Hiromi Ei

Shunji Ito

Tilings from some non-irreducible, Pisot substitutions

Keywords: Substitution, Pisot number, Pisot substitution, atomic surface, tiling, fractal, dynamical system

. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is non-irreducible for a special class of substitutions on five letters. Finally we show that the substitution dynamical systems for this class are isomorphic to induced transformations of rotations on the torus.

Introduction

In this paper we want to discuss tilings and dynamical systems generated by the following substitutions given by σ :

                   1 → K+1 times 11 • • • 1 2 2 → 3 3 → 4 4 → K times 1 • • • 1 5 5 → 1 , K ≥ 0.
(1.1)

The characteristic polynomial of the incidence matrix L σ is

x 5 -(K + 1)x 4 -Kx -1 = (x 2 -x + 1)(x 3 -Kx 2 -(K + 1)x -1). (1.2)
Since it is non-irreducible, its factor x 3 -Kx 2 -(K + 1)x -1 is a minimal polynomial of some Pisot number β. Furthermore | det(L σ )| = 1; we thus say that the above substitution σ is of the non-irreducible, Pisot, unimodular type. The aim of this paper is to discuss how we obtain tilings and dynamical systems generated by non-irreducible, Pisot, unimodular substitutions for the special class (1.1) which is coming from Pisot β-expansions. Let us recall some results in the irreducible, Pisot, unimodular case. (See [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF].) For example we consider the following substitution σ on 3 letters:

σ :    1 → 12 2 → 13 3 → 1 .
(1.

3)

The substitution σ has the incidence matrix

L σ =   1 1 1 1 0 0 0 1 0  
and its characteristic polynomial x 3 -x 2 -x-1. So this substitution is of the irreducible, Pisot, unimodular type.

For the contractive plane P of the matrix L σ , a stepped surface which is a discrete plane approximation of P is determined. Thus we have a tiling of the plane P with three prototiles, which are parallelograms, by using the projection π defined as the map from R 3 to P along the eigenvector u of L σ corresponding to the Pisot eigenvalue β.

This substitution σ has a unique fixed point and we denote it by ω = s 0 s 1 • • • s n • • • . Then we obtain the sets X i (i = 1, 2, 3) given by the closure of {π n-1 k=0 e s k | s n = i, n = 1, 2, • • • } and X = ∪ 3 i=1 X i , where {e i } i=1,2,3 is the canonical basis of R 3 . These sets X i , X are called atomic surfaces of σ.

On the other hand, it is known that the tiling and the atomic surfaces can be generated by the so called tiling substitution E * 1 (σ) on the Z-module G * 1 defined by

G * 1 =    δ∈Z 3 ×{1 * ,2 * ,3 * } n δ δ | n δ ∈ Z, # {δ ∈ Z 3 × {1 * , 2 * , 3 * } | n δ = 0} < ∞    .
Here we identify (x, i * ) ∈ Z 3 × {1 * , 2 * , 3 * } with the unit square {x + se j + te k ∈ R 3 |{j, k} = {1, 2, 3}\{i}, 0 ≤ s ≤ 1, 0 ≤ t ≤ 1}, and summation "+" in an element in G * 1 means the union of these unit squares. More generally we consider a substitution σ denoted by σ(i) = W

(i) 0 W (i) 1 • • • W (i) k • • • W (i)
l (i) -1 . By using the canonical homomorphism f from the free monoid on 3 letters to Z 3 defined by f (i) = e i (i = 1, 2, 3) and the notations P (i) k and S (i) k stand for respectively the prefix of length k and suffix of length l (i) -1k of σ(i), we define the endomorphism E * 1 (σ) on G * 1 as follows:

E * 1 (σ)(x, i * ) := 3 j=1 S (j) k :W (j) k =i (L -1 σ x + L -1 σ f (S (j)
k ), j * ) .

On this setting we can generate the stepped surface of the plane P by E * n 1 (σ)((e 1 , 1 * ) + (e 2 , 2 * ) + (e 3 , 3 * )) (n → ∞ ) and the atomic surfaces by

X i = -lim n→∞ L n σ πE * n 1 (σ)(e i , i * ) (i = 1, 2, 3),
where the right side converges in the sense of the Hausdorff metric and we denote lim n→∞ L n σ πE * n 1 (σ)(e i , i * ) by X i . Furthermore, we have the following property: Property 1.1 (1) The pieces in the union ∪ 3 i=1 X i denoted by X do not overlap up to a set of Lebesgue measure 0, (2) the following set equation holds:

L -1 σ X i = 3 j=1 P (j) k :W (j) k =i ( X j -L -1 σ πf (P (j) k )) ,
where the sets in the right side of the equation do not overlap up to a set of Lebesgue measure 0, and moreover, the transformation F : X → X given by

F (x) = L -1 σ (x + πf (P (j) k )) if x ∈ L σ X j -πf (P (j) k )
is well-defined and it is a Markov transformation with matrix structure L σ ,

(3) the transformation E : X → X, called the domain exchange transformation, given by

E(x) = x -πe i x ∈ X i
is well-defined and the transformation E is measure-theoretically isomorphic to a rotation on the 2-dimensional torus, and moreover, the orbit of the origin point by

E satisfies E k (0) ∈ X s k , k = 0, 1, • • • .
Property (2) and (3) are not known to hold for any irreducible Pisot substitution. Property (2) holds provided that (1) holds if the substitution σ has the strong coincidence condition (cf. [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF]). Moreover, for the transformation F to be well-defined, the pieces must not overlap. The isomorphism with a rotation (3) is equivalent with the tiling property. The aim of the paper is to obtain the analogous property for the non-irreducible, Pisot, unimodular substitutions given by (1.1).

This paper is sketched as follows.

In Section 2, we define a projection map π from R 5 to the contractive 2-dimensional plane of the incidence matrix L σ of σ. A substitution given by (1.1) has a unique fixed point. Therefore, using the projection we obtain atomic surfaces X and X i (i = 1, 2, 3, 4, 5) with respect to the letter i in the same way as in the irreducible case. (See Fig. 1.)

In Section 3, 4 and 6, we define the tiling substitution τ * of a substitution σ according to some modification of the method by [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF]. Since we deal with the non-irreducible case, we can not use the same method as in the irreducible case. However, we introduce new polygonal tiles instead of parallelograms, a tiling substitution τ * and the concept of a stepped surface; and try to construct atomic surfaces by using a map τ * and tilings T τ * with five polygonal prototiles and T b X τ * with the atomic surfaces X i (i = 1, 2, 3, 4, 5). (See Fig. 2 and Fig. 3.)

In Section 5, we introduce two dynamical systems on X := -X associated with non-irreducible substitutions, that is, a Markov transformation and a domain exchange transformation related to Property 1.1 (2) and (3). The main theorems in this paper are Theorem 5.2 and Theorem 5.3, and these theorems say the following:

The domain exchange transformation E : X → X is defined by

E(x) = x -πe i x ∈ X i
and the orbit of the origin point by the transformation

E satisfies E k (0) ∈ X s k , k = 0, 1, • • • .
The transformation E is not measure-theoretically isomorphic to a rotation on the 2-dimensional torus, but isomorphic to the induced transformation of a rotation on the torus.

2 Substitutions and atomic surfaces

General results on atomic surfaces

Let A be an alphabet consisting of d letters {1, 2, • • • , d} . The free monoid on the alphabet A with the empty word ǫ is denoted by A * = ∪ ∞ n=0 A n and A N denotes the collection of all right infinite sequences of symbols from A. Let σ be an endomorphism on A * such that σ(i) ∈ A * \ {ǫ} for all i ∈ A, which is called a substitution. By defining σ(U V ) = σ(U )σ(V ) for a concatenation U V of words, the substitution σ is extended to an endomorphism on A * and A N . Put

σ(i) = W (i) 0 W (i) 1 • • • W (i) l (i) -1 = P (i) k W (i) k S (i) k and σ n (i) = W (n,i) 0 W (n,i) 1 • • • W (n,i) l (n,i) -1 = P (n,i) k W (n,i) k S (n,i) k , where P (i) 0 = P (n,i) 0
= ǫ for any i ∈ A and any positive integer n. We call P (i) k (resp., S (i) k ) the k-prefix (resp., the k-suffix) of a word σ(i). We define the canonical homomorphism f :

A * → Z d by f (ǫ) = 0 and f (i) = e i (i ∈ A), where {e i } i=1,•••d is the canonical basis of R d . This is naturally extended to a map on A * by defining f (U V ) = f (U ) + f (V ) for words U, V in A * .
There is a matrix L σ on Z d for a substitution σ satisfying the following commutative diagram:

A * σ → A * f ↓ ↓ f Z d Lσ → Z d .
The matrix L σ is called the incidence matrix of the substitution and its entry L σ (i, j) is equal to the number of occurrences of the letter i in σ(j).

Recall that a Pisot number is an algebraic integer greater than 1 and whose conjugates have modulus strictly less than 1. Before discussing atomic surfaces of substitutions given by (1.1), we will define atomic surfaces on a general setting with the following assumption:

Assumption Throughout this paper, we assume that for a substitution σ

(1) W (1) 0 = 1, that is, σ(1) begins with 1, (2) σ is unimodular, that is, | det(L σ ) |= 1,
(3) the characteristic polynomial of L σ is not irreducible and is decomposed as f (x)g(x) such that f (x) is a minimal polynomial of some Pisot number β, and the roots of g(x) have modulus 1.

A substitution with Assumption (2) and ( 3) is referred to be of unimodular, non-irreducible, Pisot type. From Assumption (1), there exists a fixed point ω of the substitution σ:

ω = lim n→∞ σ n (1) = s 0 s 1 • • • s n • • • .
From Assumption (2) and (3), we define the expanding subspace L(u) spanned by the eigenvector u corresponding to the eigenvalue β and the contractive subspace P = {0}(⊂ R d ) corresponding to the other conjugate eigenvalues of β. Then we have a direct sum R d = P ⊕L(u)⊕P ′ , where P ′ corresponds to the other eigenvectors coming from g(x). Let us define the projection π : R d → P by

π(p + x + p ′ ) = p,
where p ∈ P , x + p ′ ∈ L(u) ⊕ P ′ .

Definition 2.1 From sets Z i , Z ′ i of prefixes of a fixed point ω:

Z i := {s 0 s 1 • • • s k-1 | s k = i, k = 1, 2, • • • } Z ′ i := {s 0 s 1 • • • s k | s k = i, k = 0, 1, • • • },
we define sets Y i , Y ′ i in P as follows:

Y i := πf (Z i ) = {πf (s 0 s 1 • • • s k-1 ) | s k = i, k = 1, 2, • • • } Y ′ i := πf (Z ′ i ) = {πf (s 0 s 1 • • • s k ) | s k = i, k = 0, 1, • • • } .
The sets X, X i , X ′ i in P are defined by

X := cl.(∪ d i=1 Y i ) = cl.(∪ d i=1 Y ′ i ) X i := cl.(Y i ) X ′ i := cl.(Y ′ i ) ,
where cl.S means the closure of a set S.

We call the sets X, X i (i ∈ A) atomic surfaces of σ. Note that the equality

X ′ i = X i + πe i holds from Y ′ i = Y i + πe i , where S + a = {x + a | x ∈ S}. For words U = u 0 u 1 • • • u l , V = v 0 v 1 • • • v m ∈ A * , U ≺ V denotes that l < m and v 0 v 1 • • • v l = U . The following proposition can be found in [AI01][FFIW00].
Proposition 2.1 The following set equations hold:

(1)

L -1 σ X i = d j=1 P (j) k :W (j) k =i (X j + L -1 σ πf (P (j) k )) (2) L -1 σ X ′ i = d j=1 S (j) k :W (j) k =i (X ′ j -L -1 σ πf (S (j) k )) .
Remark 1 We will see the following property in Theorem 4.1 (4): the sets X j + L -1 σ πf (P

(j) k ) (resp., X ′ j -L -1 σ πf (S (j) k )) with W (j) k
= i in the equation (1) (resp., (2)) do not overlap up to a set of Lebesgue measure 0.

