d
d
gg gg
p
g
2
ggg p
2
a
d
a
a
d
a
d
d
d
d
d
d
d
a
a
g d
g g
g g
g
g
g
g
g
g
p
2
p
2
a
d
a d
d d
d
d
Figure 17. Distinct classes of congruent vertices.
The distinct classes of congruent vertices of each f-tiling are illustrated in Figure 17.
A NOTE ON SPHERICAL F–TILINGS BY RIGHT TRIANGLES
f-tiling
Rkα
Da
Db
Dc
L
α
„
d
(k − 2)π (k + 2)π
,
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
281
√
√
√
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.
Acknowledgment. The authors would like to thank the anonymous referee
for helpful comments and suggestions.
References
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”
2. Avelino C. P. and Santos A. F., Right triangular dihedral f–tilings of the sphere: α, β,
2
“
π”
and γ, γ,
, accepted for publication in Ars Combinatoria.
2
, Right triangular spherical dihedral f–tilings with two pairs of congruent sides, ac3.
cepted 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