M.A. Rodrıguez,a J.L. Arag ónb and L. Verde-Starc. aDepartamento de Matemáticas, Escuela Superior de Fısica y Matemáticas, Instituto Politécnico Nacional.
research papers Acta Crystallographica Section A
Foundations of Crystallography ISSN 0108-7673
Received 0 XXXXXXX 0000 Accepted 0 XXXXXXX 0000 Online 0 XXXXXXX 0000
Clifford algebra approach to the coincidence problem for planar lattices M.A. Rodr´ıguez,a J.L. Aragon ´ b and L. Verde-Starc a
Departamento de Matem´aticas, Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional.
Unidad Profesional Adolfo Lopez ´ Mateos, Edificio 9. 07300 M´exico D. F., M´exico., b Centro de F´ısica Aplicada y Tecnolog´ıa Avanzada, Universidad Nacional Autonoma ´ de M´exico, Apartado Postal 1-1010, 76000 Quer´etaro, M´exico., and c Departamento de Matem´aticas, Universidad Autonoma ´ Metropolitana, Unidad Iztapalapa, Apartado Postal 55–534, CP 09340 M´exico D.F., M´exico.
c 0000 International Union of Crystallography
Printed in Great Britain – all rights reserved
The problem of coincidences of planar lattices is analyzed using Clifford algebra. We show that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows us to characterize algebraically planar coincidence lattices. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions.
1. Introduction Coincidence Site Lattice (CSL) theory has provided partial answers to the complex problem that arises in the description of grain boundaries and interfaces (see, for instance, Sutton & Baluffi, 1995). Most of the existing geometric models of grain boundaries idealize the two crystals that meet at the boundary as two interpenetrating lattices and it is assumed that grain boundaries with special properties arise when there is a high degree of good fit (or matching) between the lattices. The CSL model (Ranganathan, 1966) considers points common to both lattices (the intersection lattice) as points of good fit and assumes that special boundaries arise when the density of coincidence sites is high, because many atoms would occupy sites common to both grains. The experimental support for the CSL model is the special properties, such as changed migration rates, observed for boundaries with certain orientational relationship. Working with this model, we have to consider two identical copies of a lattice Λ (one of the 14 Bravais lattices) and brought into coincidence. Next one lattice is rotated, relative to the other, by an angle θ abut an axis through a common lattice point. Then for different values of θ two possibilities will arise: no lattice sites will coincide (except the site where the rotation axis passes through) or, due to the periodicity of Λ, a infinite number will coincide forming a lattice. Such a lattice is a CSL and the ratio of the volumes of primitive cells for the CSL and for Λ is denoted by Σ. High densities of coincident sites correspond to low values of Σ. Since the advent of quasicrystals it has been desirable to extend the mathematical theory of CSL to more dimensions. In general, the problem can be stated as follows: Let Λ be a lattice in Rn and let R ∈ O(n) be an orthogonal transformation. R is called a coincidence isometry if Λ ∩ RΛ is a sublattice of Λ. The problem is therefore to identify and characterize the coincidence isometries of a given lattice Λ. Several approaches have been used to tackle this problem. Fortes (1983) developed a matrix theory of CSL in arbitrary dimensions, including a method to calculate a basis for the co-
incidence lattice through a particular factorization of the matrix defining the relative orientation. The same matrix approach was implemented by Duneau et al. (1992), but they evaluated the parameters of the coincidence lattice using a method based on the Smith normal form for integer matrices. Baake (1997) used complex numbers and quaternions to solve the problem in dimensions up to 4. Finally, Arag´on et al. (1997) proposed a weak coincidence criterion and used four-dimensional lattices to characterize coincidence lattices in the plane. Here, we analyze the problem using Clifford algebra with a twofold purpose. First, the power of this mathematical language for expressing geometric ideas is used to solve the coincidence problem, which is merely geometric. Second, we try to show that Clifford algebra, already used as powerful language in several fields, can also be useful in geometrical crystallography. Although nothing new emerges, the results provide new insights in this and other problems in geometric crystallography and the approach could be valuable for extension to arbitrary dimensions. In this approach, reflections are considered as primitive transformations and Clifford algebra emerges as a natural tool for this problem, without using matrices and only a minimum of group theory. It is found that any arbitrary coincidence isometry can be decomposed as a product of coincidence reflections by vectors of the lattice Λ, and the group of coincidence isometries is characterized by providing a way to generate it from vectors of Γ. The paper is organized as follows. In Section 2 we provide a brief introduction to the Clifford algebra by considering the particular case of the Euclidean plane and by focusing only in results relevant to the coincidence problems. In Section 3 some basic mathematical definitions of the CSL problem are presented. Sections 4, 5, and 6 present the solution of the coincidence problem for square, rectangular and rhombic lattices, respectively. Finally, Section 7 is devoted to conclusions and discussion. 2. The Clifford algebra of the Euclidean plane
research papers Clifford algebra has proved to be a useful language in many areas of physics, engineering and computer science (see, for example, Bayro-Corrochano & Sobczyk, 2001). In crystallography, Hestenes (2002) presents a new approach to symmetry groups and Arag´on et al. (2001) use Clifford algebra to study the problem of faceting in quasicrystals and to state the basis of the CSL theory which is developed and exploited here. In what follows we introduce the basic elements of Clifford algebra by considering the particular case of the Euclidean plane with the standard scalar (inner) product.
and the outer (Grassman) product x ∧ y = (α1 β2 − α2 β1 )e1 e2 . Consequently, the geometric product (2) can be written as xy = x · y + x ∧ y.