Proof: From the property cl.(

n i=1 A i ) = n i=1 cl.(A i ), it is enough to show that f (Z i ) = d j=1 P (j) k :W (j) k =i (L σ f (Z j ) + f (P (j) k )).
(2.1)

Take s 0 s 1 • • • s n ∈ Z i with s n+1 = i, then there exists an integer m (m < n) such that σ(s 0 s 1 • • • s m-1 ) ≺ s 0 s 1 • • • s n i σ(s 0 s 1 • • • s m ).
Consequently there exists an integer t such that

s 0 s 1 • • • s n i = σ(s 0 s 1 • • • s m-1 )P (sm) t W (sm) t ,
where W (sm) t = i. Let f act on both sides of the above equality,

f (s 0 s 1 • • • s n i) = f (σ(s 0 s 1 • • • s m-1 )P (sm) t W (sm) t
).

Hence, by f

• σ = L σ • f , we have f (s 0 s 1 • • • s n ) = L σ f (s 0 s 1 • • • s m-1 ) + f (P (sm) t
) .

Thus f (s 0 s 1 • • • s n ) ∈ L σ f (Z sm ) + f (P (sm) t
) with W (sm) t = i. This shows ⊆ is true for the equality (2.1) by choosing j = s m and k = t.

On the other hand, for any s 0 s 1 • • • s m ∈ Z j with s m+1 = j and W (j) k = i, we have

L σ f (s 0 s 1 • • • s m ) + f (P (j) k ) = f (σ(s 0 s 1 • • • s m )) + f (P (j) k ) = f (σ(s 0 s 1 • • • s m )P (j) k ) .
Since s m+1 = j, then it is clear that

σ(s 0 s 1 • • • s m )P (j) k i σ(s 0 s 1 • • • s m s m+1 ) . Therefore, σ(s 0 s 1 • • • s m )P (j) k ∈ Z i .
This leads to the other direction of the equality (2.1). The second set equation is shown by (1) and X i = X ′ iπe i . ✷

Atomic surfaces generated by the labeled graph G *

To obtain a numerical representation of X or X i as we will see in Theorem 2.1, we introduce the new alphabet B and the subset B * σ of the monoid B * by using the prefix automaton as follows:

B := i t i | i ∈ A t i ∈ {0, 1, • • • , |σ(i)| -1} , B * σ := i 0 k 0 i 1 k 1 • • • i N k N ∈ B * | W (in) kn = i n-1 (n = 1, 2, • • • , N ), N = 0, 1, • • • ,
where |U | means the length of a word U . (See Fig. 4.)

σ(i n-1 )= ❅ ❅ ❅ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ σ(i n ) = i n-1 k n th ❍ ❍ ❍ ❍ ❍ ❍ ✁ ✁ ✁ σ(i n+1 )= i n k n+1 th Fig. 4: A subword `in kn ´`i n+1 k n+1 ´of a word in B * σ
We define a labeled graph G * such that the set of vertices is V = {1, 2, • • • , d} and the set of edges is

E = {0, 1, • • • , |σ(i 0 )| -1} with the largest |σ(i 0 )| for i 0 ∈ A. If W (j) k
= i, that is, the letter i occurs in σ(j) as k-th letter, then one edge from the vertex i to the vertex j named k is drawn. (See Fig. 5. cf.

B * (i) σ := i 0 k 0 i 1 k 1 • • • i N k N ∈ B * σ | W (i0) k0 = i, N = 0, 1, • • • . The set B * (i) σ
is the set of all finite paths whose initial vertex is i in G * .

Theorem 2.1 For any substitution σ satisfying Assumption, the following equalities hold:

X i = cl. N n=0 L n σ πf (P (in) kn ) | i 0 k 0 i 1 k 1 • • • i N k N ∈ B * (i) σ , P (i N ) k N W (i N ) k N ≺ ω, N = 0, 1, • • • X = cl. N n=0 L n σ πf (P (in) kn ) | i 0 k 0 i 1 k 1 • • • i N k N ∈ B * σ , P (i N ) k N W (i N ) k N ≺ ω, N = 0, 1, • • • . Proof: We show Y i = N n=0 L n σ f (P (in) kn ) | i 0 k 0 i 1 k 1 • • • i N k N ∈ B * (i) σ , P (i N ) k N W (i N ) k N ≺ ω, N = 0, 1, • • • . By the proof in Proposition 2.1, for s 0 s 1 • • • s m ∈ Z i , there exist positive integers i 0 , k 0 , m 0 such that f (s 0 s 1 • • • s m ) = L σ f (s 0 s 1 • • • s m0 ) + f (P (i0) k0 )
and

W (i0) k0 = i, s 0 s 1 • • • s m0 ∈ Z i0 . For s 0 s 1 • • • s m0 ∈ Z i0 , let us continue the same procedure. Because |s 0 s 1 • • • s m l |
is monotone decreasing with respect to l, at last we have the following equality:

f (s 0 s 1 • • • s m ) = L σ f (s 0 s 1 • • • s m0 ) + f (P (i0) k0 ) = L σ (L σ f (s 0 s 1 • • • s m1 ) + f (P (i1) k1 )) + f (P (i0) k0 ) = • • • = L N σ f (P (i N ) k N ) + L N -1 σ f (P (i N -1 ) k N -1 ) + • • • + f (P (i0) k0 )
,

where i0 k0 i1 k1 • • • i N k N ∈ B * (i)
σ . This shows

Y i ⊆ N n=0 L n σ f (P (in) kn ) | i 0 k 0 i 1 k 1 • • • i N k N ∈ B * (i) σ , P (i N ) k N W (i N ) k N ≺ ω, N = 0, 1, • • • .
We obtain the converse inclusion as in the proof of Proposition 2.1. Thus the first equality is proved. Analogously we prove the second one. ✷

From the above decomposition of f (s 0 s 1 • • • s m ), we have the following corollary:

Corollary 2.1 For any substitution which satisfies Assumption, the atomic surface X is bounded.

Atomic surfaces corresponding to a substitution given by (1.1)

From now on, let us deal with the non-irreducible, Pisot, unimodular substitutions σ given by (1.1):

σ :                    1 → K+1 times 11 • • • 1 2 2 → 3 3 → 4 4 → K times 1 • • • 1 5 5 → 1 .
In this case,

L σ =       K + 1 0 0 K 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0       .
We see that the characteristic polynomial of the incidence matrix L σ is given by (1.2) and it is also the characteristic polynomial of β (cf. [START_REF] Parry | On the β-expansions of real numbers[END_REF]). It is easy to check that the equation x 3 -Kx 2 -(K + 1)x -1 = 0 gives one real root β and two imaginary roots

β (2) , β (2) if K ≤ 2, or three real roots β, β (2) , β (3) if K ≥ 3. The five roots of x 5 -(K + 1)x 4 -Kx -1 = 0 are denoted by λ i (i = 1, 2, 3, 4, 5), where λ 1 = β (2) , λ 2 = β (2) , λ 3 = β if K ≤ 2, or λ 1 = β (2) , λ 2 = β (3) , λ 3 = β if K ≥ 3
, and λ 4 , λ 5 are the imaginary roots of x 2x + 1 = 0. Choose any eigenvector u i associated with the eigenvalue

λ i (i = 1, 2, 3, 4, 5) such that u 2 = u 1 if K ≤ 2 and u 5 = u 4 . Put v 1 = 1 2 (u 2 + u 1 ), v 2 = 1 2i (u 2 -u 1 ), v 3 = u 3 , v 4 = 1 2 (u 5 + u 4 ), v 5 = 1 2i (u 5 -u 4 ) if K ≤ 2 , v 1 = u 1 , v 2 = u 2 , v 3 = u 3 , v 4 = 1 2 (u 5 + u 4 ), v 5 = 1 2i (u 5 -u 4 ) if K ≥ 3
, and let us define the 5 × 5 matrix V by

V := (v 1 , v 2 , v 3 , v 4 , v 5 ).
The real vectors v i (i = 1, 2, 3, 4, 5) and the real matrix V satisfy the following relation:

L σ V = V       R 0 0 0 0 0 0 0 0 β 0 0 0 0 0 ℜ[λ 4 ] -ℑ[λ 4 ] 0 0 0 ℑ[λ 4 ] ℜ[λ 4 ]       ,
where ℜ[a] (resp., ℑ[a]) means the real (resp., imaginary) part of a and

R =        ℜ[β (2) ] -ℑ[β (2) ] ℑ[β (2) ] ℜ[β (2) ] if K ≤ 2 β (2) 0 0 β (3) if K ≥ 3 .
The space P <v1,v2> spanned by vectors v 1 , v 2 is an invariant contractive space of the linear transformation L σ . More precisely, we know

L σ x = (v 1 , v 2 )R x 1 x 2 , where x = x 1 v 1 + x 2 v 2 ∈ P <v1,v2> , x 1 , x 2 ∈ R. We have the direct sum R 5 = P <v1,v2> ⊕ L(v 3 ) ⊕ P ′ <v4,v5>
, where P ′ <v4,v5> is spanned by v 4 , v 5 ; and define the projection map π : R 5 → P <v1,v2> by

π(x 1 v 1 + x 2 v 2 + x 3 v 3 + x 4 v 4 + x 5 v 5 ) = x 1 v 1 + x 2 v 2 , x i ∈ R (i = 1, 2, 3, 4, 5).

It is easy to see that

πe i = v ′ 1i v 1 + v ′ 2i v 2 ,
where

V -1 = (v ′ ij ) 1≤i≤5 1≤j≤5
.

By Proposition 2.1, in this case we obtain equations in concrete terms as stated in the following corollary:

Corollary 2.2 For the substitution σ given by (1.1), the sets {X i } i=1,2,3,4,5 given in Theorem 2.1 satisfy the following set equations:

(1)

L -1 σ X 1 =      X 1 ∪ X 5 if K = 0 K n=0 (X 1 + nL -1 σ πe 1 ) ∪ K-1 n=0 (X 4 + nL -1 σ πe 1 ) ∪ X 5 if K ≥ 1 , L -1 σ X 2 = X 1 + (K + 1)L -1 σ πe 1 , L -1 σ X 3 = X 2 , L -1 σ X 4 = X 3 , L -1 σ X 5 = X 4 + KL -1 σ πe 1 , (2)
L -1 σ X ′ 1 =      (X ′ 1 -L -1 σ πe 2 ) ∪ X ′ 5 if K = 0 K n=0 (X ′ 1 -L -1 σ π(e 2 + ne 1 )) ∪ K-1 n=0 (X ′ 4 -L -1 σ π(e 5 + ne 1 )) ∪ X ′ 5 if K ≥ 1 , L -1 σ X ′ 2 = X ′ 1 , L -1 σ X ′ 3 = X ′ 2 , L -1 σ X ′ 4 = X ′ 3 , L -1 σ X ′ 5 = X ′ 4 .
For the substitution σ, the alphabet B and the graph G * are as follows:

B = 1 0 , 1 1 , • • • , 1 K + 1 , 2 0 , 3 0 , 4 0 , 4 1 , • • • , 4 K , 5 0 .
From the definition of σ given by (1.1), we have f (P

k ) = ke 1 for any j ∈ {1, 2, 3, 4, 5}. Hence it is enough to take only the path

(k 0 , k 1 , • • • , k N ) of the edges in i0 k0 i1 k1 • • • i N k N ∈ B * .
Then the set X is written as follows:

X = cl. N n=0 L n σ πf (P (in) kn ) | i0 k0 i1 k1 • • • , i N k N ∈ B * σ , P (i N ) k N W (i N ) k N ≺ ω, N = 0, 1, • • • = cl. N n=0 k n L n σ πe 1 | t(k n ) = i(k n+1 ) in G * , N = 0, 1, • • • ,
where t(e) ∈ V (resp., i(e)) means the terminal (resp., the initial) vertex of an edge e. The condition

P (i N ) k N W (i N ) k N
≺ ω in the first line of the above equality means i N k N ∈ B \ { 2 0 , 3 0 , 4 K } by the definition of σ, but we can omit this condition for the following reason: Let us consider the case i

N k N = 4 K , that is, the path in B * σ is written as i0 k0 i1 k1 • • • i N -1 k N -1 4 K .
From the graph G * in Fig. 6, the path is determined

as i0 k0 • • • 1 k N -2 5
* i0 k0 • • • 1 k N -2 1 0 1 K . Therefore we can allow i N k N = 4 K . The other cases i N k N = 2 0 or 3 0 can be shown analogously.
Moreover, from the labeled graph G * , the sets {X i } i=1,2,3,4,5 are given by

X i = cl. N n=0 k n L n σ πe 1 | i(k 0 ) = i, t(k n ) = i(k n+1 ) in G * N = 0, 1, • • • .
This leads to the following corollary:

Corollary 2.3 An element in the atomic surface X has an infinite expansion, that is,

X = ∞ n=0 k n L n σ πe 1 | t(k n ) = i(k n+1 ) in G * .
Example 2.1 Let us consider the case of K = 0. The substitution σ is σ :

           1 → 12 2 → 3 3 → 4 4 → 5 5 → 1
.

By the labeled graph G * in Fig. 6, the following admissible conditions hold for the sequence of digits {k n }:

X = ∞ n=0 k n L n σ πe 1 | t(k n ) = i(k n+1 ) in G * = ∞ n=0 k n L n σ πe 1 | k n = 1 ⇒ k n+1 = k n+2 = k n+3 = k n+4 = 0 .
3 A tiling with polygonal tiles of the plane P <v 1 ,v 2 >

A tiling substitution τ *

In this subsection we introduce a tiling substitution τ * associated with σ according to [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF]. From now on let σ be a substitution given by (1.1) in Section 1. Then we have the direct sum R 5 = P <v1,v2> ⊕ L(v 3 ) ⊕ P ′ <v4,v5> , and the map π is a projection from R 5 to the 2-dimensional plane P <v1,v2> . Lemma 3.1 The following relations hold:

πe 3 + πe 4 = πe 1 , πe 4 + πe 5 = πe 1 + πe 2 , πe 5 = πe 2 + πe 3 .

Proof:

We check that we can choose the vector t (-1, 0, 1, 1, 0) = -e 1 + e 3 + e 4 as the vector v 5 . Since v 5 is mapped to 0 by the projection π, we have

-πe 1 + πe 3 + πe 4 = 0.
From the fact that L σ • π = π • L σ , we have the second and third equalities. ✷

We introduce Z-modules F and F * using finite integer combinations of elements of Z 5 × {1, 2, 3, 4, 5} and Z 5 × {1 * , 2 * , 3 * , 4 * , 5 * } as follows:

F u :=    δ∈Z 5 ×{1,2,3,4,5} n δ δ | n δ ∈ Z, # {δ ∈ Z 5 × {1, 2, 3, 4, 5} | n δ = 0} < ∞    , F * u :=    δ∈Z 5 ×{1 * ,2 * ,3 * ,4 * ,5 * } n δ δ | n δ ∈ Z, # {δ ∈ Z 5 × {1 * , 2 * , 3 * , 4 * , 5 * } | n δ = 0} < ∞    .
From Lemma 3.1 we can introduce the equivalence relation ∼ on F u (resp., F * u ) defined by

N k=1 (x k , i k ) ∼ N k=1 (y k , i k ) if πx k = πy k for all k (resp., N k=1 (x k , i * k ) ∼ N k=1 (y k , i * k ) if πx k = πy k for all k) and we set F := F u / ∼ (resp., F * := F * u / ∼
). F will be used in Section 4 mainly. To give a geometrical meaning of (x, i * ), first we define the map π 1 : F → P <v1,v2> , which gives a one dimensional geometric representation of the symbolic object (x, i), by Proof: We show {πe 2 , πe 3 , πe 4 } is a hexa-generator. The other part can be shown analogously.

π 1 (x, i) = {πx + tπe i | 0 ≤ t ≤ 1} , where for N k=1 n k (x k , i k ) ∈ F with n k ∈ Z -{0} π 1 N k=1 n k (x k , i k ) = N k=1 π 1 (x k , i k ) .
π 1 ( 0, 4 ) π 1 ( 0, 2 ) π 1 ( 0, 5 ) π 1 ( 0, 3 ) Fig. 7: Representation of π1(0, i) (i = 1, 2, 3, 4, 5) A set consisting of three vectors {a, b, c} (a, b, c ∈ R 2 ) is called a hexa-generator if the domain {t 1 a + s 1 b, t 2 a + s 2 c, t 3 b + s 3 c | 0 ≤ t i ≤ 1, 0 ≤ s i ≤ 1, i = 1, 2, 3}
For x = x1 x2 ∈ R 2 , n(x) denotes -x2
x1 , that is, a normal vector of x. We can calculate the coordinates of πe i (i = 1, 2, 3, 4, 5) and easily check that every n(π(e 2 )) • π(e 4 ), n(π(e 4 )) • π(e 3 ) and n(π(e 3 )) • π(e 2 ) has the same signature, where a • b means the inner product for a, b ∈ R 2 . Therefore {πe 2 , πe 3 , πe 4 } is a hexa-generator. (See Fig. 8.) ✷ From Lemma 3.1 and Lemma 3.2, we can consider the map π 2 : F * → P <v1,v2> , which gives a two-dimensional geometric representation of the symbolic object (x, i * ), by

π 2 (0, 1 * ) = [π 1 (0, 2), π 1 (0, 5), π 1 (e 2 , 3)] π 2 (0, 2 * ) = [π 1 (0, 1), π 1 (0, 3), π 1 (e 3 , 4)] π 2 (0, 3 * ) = [π 1 (0, 2), π 1 (0, 4), π 1 (e 2 , 1), π 1 (e 4 , 5)] π 2 (0, 4 * ) = [π 1 (0, 3), π 1 (0, 5), π 1 (e 3 , 2)] π 2 (0, 5 * ) = [π 1 (0, 1), π 1 (0, 4), π 1 (e 4 , 3)] π 2 (x, i * ) = π 2 (0, i * ) + πx ,
N k=1 n k (x k , i k * ) ∈ F * with n k ∈ Z -{0} π 2 N k=1 n k (x k , i k * ) = N k=1 π 2 (x k , i k * ) .
(See.

k :W (j) k =i (L -1 σ x + L -1 σ f (S (j) k ), j * ) . (3.1)
We remark that the sign is plus here in this definition with respect to the sign minus in Formula (2) in Proposition 2.1. (See Remark 4 in Section 4.)

Remark 2 For the substitution σ given by (1.1), the tiling substitution τ * is defined explicitly by

τ * (x, 1 * ) = (L -1 σ x + e 1 -e 5 , 1 * ) + (L -1 σ x, 5 * ) if K = 0 K+1 n=1 (L -1 σ x + e 1 -ne 5 , 1 * ) + K n=1 (L -1 σ x + e 4 -ne 5 , 4 * ) + (L -1 σ x, 5 * ) if K ≥ 1 , τ * (x, 2 * ) = (L -1 σ x, 1 * ) , τ * (x, 3 * ) = (L -1 σ x, 2 * ) , τ * (x, 4 * ) = (L -1 σ x, 3 * ) , τ * (x, 5 * ) = (L -1 σ x, 4 * ) .
(See Fig. 10.) Remark 3 The formula of the tiling substitution τ * can be found in [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF] under the notation E * 1 (σ). But the geometrical meaning of (x, i * ) (i = 1, 2, 3, 4, 5) is different. In [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF] we mean by (x, i * ) a unit cube of dimension four in R 5 and a cube (x, i * ) projected by π is a good prototile for the tiling of R 4 for an irreducible substitution, however, for a non-irreducible substitution which we deal with now, a cube (x, i * ) projected by π does not work as prototile of a tiling of the contractive space P <v1,v2> . In this section we will see that the tiles π 2 (x, i * ) given by Fig. 9

π 2 (τ * (0, i * )) L -1 σ (π 2 (0, i * )) K times
τ * (U) ⊃ U, τ * (U ′ ) ⊃ U ′ ,
where for V, W ∈ F * , V ⊃ W means that π 2 (V) ⊃ π 2 (W), and that there exists

W ′ = N k=1 n k (x k , i * i ) ∈ F * with n i > 0 for all i such that V = W + W ′ ; moreover, τ * (U) -τ * (U ′ ) = U -U ′ and for any positive integer n τ * n (U) -τ * n (U ′ ) = U -U ′ .
Proof: From the definition of τ * , we see

τ * (U) =      U + (e 1 -e 5 , 1 * ) if K = 0 U + K+1 n=1 (L -1 σ x + e 1 -ne 5 , 1 * ) + K n=1 (L -1 σ x + e 4 -ne 5 , 4 * ) if K ≥ 1 τ * (U ′ ) =      U ′ + (e 1 -e 5 , 1 * ) if K = 0 U ′ + K+1 n=1 (L -1 σ x + e 1 -ne 5 , 1 * ) + K n=1 (L -1 σ x + e 4 -ne 5 , 4 * ) if K ≥ 1 . Therefore, we know τ * (U) -τ * (U ′ ) = U -U ′ . ✷
The replacing and re-dividing method

Observe the two domains π 2 (τ * (x, i * )) and L -1 σ (π 2 (x, i * )), then we have the following three cases (See Fig. 10.):

(1) π 2 (τ * (x, i * )) = L -1 σ (π 2 (x, i * )) (i = 2, 5) (2) π 2 (τ * (x, i * )) ⊂ L -1 σ (π 2 (x, i * )) (i = 3, 4) (3) π 2 (τ * (x, i * )) ⊃ L -1 σ (π 2 (x, i * )) (i = 1)
In the case where i = 1, 3, 4, each of these domains π 2 (x, i * ) (i = 1, 3, 4) contains at least one edge of the form π 1 (y, 2) and one edge of the form π 1 (y ′ , 5), so we introduce the following "replacing and re-dividing" method to get the domain π 2 (τ * (x, i * )) from the domain L -1 σ (π 2 (x, i * )).

First, we "replace" every edge L -1 σ π 1 (y, 2) on L -1 σ (π 2 (x, i * )) (i = 1, 3, 4) with

π 1 {(L -1 σ y, 1) + K+1 n=1 (L -1 σ y + e 1 -ne 5 , 5)},
what is more, in the case of K ≥ 1, replace every edge

L -1 σ π 1 (y ′ , 5) on L -1 σ (π 2 (x, i * )) (i = 1, 3, 4) with π 1 {(L -1 σ y ′ , 4) + K n=1 (L -1 σ y ′ + e 4 -ne 5 , 5)}.
By this procedure, we have π 2 (τ * (x, i * )) in the case of i = 3, 4.

Secondly, we "re-divide" the domain in the case of i = 1 (i.e. π 2 (τ * (x, 1 * ))). Then, we have π 2 (τ * (x, i * )). (See Fig. 11.)

K times replacing re-dividing ( , 2 ) 1 π 0
( , 5 ) 1 π 0 -1 L σ -1 L σ -1 L σ
Fig. 11: The replacing and re-dividing method for the domain L -1 σ (π2(x, 1 * ))

3.2 A tiling of the plane P <v 1 ,v 2 > generated by τ *

We construct a first quasi-periodic tiling of P <v1,v2> with five polygonal prototiles generated by the tiling substitution τ * associated to σ. (See Fig. 2 in Section 1.) This tiling corresponds to the projection of a discrete plane approximation (stepped surface) in the Pisot case. First we will see the property of non-overlap.

Proposition 3.2

The sets π 2 (τ * n (U)) (resp., π 2 (τ * n (U ′ ))) consisting of five prototiles of the form π 2 (x, i * ) (i = 1, 2, 3, 4, 5) do not overlap for any positive integer n.

Proof: Suppose that the pieces of π 2 (τ * n (U)) do not overlap, then the pieces of L -1 σ π 2 (τ * n (U)) do not overlap. By the replacing and re-dividing method, we obtain π 2 (τ * n+1 (U)) from L -1 σ π 2 (τ * n (U)). Therefore, to show the pieces of π 2 (τ * n+1 (U)) do not overlap, it is enough to show that replaced edges on L -1 σ π 2 (τ * n (U)) do not cause overlap. From (3) in the replacing and re-dividing method, it is possible that overlaps occur by replacing edges on π 2 (x, 1 * ). We list the pairs of tiles which are just touching with the segment π 1 (x, 2) or π 1 (x, 5) on π 2 (x, 1 * ), which are given by {(x, 1 * ), (xe 3 , 4 * )}, {(x, 1 * ), (x, 3 * )}, {(x, 1 * ), (x, 4 * )}, {(x, 1 * ), (xe 4 , 3 * )} .

We deal with the pair {(x, 1 * ), (x-e 3 , 4 * )}. The edge which substitutes for L -1 σ π 1 (x, 2) on L -1 σ π 2 (x, 1 * ) is not included in the domain L -1 σ π 2 (x, 1 * ), but in the adjoining domain L -1 σ π 2 (xe 3 , 4 * ). Moreover, the edge does not cross the other edges. Thus there is no overlap by replacing edges. We see other cases analogously. (See Fig. 12.)

In the case where π 1 (x, i) (i = 2, 5) in π 2 (x, 1 * ) is a part of the boundary of π 2 (τ * n (U)), it is easy to see that replacing edges do not cause overlap. ✷ ( , 2 ) Secondly we consider the covering of the plane P <v1,v2> by π 2 (τ * n (U)).