From the above expressions, the inner and outer products can be defined in terms of the geometric product as
Definition 1 The real associative and distributive algebra generated by the Euclidean plane and the product rules e2i = 1, i = 1, 2, e1 e2 + e2 e1 = 0,
As a vector space, the geometric algebra R2,0 is fourdimensional, with basis {1, e1 , e2 , e1 e2 }, where e1 e2 is called bivector. A general element of this vector space is formed by an arbitrary linear sum over the four basis elements and is called multivector. If a1 , . . . , a4 are scalars, an arbitrary multivector is then A = a 1 1 + a 2 e2 + a 3 e3 + a 4 e1 e2 . From Definition 1, we have that if x, y ∈ R2 , i.e., x = α1 e1 + α2 e2 and y = β1 e1 + β2 e2 , for real numbers α1 , α2 , β1 and β2 , then the geometric product of x and y is xy = α1 β1 + α2 β2 + (α1 β2 − α2 β1 ) e1 e2
(2)
The multiplication rules (1) lead also to the so-called fundamental axiom of geometric algebra: Proposition 1 If x, y ∈ R2 , then x2 , y2 ≥ 0 and xy + yx ∈ R. Even more x2
= x · x, 1 (xy + yx) = x · y, 2 where x · y is the Euclidean inner product in R2 .
Proof. Let x = α1 e1 + α2 e2 and y = β1 e1 + β2 e2 , be vectors in R2 . Then, from (2) we have that 2
x2 = (α1 e1 + α2 e2 ) = α21 + α22 ≥ 0, and
xy + yx = 2 (α1 β1 + α2 β2 ) ∈ R.
The scalar and bivector part of the geometric product (2) are associated, respectively, with the inner product x · y = α 1 β1 + α 2 β2 ,
x∧y = x·y =
(1)
where {e1 , e2 } is the standard canonical basis of R2 , is called universal Clifford algebra or geometric algebra of the plane, and is denoted by R2,0 .
(3)
1 (xy − yx), 2 1 (xy + yx), 2
(4) (5)
which are quite convenient coordinate-free definitions. Notice that since x · y = y · x, and x∧y=
1 1 (xy − yx) = − (yx − xy) = −y ∧ x, 2 2
the geometric product (2) is formed by a symmetric (x · y) and an antisymmetric (x ∧ y) part. In geometric algebra, the inverse of a general multivector can be defined (Hestenes & Sobczyk, 1985). In particular, any vector x ∈ R2 , x 6= 0, has the inverse x−1 = 2.1.
x . x2
(6)
Geometric interpretation
Bivectors have an interesting geometric interpretation. Just as a vector describes an oriented line segment, with the direction of the vector represented by the oriented line and the magnitude of the vector is equal to the length of the segment, so a bivector x ∧ y describes an oriented plane segment, with the direction of the bivector represented by the oriented plane and the magnitude of the bivector, measuring the area of the plane segment. The same interpretation is extended to higher-order terms not discussed here: x ∧ y ∧ z represents an oriented volume, and so on. The bivector x ∧ y then defines an oriented parallelogram with sides x and y (Fig. ??) and area given by |x ∧ y| = |α1 β2 − α2 β1 |. Notice also that x ∧ y and y ∧ x have the same magnitude but opposite directions, as illustrated in Fig. ??. The geometric product (2) relates algebraic operations with geometrical properties. In particular, we have the following important geometric results: xy = yx ⇐⇒ x k y ⇐⇒ x ∧ y = 0 ⇐⇒ xy = x · y, xy = −yx ⇐⇒ x ⊥ y ⇐⇒ x · y = 0 ⇐⇒ xy = x ∧ y. That is, parallel vectors commute under the geometric product and perpendicular vectors anticommute.
research papers 2.2. Reflections and rotations
The algebraic properties of geometric algebra provide us with a convenient way of representing reflections and rotations. Suppose a ∈ R2 is a non-zero vector and let Ha be its orthogonal complement, i.e., Ha = {x ∈ R2 | a · x = 0}. A reflection of a with respect to Ha is an orthogonal transformation with the following properties ϕ (a) ϕ (w)
= −a, = w if w ∈ Ha .
(7)
Now we have finally a result that will be extensively used in the next Sections. It asserts that given an arbitrary rotation T we can always find a vector a ∈ R2 such that T can be viewed as a reflection by the basis vector e2 followed by a reflection by the vector a. In other words, the next Proposition describes a method to decompose an arbitrary rotation T into a product of reflections. Proposition 2 Let T : R2 → R2 be a nontrivial rotation. Then, there exists a ∈ R2 , such that
The transformation ϕ is called reflection by the line Ha .