Proposition 3.3 ∪ ∞ n=1 π 2 (τ * n (U)) = P <v1,v2>
, that is, π 2 (τ * n (U)) covers the plane as n goes to ∞. The proof of this proposition is long and not easy. So the detail of this proof is put in Section 6. In the irreducible case, we can prove the property in Proposition 3.3 by using the notion of stepped surface of a substitution σ (cf. [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF]), but here we must prove it without such a notion. That is the reason why the proof is difficult.

The following proposition is deduced from Proposition 3.2 and Proposition 3.3:

Proposition 3.4 The sets τ * n (U) generate a tiling of the plane P <v1,v2> , that is,

T τ * := {π 2 (x, i * ) | (x, i * ) ⊂ τ * n (U) f or some n}
is a tiling of P <v1,v2> . (See Fig. 2 in Section 1.)

Finally we discuss the periodicity of the tiling T τ * . Let us introduce the following notation:

G * := {(x, i * ) ∈ F * | (x, i * ) ⊂ τ * n (U) f or some n}, that is, T τ * = {π 2 (x, i * ) | (x, i * ) ∈ G * },
and we also introduce the new prototiles (x, i * ) based on πx and the tiling associated to G * defined by

(x, i * ) := τ * (L σ x, i * ) for (L σ x, i * ) ∈ G * , G * := {(x, i * ) | (L σ x, i * ) ∈ G * } .
So we have the ordinary tiling T τ * by dividing the prototiles in G * following the method in Subsection 3.1. We say G * (or T τ * ) is periodic if there exists at least one non-zero period p ∈ R 5 such that (x, i * ) ∈ G * implies (x + p, i * ) ∈ G * .

Lemma 3.3 If p is a non-zero period of G * , then G * also has the period p.

Proof: Assume G * has a non-zero period p. At first we consider the periodicity of the tiles of the form (x, 1 * ) in G * . Note that the only τ * (x, 1 * ) in the images by τ * includes a tile of the form (y, 5 * ), that is, (x, 1 * ) ∈ G * if and only if (x, 5 * ) ∈ G * . Suppose (x, 1 * ) ∈ G * , then (x, 5 * ) ∈ G * , and so (x + p, 5 * ) ∈ G * by the assumption, and finally we have (x + p, 1 * ) ∈ G * . It means that p is also a period for the tiles of the form (x, 1 * ). Now we want to observe the periodicity for the tiles in G * except the tiles (x, 1 * ). Put

D := {(y, j * ) ∈ G * | (y, j * ) ⊂ (x, 1 * ) f or some (x, 1 * ) ∈ G * } D := {(x, 1 * ) | (x, 1 * ) ∈ G * } .
From the above discussion we know that D has a period p. This means that D is closed for the translation by p, and has the same period p. Thus G * -D also has a period p by the assumption. After projection by π 2 , G * -D and G * -D provide the same covering of P <v1,v2> with many holes by the equality (x, i * ) = (x, (i -1) * ) (i = 2, 3, 4, 5) .

So G * -D has a period p. Therefore, G * has a period p. ✷ Theorem 3.1 The tiling T τ * is not periodic.

Proof: Suppose the tiling T τ * is periodic, that is, G * is periodic. Since G * is a discrete set, there exists a non-zero and minimum period p of G * , where a minimum period is a period whose norm πp is minimum. From Lemma 3.3, p is also a period of G * . Define the map ι * :

G * → G * by ι * (x, i * ) = (L σ x, i * ) (i = 1, 2, 3, 4, 5) .
By the definition of G * , ι * (x, i * ) is in G * , that is, ι * is well-defined. Moreover, it is a bijection and the inverse is given by ι

* -1 (y, i * ) = (L -1 σ y, i * ) for (y, i * ) ∈ G * . Hence, (x, j * ), (x+p, j * ) ∈ G * implies ι * (x, j * ) = (L σ x, j * ) ∈ G * , ι * (x+p, j * ) = (L σ x+L σ p, j * ) ∈ G * .
This means π(L σ p) is a period of the tiling T τ * . On the other hand, L σ is contractive on P <v1,v2> . Therefore, it contradicts the minimality of the period p. ✷ Definition 3.1 A tiling T = {T λ | λ ∈ Λ, T λ is a tile on P } of the space P is called a quasi-periodic tiling if for any r > 0 there exists R > 0 such that any patch γ = λ ′ ∈Λ ′ ⊂Λ # Λ ′ <∞ T λ ′ whose diameter is smaller than r occurs somewhere in a neighbourhood of radius R of any point.

For γ, δ ∈ F * , γ ≻ δ denotes that there exists z ∈ Z 5 such that M z γ ⊃ δ, where M z is the translation map given by

M z ( N k=1 n k (x k , i k * )) = N k=1 n k (x k + z, i k * ) for N k=1 n k (x k , i k * ) ∈ F * .
Theorem 3.2 The tiling T τ * is quasi-periodic.

Proof: Take any r > 0. There exists a positive integer N such that τ * N (U) ≻ γ, for any γ ∈ G * satisfying the diameter of π 2 (γ) is smaller than r, because the number of such γ's is finite. From τ * 8 (e i , i * ) ⊃ τ * 4 (e 1 , 1 * ) ⊃ U for any i = 1, 2, 3, 4, 5, putting M=N+8, we have

τ * M (e i , i * ) ≻ γ (i = 1, 2, 3, 4, 5).
By the definition of G * , for any (x, i * ) ∈ G * there exists (y, j * ) ∈ G * such that (x, i * ) ⊂ τ * M (y, j * ).

Therefore, we have

U R (x) ⊃ π 2 (τ * M (y, j * )) ,
where R = max i=1,2,3,4,5 diam.(π 2 (τ * M (e i , i * ))) and U R (x) means the neighbourhood of x with the radius R. Thus, U R (x) contains any configuration of π 2 (γ) whose diameter is smaller than r. ✷ 4 Atomic surfaces given by τ * and a second tiling

In Section 2 we constructed atomic surfaces X, X i (i = 1, 2, 3, 4, 5) from the fixed point of a substitution and the projection map π. In Subsection 4.1 we generate the atomic surfaces by using the tiling substitution τ * ; and by the virtue of this construction, we can observe the boundaries of atomic surfaces in Subsection 4.2; and in Subsection 4.3 we obtain a second tiling with atomic surfaces by replacing the polygonal tiles on the first tiling T τ * by atomic surfaces.

Atomic surfaces given by

n , (resp.,

D n ′ , D (i) n ′
) as follows:

D n := π 2 (τ * n (U)) D (i) n := π 2 (τ * n (e i , i * )) D n ′ := π 2 (τ * n (U ′ )) D (i) n ′ := π 2 (τ * n (0, i * )).
Theorem 4.1 We take a renormalization of the domains D n , D

n , then (1) the following limit sets exist in the sense of the Hausdorff metric:

X i := lim n→∞ L n σ D (i) n X ′ i := lim n→∞ L n σ D (i) n ′
and they satisfy the relations:

X i = X ′ i + πe i , ( X :=) ∪ 5 i=1 X i = ∪ 5 i=1 X ′ i ,
(2) the following inequality holds:

L σ       µ( X 1 ) µ( X 2 ) . . . µ( X 5 )       ≥ β       µ( X 1 ) µ( X 2 ) . . . µ( X 5 )       ;
moreover, the vector of volumes t (µ( X 1 ), µ( X 2 ), • • • , µ( X 5 )) is an eigenvector of L σ with respect to the maximum eigenvalue β, where µ is the Lebesgue measure,

(3) the following set equations hold: The proof of the theorem can be obtained by a quite similar way as in [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF] following Lemma 11, Lemma 12 and Corollary 2.

L -1 σ X i = 5 j=1 P (j) k :W (j) k =i ( X j -L -1 σ πf (P (j) k )) L -1 σ X ′ i = 5 j=1 S (j) k :W (j) k =i ( X ′ j + L -1 σ πf (S (j) k )) , (

Remark 4

We can find a relationship between X i and the atomic surfaces X i . By Proposition 2.1, we have the following equation for the atomic surfaces X i :

L -1 σ (-X i ) = 5 j=1 P (j) k :W (j) k =i ((-X j ) -L -1 σ f (P (j) k )).
This means -X i and X i (i = 1, 2, 3, 4, 5) satisfy the same set equations. Since L σ is a contractive transformation on the plane P <v1,v2> and from the uniqueness of self-affine sets (See Theorem 1 in [START_REF] Mauldin | Hausdorff dimension in graph directed constructions[END_REF] for the uniqueness.), we have the following relation

-X i = X i (i = 1, 2, 3, 4, 5).
Remark 5 We are interested in the disjointness of the partitions X i (i = 1, 2, 3, 4, 5) of X. From the property τ * 4 (e 1 , 1 * ) ⊃ U, any X i is included in the right side of the equation

L -4 σ X 1 = 5 j=1 P (4,j) k :W (4,j) k =1 ( X j -L -4 σ πf (P (4,j) k )) ,
thus by Theorem 4.1 (4), for any i, j

(i = j) µ( X i ∩ X j ) = 0 .
This property of disjointness holds for substitutions satisfying the strong coincidence property in [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF].

Boundaries of atomic surfaces

In this subsection we will observe the boundaries of atomic surfaces X, X i . One of our aims here is to obtain Proposition 4.1, which says that the origin is an inner point of the domain X 1 . For that we will show that the distance between the origin point and the boundary is positive by studying the boundary.

We introduce an endomorphism τ on F as follows (See Fig. 13.):

τ (x, 1) = (L -1 σ x, 5) τ (x, 2) = (L -1 σ x, 1) - K+1 n=1 (L -1 σ x + e 1 -ne 5 , 5) τ (x, 3) = (L -1 σ x, 2) τ (x, 4) = (L -1 σ x, 3) τ (x, 5) = (L -1 σ x, 4) if K = 0 (L -1 σ x, 4) - K n=1 (L -1 σ x + e 4 -ne 5 , 5) if K ≥ 1 Define the boundary map ∂ F * : F * → F as follows: ∂ F * (x, 1 * ) = -(x, 2) + (x, 5) -(x + e 2 , 3) ∂ F * (x, 2 * ) = (x, 1) -(x, 3) -(x + e 3 , 4) ∂ F * (x, 3 * ) = (x, 2) -(x, 4) + (x + e 2 , 1) -(x + e 4 , 5) ∂ F * (x, 4 * ) = (x, 3) -(x, 5) + (x + e 3 , 2) ∂ F * (x, 5 * ) = -(x, 1) + (x, 4) + (x + e 4 , 3) . K+1 K times times ( ) 0 , 1 2 ( ) 0 , τ - τ - τ - τ - τ - 3 ( ) 0 , 4 ( ) 0 , 5 ( ) 0 , 1 π 5 ( ) 0 , 1 π 2 ( ) 0 , -1 L σ -1 L σ

Fig. 13: Geometrical meaning of τ

The following diagram is commutative:

F * τ * → F * ∂ F * ↓ ↓ ∂ F * F τ → F
, and from the definition of the maps π 1 , π 2 , we have

F * π2 → P <v1,v2> ∂ F * ↓ ↓ ∂ F π1 → P <v1,v2>
, where ∂D denotes the boundary of the domain D.

From the two diagrams, we deduce that

∂(L n σ D (i) n ) = ∂(L n σ π 2 (τ * n (e i , i * ))) = L n σ (∂ π 2 (τ * n (e i , i * ))) = L n σ π 1 (τ n (∂ F * (e i , i * ))) . Let B (i) n denote the set of vertices on ∂(L n σ D (i) n ). Lemma 4.1 The sequences of sets {∂(L n σ D (i) n )} and {B (i)
n } converge towards the same set as n goes to ∞ in the sense of the Hausdorff metric.

Proof:

Existence of the limit set of {∂(L n σ D

(i) n )} ∞ n=1 .
It is enough to show that the following limit set exists:

I j = lim n→∞ L n σ π 1 (τ n (0, j)) for any j = 1, 2, 3, 4, 5 . Put c 0 := max i=1,2,3,4,5 d H ( L σ π 1 (τ (0, i)), π 1 (0, i) ),
where d H is the Hausdorff metric. In general the following property holds:

d H (A ∪ B, C ∪ D) ≤ max(d H (A, C), d H (B, D)) ,
for sets A, B, C, D. It is easy to check that τ n (x, i) does not have cancellation for any (x, i) and any positive integer n. Therefore we see

d H ( L σ π 1 (τ n+1 (0, i)), π 1 (τ n (0, i)) ) ≤ c 0 . Hence, d H ( L n+1 σ π 1 (τ n+1 (0, i)), L n σ π 1 (τ n (0, i)) ) ≤ c 0 β 0 n , (4.1)
where

β 0 = 1 √ β K ≤ 2 max{|β (2) |, |β (3) |} K ≥ 3 .
This means the sequence {L n σ π 1 (τ n (0, i))} ∞ n=1 is a Cauchy sequence and it has a limit set in the sense of the Hausdorff metric.

Analogously we see that the sequence

B (i) n converges. By the construction of B (i)
n and a simple approximation argument, we see these limit sets are equal. ✷ Let B (i) (resp., B) denote the limit set

lim n→∞ ∂(L n σ D (i) n )(= lim n→∞ B (i) n ) (resp., lim n→∞ ∂(L n σ D n )).
Lemma 4.2 µ( X i ∩ U r (x)) > 0 (i = 1, 2, 3, 4, 5) for any x ∈ X i and any r > 0.

Proof: From Theorem 4.1 (3),

X i = 5 j=1 P (n,j) k :W (n,j) k =i (L n σ X j -πf (P (n,j) k ))
.

By the boundedness of X i (See Corollary 2.1.), for any x ∈ X i and any r > 0, there exist positive integers n, j, k such that x ∈ L n σ X jπf (P

(n,j) k ) ⊂ U r (x) .
Thus we have

µ( X i ∩ U r (x)) ≥ µ( X i ∩ (L n σ X j -πf (P (n,j) k ))) = µ(L n σ X j ) > 0 . ✷ Lemma 4.3 We have ∂ X i = B (i) for i = 1, 2, 3, 4, 5.
Lemma 4.3 says that the boundary of X i is constructed by the map τ . Proof:

We show that ∂ X 1 = B (1)
and the other cases are shown analogously. To see

B (1) ⊂ ∂ X 1 , it is sufficient to see B (1)
n ⊂ ∂ X 1 for any n. Let N be the collection of tiles which consists of (e 1 , 1 * ) and its neighbour tiles:

N := (e 1 , 1 * ) + (e 3 , 3 * ) + (e 4 , 4 * ) + (e 5 , 5 * ) . By N ⊂ τ * 4 (e 1 , 1 * ) and Theorem 4.1 (3), µ( X 1 ∩ X i ) = 0 for (e i , i * ) ⊂ N , i = 1 . Take x ∈ B (1)
n , then there exists (e j , j * ) ⊂ N , j = 1 such that x ∈ X 1 and x ∈ X j . Suppose that x is an inner point of X 1 , that is, there is r > 0 such that U r (x) ⊂ X 1 . By x ∈ X j and Lemma 4.2, we have

µ( X j ∩ U r (x)) > 0, therefore, for (e i , i * ) ∈ N µ( X 1 ∩ X j ) ≥ µ(U r (x) ∩ X j ) > 0 ,
which leads to a contradiction and it implies B

(1) n ⊂ ∂ X 1 for any n. Conversely, suppose that x ∈ ∂ X 1 ⊂ X 1 . Then there are sequences {x n } ∞ n=1 with x n ∈ L n σ D

(1)

n and {y n } ∞ n=1 with y n ∈ X 1 such that d(x, x n ) < 1 n and d(x, y n ) < 1 n for any positive integer n, where d is the usual Euclidean distance on the plane P <v1,v2> . By y n ∈ X 1 , we have sequences {y n,m } ∞ m=1 : lim m→∞ y n,m = y n , and for any m there exists M ≥ m such that y n,M ∈ L M σ D

(1)

M . Therefore, we can choose y n,kn ∈ L kn σ D

(1)

kn so that d(y n,kn , y n ) < 1 n and k 1 < k 2 < • • • . Thus the sequence {y n,kn } ∞ n=1 satisfies d(y n,kn , x) ≤ d(y n,kn , y n ) + d(y n , x) = 2 n ,
and this means lim n→∞ y n,kn = x. On the segment between x kn ∈ L kn σ D

(1) kn and y n,kn ∈ L kn σ D

(1) kn , there exists c kn ∈ ∂L kn σ D

(1)

kn , and lim n→∞ c kn = x. This implies

x ∈ lim n→∞ ∂L kn σ D (1) kn = lim n→∞ ∂L n σ D (1)
n and ∂ X 1 ⊂ B (1) . ✷ Proposition 4.1 The origin is an inner point of X 1 .

Proof: By Lemma 4.3, we see

L -N σ ∂( X i ) = lim n→∞ L n σ π 1 ( τ n (∂ F * (τ * N (e i , i * ))) ) ,
for any positive integer N . This means that L -N σ ∂( X i ) is constructed by replacing each edge π 1 (x, j) (j = 1, 2, 3, 4, 5) on ∂D (i) N with I j + x, where I j = lim n→∞ L n σ π 1 (τ n (0, j)). By the inequality (4.1), we have the following inequality

d H (L n σ π 1 (τ n (0, i)), π 1 (0, i)) ≤ n-1 j=0 c 0 β 0 j < c for any n,
where c is some positive number. Therefore,

d H (I i , π 1 (0, i)) = d H lim n→∞ L n σ π 1 (τ n (0, i)), π 1 (0, i) ≤ c . (4.2)
By the fact that π 2 (τ * n (U)) is covering the plane P <v1,v2> as n goes to ∞ (See Proposition 3.3 and Section 6.), there exists a positive integer N such that

inf d(0, x) | x ∈ ∂π 2 (τ * N (e 1 , 1 * )) > 2c .
From the construction of L -N σ ∂( X 1 ) and (4.2), we have

inf d(0, x) | x ∈ L -N σ ∂( X 1 ) > c .
Thus we see that for some positive number c

′ inf d(0, x) | x ∈ ∂( X 1 ) > c ′ > 0 .
This implies the origin is an inner point of X 1 . ✷ Proposition 4.2 The Hausdorff dimension of the limit set B of the boundaries

lim n→∞ ∂(L n σ D n ) satisfies (dim H ∂ X =) dim H B ≤ 2 log λ θ log β ,
where

M θ =       0 0 0 0 1 1 0 0 0 K + 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 K      
and λ θ is the maximum eigenvalue of the matrix M θ , that is, the maximum solution of the equation x 5 -Kx 4 -(K + 1)x -1 = 0.

Proof: At first we show that T b X τ * is a tiling of P <v1,v2> . From Proposition 4.1, there exists a positive number δ such that U δ (0) ⊂ X 1 .

Notice that for any positive integer n U β0 -n δ (0) ⊂ L -n σ U δ (0).

By Theorem 4.1 (3)

U β0 -n δ (0) ⊂ L -n σ X 1 = 5 j=1 P (n,j) k :W (n,j) k =1 ( X j -L -n σ πf (P (n,j) k )) ,
where the tiles ( X j -L -n σ πf (P (n,j) k

)) do not overlap for any k, j. From

T τ * = ∪ ∞ n=0 π 2 (τ * n (U)) = ∪ ∞ n=0 π 2 (τ * n (e 1 , 1 * )),
the union of tiles

5 j=1 P (n,j) k :W (n,j) k =1 ( X j -L -n σ πf (P (n,j) k )) is a part of T b X τ * .
This means that the pieces of T b X τ * do not overlap and cover U β0 -n δ (0) for any n. Therefore T b X τ * is a tiling of P <v1,v2> by taking n → ∞. On the other hand, from the quasi periodicity of the tiling T τ * , T b X τ * is also quasi periodic. ✷ At the beginning of this paper, we started from the substitution σ given by (1.1). And we obtained atomic surfaces {X i } i=1,2,3,4,5 (= {-X i } i=1,2,3,4,5 ) and the tiling T b X τ * . We also get tiles {T i } i=1,2,3,4,5 and a tiling T β by using the numeration system related to a Pisot number β as in [START_REF] Akiyama | Self affine tiling and Pisot numeration system[END_REF], [START_REF] Thurston | Groups, tilings and finite state automata[END_REF]. We plan to make the relation between {X i } i=1,2,3,4,5 (resp. T b X τ * ) and {T i } i=1,2,3,4,5 (resp. T β ) explicit with the subdivision rule in [START_REF] Ei | Atomic surfaces and self similar tiling II: Non-irreducible case[END_REF].

Dynamical systems

We introduce two types of measured dynamical systems on X a Markov transformation and a domain exchange transformation with σ-structure.

From Theorem 4.1 (3), x ∈ X i implies that there exist integers j, k such that

L -1 σ x ∈ X j -L -1 σ πf (P (j) k ).
Therefore we get the division of X i :

X i = j k :W (j) k =i X i j k ,
where

X i j k := {x ∈ X i | L -1 σ x ∈ X j -L -1 σ πf (P (j) k )}.
Here we have the following theorem which provides a Markov transformation.

Theorem 5.1 Let us define the map F : X → X by

F (x) = L -1 σ x + L -1 σ πf (P (j) k ) if x ∈ X i j k ,
then the map F is well-defined and

F ( X i j k ) = X j .
The transformation F is well-defined because of Theorem 4.1 (4), and it is called a Markov transformation with matrix structure L σ with respect to partitions { X i } i=1,2,3,4,5 .

From now on we consider a domain exchange transformation with σ-structure.

Definition 5.1 Let (X, T, µ) be a measured dynamical system, σ a substitution over the alphabet A such that

σ(i) = W (i) 0 W (i) 1 • • • W (i) l (i) -1 ,
and consider a measurable partition {X (i) | i ∈ A} of X, a subset A of X and a measurable partition

{A (i) | i ∈ A} of A.
We say that the transformation T has σ-structure with respect to the pair of partitions {X (i) }, {A (i) } if T satisfies the following condition:

T k A (i) ⊂ X (W (i) k ) f or all i ∈ A , k = 0, 1, • • • , l (i) -1 T l (i) A (i) ⊂ A f or all i ∈ A X = i∈A 0≤k≤l (i) -1 T k A (i) (non -overlapping) .
For the transformation T with σ-structure, the induced transformation T | A on A is defined by

T | A (x) = T l (i) (x) f or x ∈ A (i) .
Proposition 5.1 Let σ be a substitution satisfying (1.1) and put

L -n σ = f (n) 1 , f (n) 2 , f (n) 3 , f (n) 4 , f (n) 5
.

(1) The transformation E n : D n → D n given by

E n (x) = x -πf (n) i if x ∈ D (i) n (i = 1, 2, 3, 4, 5)
is well-defined and preserves the Lebesgue measure µ. (See Fig. 14.)

(2) The transformation E 1 : D 1 → D 1 has σ-structure with respect to the pair of partitions {D

E(x) = x -πe i x ∈ X i .
The transformation E, which preserves the Lebesgue measure µ, is well-defined on X. And E has σstructure with respect to the pair of partitions { X i }, {L σ X i }, moreover, E has σ n -structure with respect to the pair of partitions { X i }, {L n σ X i } for all n ∈ N.

Proof: From Theorem 4.1 (1), the transformation E is well-defined. From the equation given by Theorem 4.1 (3) and (4), we have

X i = 5 j=1 P (j) k :W (j) k =i (L σ X j -πf (P (j) k )) , L σ X i ⊂ X W (i) 0 . Hence, E(L σ X i ) = L σ X i -πf (W (i) 0 ) ⊂ X W (i) 1 . Analogously we say E k (L σ X i ) = L σ X i -πf (P (i) k ) ⊂ X W (i) k for k = 0, 1, • • • , l (i)-1 and E l (i) (L σ X i ) = L σ X i -πf (σ(i)) = L σ ( X i -πe i ) = L σ X i ′ .
This means E has σ-structure with respect to the pair of partitions { X i }, {L σ X i }. By induction, we can show the second statement. ✷

This transformation E : X → X is also called the domain exchange transformation associated with a substitution σ. From this theorem and Proposition 4.1 we have the following corollaries:

Corollary 5.1 For k = 0, 1, • • • , we have E k (0) ∈ X s k , where ω = lim n→∞ σ n (1) = s 0 s 1 • • • s n • • • .
From Corollary 5.1 we have the following corollary: (See Lemma 6 in [START_REF] Arnoux | Pisot substitutions and Rauzy fractals[END_REF] and [START_REF] Berth É | Substitutions in dynamics, arithmetics and combinatorics[END_REF].) Corollary 5.2 Let (Ω σ , S) be the substitution dynamical system generated by a substitution σ given by (1.1). The dynamical system ( X, E) is a realization of (Ω σ , S), and the realization map φ from Ω σ to X is given by using φ(S k (s 0 s 1 • • • )) = E k (0) for all positive integers k.

Finally to observe ergodic property of the domain exchange transformation E, we define new domains D n (resp., X) and domain exchange transformations E n (resp., E) on the domains which are measuretheoretically isomorphic to a rotation on the 2-dimensional torus.

Since the domain π 2 (U) is not a 2-dimensional fundamental domain, we introduce the symmetrical image of U denoted by U and the union of U and U denoted by U (cf. Fig. 15 and Fig. 16), which will be a 2-dimensional fundamental domain as follows:

U := {(e 4 , 1 * ) + (e 4 , 3 * ) + (e 4 , 5 * ) + (2e 4 , 1 * ) + (e 2 + e 4 , 5 * )},

U := U + U, L := {m(e 2 -e 4 ) + n(e 1 + e 2 -2e 3 ) | m, n ∈ Z}, D 0 := π 2 ( U) = D 0 ∪ {-D 0 + (πe 1 + πe 2 + πe 4 )}, π(L) := {πz | z ∈ L}.
Then we have the following lemma:

Lemma 5.1 The domains L n σ π 2 (τ * n ( U)) are 2-dimensional fundamental domains for the lattice π(L), that is, L n σ π 2 (τ * n ( U)) = R 2 /L
for any non negative integer n. In other words, we have periodic tilings {L n σ π 2 (y, j * ) | (x, i * ) ⊂ M z ( U) for some z ∈ L, (y, j * ) ⊂ τ * n (x, i * )} for any non negative integer n.

Proof: Replace the segment {tπ(e 2e 4 ) | 0 ≤ t ≤ 1}, which is an edge of the fundamental domain {sπ(e 2e 4 ) + tπ(e 1 + e 2 -2e 3 ) | 0 ≤ s ≤ 1, 0 ≤ t ≤ 1} for the lattice π(L), with π 1 ((-e 4 , 4) + (-e 4 , 2)); and the segment {tπ(e 1 + e 2 -2e 3 ) | 0 ≤ t ≤ 1} with π 1 ((-e 3 , 3) + (-e 3 , 1) + (e 1 -2e 3 , 3) + (e 1 -2e 3 , 2)). We replace the other sides of these edges analogously. Then we have the domain π 2 (M -e1 ( U)). Thus π 2 ( U) is a 2-dimensional fundamental domain. By replacing every edge π 1 (x, i) on π 2 ( U) with L n σ π 1 (τ n (x, i)), we can also say that L n σ π 2 (τ * n ( U)) is a 2-dimensional fundamental domain. (See Fig. 15.) ✷

Let us define the map E 0 : D 0 → D 0 by E 0 (x) = xπe 2 (mod π(L) ) .

For example, the domain π 2 (e 1 , 1 * ) is mapped by E 0 as follows:

π 2 (e 1 , 1 * ) f E0 → π 2 (2e 4 , 1 * ) f E0 → π 2 (e 4 , 1 * ) f E0
→ π 2 (0, 1 * ).

Therefore, E 0 is well-defined and measure-theoretically isomorphic to a rotation on the 2-dimensional torus, moreover, we have

E 0 | D0 = E 0 .
π (e -e ) Let us define the map E : X → X by E(x) = xπe 2 (mod π(L) ) , then it is well-defined and measure-theoretically isomorphic to a rotation on the 2-dimensional torus, moreover, we have E| b X = E .

6 Appendix: The proof of Proposition 3.3

In this section we prove the following proposition stated in Section 3: And for (x, i * ), (y, j * ) there exist (x ′ , i ′ * ), (y ′ , j ′ * ) ⊂ z∈L M L -n σ z (τ * n ( U)) such that (x, i * ) ⊂ τ * (x ′ , i ′ * ) and (y, j * ) ⊂ τ * (y ′ , j ′ * ).

From Fig. 15 we have the following properties:

(1) U, U are C-covered,

(2) if z, z ′ ∈ L (z = z ′ ) satisfy π 2 (M z ( U)) ∩ π 2 (M z ′ ( U)) = ∅, then M z ( U) + M z ′ ( U) is C-covered,

1365-8050 c 2005

 2005 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Fig. 1: The atomic surface X of the substitution σ in the case of K = 0

Fig. 5 :

 5 Fig. 5:The edge k from the vertex i to the vertex j (W(j) k = i)

0 4 KFig. 6 :

 46 Fig. 6: The graph G *

  is a hexagon. (See Fig. 8.) Lemma 3.2 {πe 2 , πe 3 , πe 4 } and {-πe 1 , -πe 5 , -πe 1πe 2 } are hexa-generators.

Fig. 8 :

 8 Fig. 8: Hexa-generators {πe2, πe3, πe4} and {-πe1, -πe5, -πe1 -πe2}

FigFig. 9 :

 9 Fig. 9: Representation of (0, i * ), (ei, i * ) and U ′ , U

Fig. 10 :

 10 Fig. 10: The tiling substitution τ *

  work well. We define U = 5 i=1 (e i , i * ), U ′ = 5 i=1 (0, i * ) as the union of these five elements in F * . (See Fig. 9.) Proposition 3.1 The following relations hold:

Fig. 12 :

 12 Fig. 12: Four pairs including (x, 1 * )

τ * Definition 4 . 1

 41 Define the domains D n , D

  4) the sets in the right side of the equation in (3) do not overlap up to a set of Lebesgue measure 0.

Fig. 14 :

 14 Fig. 14: Domain exchange transformations Ei on Di

Fig. 15 : 2 Fig. 16 :

 15216 Fig. 15: The periodic tiling {π2(x, i * ) | (x, i * ) ⊂ Mz( e U ) for some z ∈ L}In Lemma 5.1 we obtained 2-dimensional fundamental domains and they were constructed by replacing every edge π 1 (x, i) on π 2 ( U) with L n σ π 1 (τ n (x, i)). Moreover, in the same way as in the proof of Proposition 4.1, we also obtain a new 2-dimensional fundamental domain by replacing every edge π 1 (x, i) on the boundary of the domains D 0 and {-D 0 + (πe 1 + πe 2 + πe 5 )} with I i + πx. Then we have the following theorem, which says the domain exchange transformation E is measure-theoretically isomorphic to the induced transformation of a rotation on the 2-dimensional torus:

Proposition 3. 3

 3 ∪ ∞ n=1 π 2 (τ * n (U)) = P <v1,v2> , that is, π 2 (τ * n (U)) covers the plane as n goes to ∞. First we introduce the notion of C-covered property of a set ∆ ∈ F * associated with the connectivity of the domain π 2 (∆) as in [IO93]. Define the subset C 0 of F * and C consisting of translated elements of C 0 by C 0 := {(e 1 , 1 * ) + (e 4 , 4 * ), (e 1 , 1 * ) + (e 3 , 3 * ), (e 2 , 2 * ) + (e 5 , 5 * ), (e 1 , 1 * ) + (e 5 , 5 * ), (0, 1 * ) + (e 2 , 4 * ), (0, 1 * ) + (0, 3 * ), (0, 2 * ) + (0, 5 * ), (0, 1 * ) + (0, 4 * ), (0, 3 * ) + (0, 5 * ), (0, 2 * ) + (0, 4 * )} , C := {M z ξ ∈ F * | ξ ∈ C 0 , z ∈ Z 5 }.

Fig. 17 :Fig. 18 :

 1718 Fig. 17: The elements of C0

Acknowledgements

The authors would like to express our deep gratitude to Prof. Hui RAO for reading through our preprint and giving advice. Big thanks go to the referees who gave us a lot of useful comments.

Proof:

We define the endomorphism θ on the free group < 1 ±1 , 2 ±1 , 3 ±1 , 4 ±1 , 5 ±1 > associated with τ :

We see that θ n (15 -1 34 -1 23 -1 ) has no cancellation for all n. Therefore, from the method of [START_REF] Bedford | Dimension and dynamics for fractal recurrent sets[END_REF], [START_REF] Dekking | Replicating superfigures and endomorphisms of free groups[END_REF], [START_REF] Ito | Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms[END_REF], [START_REF] Ito | On Rauzy fractal[END_REF], we have this result. ✷ Corollary 4.1 For any K ≥ 0 the Hausdorff dimension of B satisfies the inequality:

In the case of

By a simple computation, one gets that

This means K + 1 < β < K + 1 + 1 3K+2 and there is some solution λ ′ of the equation g(x) = 0 such that K+2 2 < λ ′ < K + 1. From g ′ (K + 1) > 0 and g ′′ (x) > 0 with x ≥ K + 1, we know there is no solution of the equation g(x) = 0 with x ≥ K + 1, that is, λ ′ = λ θ and K+2 2 < λ θ < K + 1. Using the above two inequalities we have the conclusion. ✷

A second tiling given by τ * with atomic surfaces

In Section 3 we introduce a first tiling T τ * with five polygonal prototiles π 2 (x, i * ) (i=1,2,3,4,5). Here we consider a second tiling with five prototiles X i with fractal boundary by replacing each prototile

is a quasi periodic tiling with five prototiles X i (i = 1, 2, 3, 4, 5) of the plane P <v1,v2> .

(3

Hence, there exists a finite subset

The second condition implies

On the other hand, τ * (Γ) := {τ * (γ t ) | t = 1, 2, • • • , N } has the following property:

and τ * (γ t ) is C-covered by Lemma 6.1 for any t, that is, π 2 (x, i * ) and π 2 (y, j * ) are connected by using elements of C. This provides

Finally we want to show that π 2 (τ * n (U)) is expanding on P <v1,v2> as n goes to ∞.

(3) there exists (x, i * ) ⊂ ∆ satisfying

(4) π 2 (Γ) is an annulus and π 2 (Γ) bd ⊃ π 2 (τ * n (U)), where A bd means the bounded domain of the complement of an annulus A.

Γ is called an n-th C-belt of ∆ .

, and replace every edge on these domains by using the replacing and redividing method. Then we have π 2 (Γ ′ ) and π 2 (τ * n+1 (U)). By the property (3) of Definition 6.3, π 2 (Γ ′ ) is an annulus. From the relation From Lemma 6.3, τ * 8n (U) has an 8(n -1)-th C-belt Γ 8(n-1) for any n such that π 2 (τ * 8n (U)τ * 8(n-1) (U)) ⊃ π 2 (Γ 8(n-1) ).

Thus the distance of the boundary of π 2 (τ * n (U)) from the origin tends to ∞; and from Lemma 6.2, π 2 (τ * n (U)) is a topological cell for any n. Therefore, π 2 (τ * n (U)) is covering the plane P <v1,v2> as n goes to ∞. ✷


Ana M Breda
email: ambreda@mat.ua.pt

Altino F Santos

Dihedral f-tilings of the sphere by rhombi and triangles

Keywords: dihedral tilings, isometric foldings, spherical trigonometry, WCSQ

.

Introduction

An isometric folding is a non-expansive locally isometry that sends piecewise geodesic segments into piecewise geodesic segments of the same length. An isometric folding is a continuous map that need not to be differentiable. The points where it is not differentiable are called singular points. The foundations of isometric foldings of Riemannian manifolds are introduced by [START_REF] Grünbaum | Isometric foldings of riemannian manifolds[END_REF].

For surfaces, the set of singular points gives rise to two colored tilings (called f-tilings) with the property that each vertex has even valency and obey the angle folding relation, i.e., the sums of alternating angles around each vertex is π.

The classification of f-tilings started out by [START_REF] Breda | A class of tilings of s 2[END_REF], where a complete classification of all monohedral f-tilings of the sphere by triangles was done.

The complete classification of monohedral tilings of the sphere by triangles (which, obviously, includes the monohedral triangular f-tilings) was made clear by [START_REF] Ueno | Classification of tilings of the 2-dimensional sphere by congruent triangles[END_REF]. This classification was partially done by [START_REF] Sommerville | Division of space by congruent triangles and tetrahedra[END_REF], and an outline of the proof was provided by [START_REF] Davies | Packings of spherical triangles and tetrahedra[END_REF].

For additional information on tilings, see [START_REF] Grünbaum | Isometric foldings of riemannian manifolds[END_REF].

Our interest focuses in dihedral spherical f-tilings.

Preliminaries

Definition 2.1 Let S 2 be the euclidean 2-sphere of radius 1. A spherical moon L is said well centered if its vertices belong to the great circle S 2 ∩ {(x, y, z) ∈ R 3 | x = 0} and the semi-great circle bisecting L contains the point (1, 0, 0). If L 1 and L 2 are two well centered spherical moons with orthogonal vertices, then L 1 and L 2 are said to be orthogonal.

Definition 2.2 By a well centered spherical quadrangle (WCSQ) we mean a spherical quadrangle which is the intersection of two well centered spherical moons with distinct vertices.

The properties of WCSQ was described by [START_REF] Breda | Well centered spherical quadrangles[END_REF], where was established that any spherical quadrangle with congruent opposite internal angles is congruent to a WCSQ. Definition 2.3 Let Q and T be, respectively, a spherical triangle and a spherical quadrangle. A dihedral tiling of the sphere S 2 with prototiles Q and T is an edge-to-edge polygonal subdivision τ of S 2 , such that each tile (face) of τ is isometric either to Q or T .

In addition, if the number of geodesic rays (edges) emanating from each vertex of τ is even, making alternated angles

α 1 , β 1 , α 2 , β 2 , . . . , α n , β n (n ≥ 2), where n j=1 α j = n j=1 β j = π (angle folding relation)
then τ is called a dihedral folding tiling or dihedral f-tiling for short.

In short, dihedral means that every tile of τ is congruent to one of the fixed sets Q and T , called the prototiles of τ . Definition 2.4 Let v be a vertex of a (dihedral) f-tiling. The number of edges emanating from v is called the valency of v.

In Fig. 1 is illustrated a vertex v obeying the angle folding relation (α In [START_REF] Breda | Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles[END_REF] the classification of dihedral f-tilings of S 2 by spherical triangles and equiangular spherical quadrangles (i.e., spherical quadrangles that are congruent to a WCSQ which is the intersection of two orthogonal well centered spherical moons) was presented.

1 + α 2 + α 3 = β 1 + β 2 + β 3 = π) with valency six. b 1 a 2 b 2 a 3 b 3 a 1 v
Here we extend the study of f-tilings presenting the classification of all dihedral f-tilings by spherical triangles and non-equiangular spherical rhombi, in other words, spherical quadrangles whose edges have the same length and with distinct pairs of opposite internal angles.

In [START_REF] Breda | Well centered spherical quadrangles[END_REF] it was shown that any non-equiangular spherical rhombus Q is congruent to a WCSQ which is the intersection of two non-orthogonal well centered spherical moons L 1 and L 2 with the same angle measure θ (Fig. 2).

Q =L 1 L 2 a 1 a 1 a 2 a 2 q q q q L 1 L 1 L 2 L 2
Fig. 2: A non-equiangular spherical rhombus Q.

Definition 2.5

The spherical tilings τ 1 and τ 2 are isomorphic if there exists an isometry φ of S 2 , such that φ(τ 1 ) = τ 2 .

Notation:

We shall denote by Ω(Q, T ) the set, up to an isomorphism, of all dihedral f-tilings of S 2 , whose prototiles are a non-equiangular spherical rhombus Q and a spherical triangle T . Proposition 2.1 (Breda and Santos (2004)) Let τ ∈ Ω(Q, T ). If M > 0 and N > 0 denote, respectively, the number of spherical quadrangles congruent to Q and the number of spherical triangles congruent to T of τ , and E and V denote, respectively, the number of edges and vertices of τ , then:

i) N + 2M = 2V -4 , 3V = 6 + M + E;
ii) there are at least 6 + M vertices of valency four; iii) π < β + γ + δ < 3π 2 and α 1 + α 2 > π, where α 1 and α 2 are the pairs of opposite internal angles of Q, and β, γ and δ are the internal angles of T .

We shall describe Ω(Q, T ), considering separately different cases depending on the nature of Q and T . In order to get the dihedral f-tilings of Ω(Q, T ) we find useful to start by considering one of its planar representation (PR), beginning with a common vertex to a spherical quadrangle and a spherical triangle in adjacent positions (it must be pointed out that there is always a vertex satisfying this condition).

Whenever a tiling is drown in one hemisphere, it means that the plane x = 0, say, is a plane of symmetry. The whole tiling is then obtained by reflecting the picture in this plane.

Spherical trigonometry is based on the following cosine rules: The internal angles α, β and γ of a spherical triangle T obey:

cos α = cos a -cos b cos c sin b sin c and cos a = cos α + cos β cos γ sin β sin γ , (2.1)
where a, b and c are the lengths of the edges opposite to α, β and γ, respectively. This formulas are of crucial interest to prove the results that follow. For a detailed discussion on spherical trigonometry, see [START_REF] Berger | Geometry[END_REF].

3 Prototiles: Non-equiangular rhombus and equilateral triangle Proposition 3.1 If Q and T are, respectively, a non-equiangular spherical rhombus and an equilateral spherical triangle, then Ω(Q, T ) is the empty set.

Proof: Let Q be a non-equiangular spherical rhombus with pairs of opposite internal angles α 1 and α 2 (α 1 > α 2 ), and let T be an equilateral spherical triangle with internal angle β.

Suppose that Ω(Q, T ) is non empty. Let v be a vertex of Q surrounded by α 1 . As

α 1 + α 2 > π (α 1 > π
2 ) and β > π 3 , then v has valency four, and the cells surrounding v have, in cyclic order, angles measure

(α 1 , α 1 , β, β), with α 1 + β = π. A planar representation (PR) of a f-tiling τ ∈ Ω(Q, T ) near v is illustrated in Fig. 3-I. It follows that α 2 + β < π and α 2 + kβ > (k + 1)β > π, for k ≥ 2.
Hence in order to have the angle folding relation satisfied we must have α 2 + α 2 = π, i.e., α 2 = π 2 . This information allows us to adjust some more cells to the initial PR leading to the PR illustrated in Fig. 3-II. In order to go on with the PR of τ we must analyze the angles behavior around v 1 . We already know that the cyclic sequence of their positions contains the sequence (. . . , β, α 2 , β, . . . ). As 2β < π and 2β + ρ > π for all ρ ∈ {β, α 1 , α 2 }, then there is no way to position another cells around v 1 in order to have the angle folding relation full filled. And so

Ω(Q, T ) = ∅. a 2 a 1 a 2 b b b b b a 2 a 1 a 1 a 1 a 2 b a 1 a 1 a 2 b b b b b b b b b b a 2 a 1 a 1 a 1 a 2 a 2 a 2 a 2 a 2 a 1 I II v v 1 v 1 v Fig. 3: Planar representations.
✷ The argumentation used to prove the Propositions stated in the following Sections are somehow similar to the one used on proof of Prop. 3.1. Thus, we decided to present only the proofs we consider more representative. A detailed study containing the proofs of those results can be found in a technical report placed in the web site http://www.mat.ua.pt/ambreda/rhombi.pdf 4 Prototiles: Non-equiangular rhombus and isosceles triangle

Through this section Q is a non-equiangular spherical rhombus with pairs of opposite internal angles α 1 and α 2 (α 1 > α 2 ), and T is an isosceles spherical triangle with internal angles β, γ, γ (β = γ).

Any element of Ω(Q, T ) has, at least, two cells congruent, respectively, to Q and T , such that they are in adjacent positions in one of the situations illustrated in Fig. 4. Next, by "unique f-tiling" we mean unique up to an isomorphism.

g a 1 a 2 a 1 a 2 g b a 2 a 1 a 1 a 2 b g g a 2 a 1 a 1 a 2 g b g B A C
α 1 + γ = π. Moreover, i) if α 2 = β, then Ω(Q, T ) = ∅ iff α 2 + β = π (β = arccos (1 -2 cos π k )
) and γ = π k , for some k ≥ 4. In this situation for each k ≥ 4, Ω(Q, T ) contains a unique dihedral f-tiling, denoted by R k , such that the angles around vertices are the ones represented at the head of the Fig. 7

(consider k ≥ 4); ii) if α 2 = β, then Ω(Q, T ) = ∅ iff α 1 = 2π 3 , α 2 = π 2 and γ = π 3 . Besides, Ω(Q, T ) = {R 3 , U 1 , U 2 , U 3 , U 4 }
, where R 3 and U i , 1 ≤ i ≤ 4 are non-isomorphic dihedral f-tilings, such that the angles around vertices are the ones represented in Fig. 7.

Proof: We only present the proof of the i) case.

In order to obtain a complete PR of an element τ ∈ Ω(Q, T ), the PR illustrated in Fig. 4-C is extended in a unique way to get the PR illustrated in Fig. 5, where α 1 + γ = π, α 2 + β = π and kγ = π, for some k ≥ 4. (Note that the case k = 3 implies α

2 = β = π 2 .) a 2 a 1 a 2 b g g g b g a 1 a 2 a 2 g a 2 a 1 a 2 b g a 1 g g g g b b b g a 1 a 1 g Fig. 5: PR of R k , k ≥ 4.
The side length of T opposite to γ and the side length of Q are the same. Using the formulas (2.1) we may show that

β = arccos (1 -2 cos π k ) = 2 arccos 1 -cos π k , k ≥ 4. (4.1)
3D representations for k = 4 and k = 5 are illustrated in Fig. 6.

5 4 R R b g a 2 a 1 g a 1 a 2 b g g g a 1 g a 2 b g b a 2
a 1 g Fig. 6: 3D representations. Fig. 7 exhibits the vertices in the dihedral f-tilings corresponding to the i) and ii) cases. In Fig. 8 a 5 Prototiles: Non-equiangular rhombus and scalene triangle

1 a 1 g g a 2 a 2 b b g g g g g g a 1 a 1 g g a 2 a 2 b a 2 1 U a 1 a 1 g g a 2 b a 2 b a 2 a 2 b a 2 U 2 g g g g g g a 1 a 1 g g b b a 2 a 2 b b b a 2 a 2 a 2 b b a 1 a 1 g g a 2 a 2 a 2 b a 2 a 2 b a 2 g g g g g . . . 3 U U 4 R k k , > 3 a 2 a 2 b b a 2 b b b k g = p
Here Q is a non-equiangular spherical rhombus with pairs of opposite internal angles α 1 and α 2 (α 1 > α 2 ), and T is a scalene spherical triangle with internal angles β, γ and δ, such that β > γ > δ.

If τ ∈ Ω(Q, T ), then there are necessarily two cells of τ congruent, respectively, to Q and T , such that they are in adjacent positions in one of the situations illustrated in Fig. 9. Suppose that the PR illustrated in Fig. 10 is contained in a complete PR of any element τ ∈ Ω(Q, T ). It follows that α 2 + γ ≤ π. We shall consider separately the cases α 2 + γ < π and α 2 + γ = π.

B A C a 2 a 1 a 1 a 2 g b d E D F a 2 a 1 a 1 a 2 b d g a 2 a 1 a 1 a 2 b g d b a 1 a 2 a 1 a 2 d g a 2 a 1 a 1 a 2 d g b d a 1 a 2 a 1 a 2 g b
Proposition 5.3 Let α 2 + γ < π. i) If α 2 > γ, then τ = M 4 ; ii) If α 2 = γ, then τ = E 3 or τ = S 3 ; iii) If α 2 < γ, then τ = M k+1 or τ = E k or τ = S k , k ≥ 4,
where the internal angles of the prototiles present in the dihedral f-tilings are:

α 1 = (k -1)π k α 2 = π -2γ β = π 2 cos γ = 1 -cos π k δ = π k M k , k ≥ 4; α 1 = π -δ α 2 = π 2k -3 β = π 2 γ = (k -2)π 2k -3 cos δ = cos 2 π 4k -6 E k , k ≥ 3; α 1 = π -δ α 2 = π k β = π 2 γ = (k -1)π 2k cos δ = cos 2 π 2k S k , k ≥ 3.
The angles around vertices are positioned as illustrated in Fig. 11. Proof: We only present the proof of the i) case.

b b b b a 1 a 1 d d k d = p d d d d d d d d d . . . b b b b a 1 a 1 d d a 2 g g g a 2 g k = p a 2 . a 2 a 2 a 2 a 2 a 2 . . a 2 a 2 b b b b a 1 a 1 d d k = p 1 ( )a 2 g a 2 g g g a 2 g . a 2 a 2 a 2 a 2 a 2 g g . . g g g g a 2 a 2 M k k , > 4 S k k , > 3 E k k , > 3
The information given by the PR illustrated in Fig. 10 leads to conclude that

α 1 > β = π 2 > α 2 > γ > δ.
(5.1)

Consider a vertex v surrounded by the cyclic sequence (α 2 , γ, γ, . . . ). Then the sum of alternating angles around v containing α 2 must be α

2 + γ + α 2 = π or α 2 + 2γ = π. 1. α 2 + γ + α 2 = π
In this case the PR illustrated in Fig. 10 is extended as shown in Fig. 12 (observe that all cells labelled from 2 to 8 are uniquely determined, since α 2 +γ < α 2 +2γ < π = 2α 2 +γ < α 2 +γ+2δ). This leads to a vertex surrounded by the cyclic sequence of angles (γ, γ, α 2 , γ, γ, . . . ), where 2γ+α 2 < π < 2γ+α 2 +ρ, for all ρ ∈ {α 1 , α 2 , β, γ, δ}. And so there is no way to obey the angle folding relation at that vertex.

α 2 + 2γ = π

In this case the cyclic sequence of angles around v is either (α 2 , γ, γ, α 2 , γ, γ) or (α 2 , γ, γ, γ, γ, α 2 ).

2.1. The first situation leads us immediately to a contradiction as illustrates Fig. 13. In fact, this case gives rise to a vertex w surrounded by the cyclic sequence (α 2 , α 2 , α 2 , α 2 , . . . ). As α 2 > γ, then 2α 2 +γ > π = α 2 +2γ > 2α 2 . And so the sums of alternating angles around w must obey 2α 2 +δ = π, as illustrated in the figure. This procedure gives rise to a vertex surrounded by α 1 and γ in adjacent positions, leading us to an incompatibility between angles around this vertex (see side length of the prototiles). 2.2. In the second case we may extend without more restrictions the PR illustrated in Fig. 10 to get the one illustrated in Fig. 14. This construction leads to two antipodal vertices surrounded by at least six δ angles. It is a straightforward exercise to show that those vertices are enclosed exclusively by δ angles. Therefore kδ = π, for some k ≥ 4. On the other hand α 2 + 2γ = π with γ < α 2 . Hence γ < π 3 < α 2 and so δ = π 4 or δ = π 5 , as γ + δ > π 2 . Firstly, consider that δ = π 4 . Then α 1 = 3π 4 and the PR illustrated in Fig. 14 as shown in Fig. 15. The sum of alternating angles around vertices verify (5.2)

α 1 + δ = π, β = π 2 , α 2 + 2γ = π (α 2 > γ) and δ = π 4 . b a 1 a 2 g g d b b b d a 1 a 1 a 2 a 2
On the other hand b is also the side length of Q, therefore

cos b = cos α2 2 (1 + cos α 1 ) sin α2 2 sin α 1 = cos( π 2 -γ)(1 + cos α 1 ) sin( π 2 -γ) sin α 1 = tan γ 1 -cos π 4 sin π 4 .
(5.3) By (5.2) and ( 5.3) we conclude that

tan 2 γ = cos π 4 1 -cos π 4 .
And so

γ = arccos 1 -cos π 4 = 1 2 arccos (1 - √ 2) ≈ 57 • .
To complete the proof we shall consider now that δ = π 5 (Fig. 14). As before the side length of T , b, opposite to β is common to Q, leading to conclude that γ = arccos 1cos π 5 = arccos

√ 3- √ 5 2 ≈ 64 • > π
3 , contradicting (5.1). 3D representations of some of these f-tilings are illustrated in Fig. 16. For additional information see also Tab. 1.

✷

Remark: A 3D representation of the dihedral f-tilings M k , k ≥ 4 is obtained from a 3D representation of R k , k ≥ 4 (Prop. 4.2 i)
) by adding some edges bisecting by β = 2 arccos 1cos π k any isosceles triangle of R k and preserving the angle-folding relation. Compare, for instance, the f-tilings M 5 (above, Fig. 16) and R 5 (Fig. 6).

Proposition 5.4 If α

2 + γ = π, then Ω(Q, T ) is the family of f-tilings denoted by R 2 δ δ∈]0, π 3 [ , where cos γ + cos δ = 1. (Its 3D representation is illustrated in Fig. 17.) Remark: If δ = π
3 , then γ = π 3 and so α 1 = α 2 = 2π 3 , giving rise to a dihedral f-tiling, which was established in [START_REF] Breda | Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles[END_REF]. Proposition 5.6 If the PR illustrated in Fig. 18 is contained in a complete PR of an element τ ∈ Ω(Q, T ), then:

i) If α 2 ≥ π 2 , then α 1 = 2π 3 , α 2 = π 2 = β, γ = π 3 and δ = π 4 ; ii) If α 2 < π 2 , then α 1 = 2π 3 , α 2 = 2π 5 , β = π 2 , γ = π
3 and δ = π 5 . Moreover, the first situation leads to two dihedral f-tilings denoted by M 3 and S 2 , and in the second case one gets three dihedral f-tilings denoted by G 1 , G 2 and G 3 , respectively. In any case Q is decomposed in four triangles congruent to T , bisecting Q twice, by α 1 and by α 2 . The sum of alternating angles around vertices are represented in Fig. 19. For a 3D representation of the mentioned f-tilings, see Fig. 20.

Finally we present the f-tilings obtained from the adjacency illustrated in Fig. 9-E Any planar representation of R k φ has the configuration of Fig. 5 (replacing one of the γ angles by δ) and the sum of alternating angles around vertices are given by: i) α 1 + δ = π, α 2 + β = π (β > α 2 ) and γ = π k , k ≥ 3, when considered the adjacency illustrated in Fig. 9-E. In this case φ = δ ∈]0, π k [; In both cases cos β + cos γ + cos δ = 1.

g g g g g g a 1 a 1 g g d d d d a 2 a 2 b b b b M 3 a 1 a 1 g g b b b b S 2 a 2 a 2 a 2 a 2 d d d d a 2 a 2 1 G b b b b a 1 a 1 g g g g g g g g d d a 2 a 2 a 2 a 2 d a 2 a 2 d d d d d d a 2 a 2 d d d d d b b b b 2 G g g g g g g a 1 a 1 g g d d a 2 a 2 a 2 a 2
Proof: We shall suppose that τ ∈ Ω(Q, T ) has two cells in adjacent positions as illustrated in Fig. 9-E (the other case is analogous). Using similar argumentation to the one used before we obtain in a unique way a planar representation with the configuration of Fig. 5 (replacing one of the γ angles by δ), where α 1 + δ = π, α 2 + β = π and γ = π k , for some k ≥ 3. Now, as Q has all congruent sides and distinct pairs of opposite internal angles, then Q is congruent to a WCSQ, as seen in [START_REF] Breda | Well centered spherical quadrangles[END_REF]. This means that Q = L 1 ∩ L 2 , for some non-orthogonal well centered spherical moons L 1 and L 2 , with angle measure γ = π k (k ≥ 3), see Fig. 21.

x p 2 x T T T T d Q =L 1 L 2 a 1 a 1 a 2 a 2 b b d p k p k p k p k a 1 d a 2 b k ( 1 p ) k 2 Fig. 21: A rhombus, Q = L1 ∩ L2.
With the labelling of Fig. 21, one gets

cos β = 1 -cos π k 2 - 1 + cos π k 2 sin x (5.4) and cos δ = 1 -cos π k 2 + 1 + cos π k 2 sin x, (5.5)
for some x ∈]0, π 2 [ and k ≥ 3. And so cos π k + cos β + cos δ = 1. The relations (5.4) and (5.5) define, respectively, an increasing continuous bijection between

x ∈]0, π 2 [ and β ∈] arccos 1-cos π k 2 , (k-1)π k [, and a decreasing continuous bijection between x ∈]0, π 2 [ and δ ∈ ]0, arccos 1-cos π k 2 [. Note that γ ≤ π 3 < arccos 1-cos π k 2 < π 2 , for all k ≥ 3. Now, as β > γ = π k > δ, then exists x k ∈]0, π 2 [, such that 0 < δ = δ(x) < π k , for all x ∈]x k , π 2 [ and k ≥ 3. In fact, δ < π k ⇔ cos δ(x) > cos π k ⇔ sin x > -1 + 3 cos π k 1 + cos π k .
And so

x k = arcsin 3 - 4 1 + cos π k . Taking β k = β(x k ) (i.e., δ = π k ), then one gets β k = 2 arccos 1 -cos π k = arccos(1 -2 cos π k ). One has β 3 = π 2 and π 2 < β k < π, if k ≥ 4. And so β = β(x) > β k ≥ π 2 > α 2 , for all x ∈]x k , π 2 [ and k ≥ 3.
The graphic representation of β and δ is shown in Fig. 22. and so α 1 = α 2 , giving rise to a dihedral f-tiling established in [START_REF] Breda | Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles[END_REF]. The case x = x k (k ≥ 3) leads to the f-tiling R k obtained in Prop. 4.2 (see (4.1)).

In Tab. 1 it is shown a complete list of all dihedral f-tilings, whose prototiles are a spherical triangle T of angles β, γ, δ (β > γ > δ) if T is scalene, or angles β, γ, γ (β = γ) if T is isosceles, and, a non-equiangular spherical rhombus Q with pairs of opposite internal angles α 1 and α 2 (α 1 > α 2 ). We have used the following notation:

• β k = arccos(1 -2 cos π k ), β * k = arccos 1-cos π k 2
, and δ k = arccos(cos 2 π 2k );

• At f-tilings R 2 δ , R k φ (φ = δ or φ = γ) one has cos β + cos γ + cos δ = 1;

• |V | is the number of distinct classes of congruent vertices (we shall say that vertices v 1 and v 2 are congruent if and only if there is a spherical isometry that maps the faces incidents to v 1 into the faces incidents to v 2 );

• M and N are, respectively, the number of quadrangles congruent to Q, and the number of triangles congruent to T , used in such dihedral f-tilings. In Fig. 24 it is shown a 3D representation of the dihedral f-tilings described on Tab. 1. Observe that for the families M k , E k and S k-1 , k ≥ 3 we have chosen M 5 , E 3 and S 3 as representatives for their 3D representations.

R k , k ≥ 3 (k-1)π k π -β k β k π k - 3 2k 4k R 2 δ π -δ π -γ π 2 ] π 3 , π 2 [ ]0, π 3 [ 3 4 8 R k δ , k ≥ 3 π -δ π -β ]β k , (k-1)π k [ π k ]0, π k [ 3 2k 4k R k γ , k ≥ 3 π -γ π -β ]β * k , β k [ ] π k , β * k [ π k 3 2k 4k M 3

Fig. 1 :

 1 Fig. 1: The angle folding relation.

Fig. 4 :

 4 Fig. 4: Distinct cases of adjacency.

Proposition 4. 1

 1 If Ω(Q, T ) = ∅, then there cannot be a pair of tiles in the position of Fig.4-Aor Fig. 4-B. Proposition 4.2 If a f-tiling contains two cells in the position of Fig. 4-C, then

Fig. 7 :Fig. 8 :

 78 Fig. 7: Distinct classes of congruent vertices.

Fig. 9 :Fig. 10 :

 910 Fig. 9: Distinct cases of adjacency.

Fig. 11 :

 11 Fig. 11: Distinct classes of congruent vertices.

Fig. 12 :

 12 Fig. 12: Planar representation.

Fig. 13 :

 13 Fig. 13: Planar representation.

Fig. 14 :

 14 Fig. 14: Planar representation.

Fig. 15 :

 15 Fig. 15: Extended planar representation.

Fig. 16 :

 16 Fig. 16: 3D representations.

Proposition 5 . 5 Fig. 18 :

 5518 Fig. 17: 3D representation.

  or Fig. 9-F. Proposition 5.7 Let τ be an element of Ω(Q, T ), such that τ has two cells in adjacent positions as illustrated in Fig. 9-E or Fig. 9-F, then τ = R k φ , for some φ ∈ 0, arccos 1cos π k 2 \ {π/ k} and k ≥ 3.

Fig. 19 :Fig. 20 :

 1920 Fig. 19: Distinct classes of congruent vertices.

Fig. 22 :

 22 Fig. 22: Graphic representation of β and δ.

  Fig. 23: 3D representations.

  Dihedral spherical f-tilings by triangles and non-equiangular rhombi.

† Supported in part by UI&D Matemática e Aplicac ¸ões of University of Aveiro, through Program POCTI of FCT cofinanced by the European Community fund FEDER. ‡ Supported in part by UI&D Matemática e Aplicac ¸ões of University of Aveiro, through Program POCTI of FCT cofinanced by the European Community fund FEDER. 1365-8050 c 2005 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Acknowledgements

The authors are grateful to the referees for their helpful comments and improvements.