T (x) = ae2 xe2 a−1 = Ta Te2 (x).
Lemma 1 The transformation Ta : R2 → R2 , defined by
Proof. Let T : R2 → R2 be a rotation through an angle θ, then
Ta (x) = −axa−1 ,
(8)
is orthogonal and represents the reflection of the vector x by the line Ha , the orthogonal complement of a. Proof. Since the Clifford algebra is distributive, we can easily see that Ta is linear. Now, it remains to prove that Ta has the properties (7). First, notice that
Now, since a · w = 0 for all w ∈ Ha , then aw = −wa and we get
and this completes the proof.
w ∈ Ha ,
T (e2 )
= − sin θe1 + cos θe2 .
Consider a ∈ R2 , defined by a = T (e1 ) − e1 = (cos θ − 1) e1 + sin θe2 6= 0. According to (6), the inverse of this vector is
Ta (x) = −axa−1 = −a−1 xa = Ta−1 (x), and the following is also true for λ ∈ R, λ 6= 0.
a (cos θ − 1) e1 + sin θe2 . = a2 2 (1 − cos θ)
Now, we define ϕ(x) = ae2 xe2 a−1 , and apply this map to e1 to get ϕ(e1 ) = ae2 e1 e2 a−1
Remark 1 Since the inverse of a reflection is the reflection itself, we can easily prove that
Ta (x) = Tλa (x),
= cos θe1 + sin θe2 ,
a−1 =
Ta (a) = −aaa−1 = −a.
Ta (w) = −awa−1 = − (−wa) a−1 = w,
T (e1 )
(9)
The representation of the reflection of a vector x given by (8) is also valid in Rn . In this case, Ha is the hyperplane orthogonal to a. A rotation is the result of two successive reflections. If the transformation (8) is followed by the transformation Tb then we have Tb Ta (x) = Tb −axa−1 = baxa−1 b−1 . Lemma 2 The transformation T : R2 → R2 , defined by T (x) = RxR−1 , where R = ab, is orthogonal and represents the rotation of the vector x through the angle 2θ, where θ is the angle formed by the vectors a and b.
= −ae1 a−1 ae1 a = − 2 (1 − cos θ) [(cos θ − 1) + sin θe2 e1 ] a = − 2 (1 − cos θ) h i 2 (cos θ − 1) − sin2 θ e1 + 2 (cos θ − 1) sin θe2 . = − 2 (1 − cos θ) Since (cos θ − 1)2 − sin2 θ = 2 cos θ(cos θ − 1), we have that
−2 cos θ + 2 cos2 θ
=
ϕ (e1 ) = T (e1 ) . Now we apply ϕ to e2 ϕ (e2 ) = ae2 e2 e2 a−1 = ae2 a−1 ae2 a = 2 (1 − cos θ) [(cos θ − 1) e1 e2 + sin θ] a = 2 (1 − cos θ) h i 2 (cos θ − 1) sin θe1 + sin2 θ − (cos θ − 1)2 e2 = 2 (1 − cos θ) = − sin θe1 + cos θe2 .
research papers We have proved that ϕ(ei ) = T (ei ) for i = 1, 2, and consequently we conclude that ϕ(x) = T (x).
Proposition 4 If T is a reflection with respect to a hyperplane such that T ∈ OC (Zn ), then there exists a ∈ Zn such that
3. Some mathematical definitions for the CSL problem Here we present a brief summary of basic concepts related to coincidence lattices. For this purpose we adopt definitions and notation given by Baake (1997), who formulated the CSL problem in more mathematical terms.
T (x) = Ta (x) = −axa−1 .
Definition 2 A discrete subset Λ ⊂ Rn is called a lattice, of dimension n, if it is spanned as Λ = Za1 ⊕ Za2 ⊕ · · · ⊕ Zan , where {a1 , a2 , . . . , an } is a set of linearly independent vectors in Rn . These vectors form a basis of the lattice. The lattice Λ is isomorphic to the free Abelian group of order n. It lead us to define the concept of sublattice: Definition 3 Let Λ be a lattice in Rn . A subset Λ0 ⊂ Λ is called a sublattice of Λ if it is a subgroup of finite index, i.e., [Λ : Λ0 ] < ∞ (the number of right lateral classes is finite). It is also said that Λ is a superlattice of Λ0 . The following two definitions are central for the coincidence problem: Definition 4 Two lattices Λ1 and Λ2 are called commensurate, denoted by Λ1 ∼ Λ2 , if and only if Λ1 ∩ Λ2 is a sublattice of both Λ1 and Λ2 . Definition 5 Let Λ be a lattice in Rn . An orthogonal transformation R ∈ O(n) is called a coincidence isometry of Λ if and only if RΛ ∼ Λ. The integer Σ(R) := [Λ : Λ ∩ RΛ] is called the coincidence index of R with respect to Λ. If R is not a coincidence isometry then Σ(R) := ∞. Two useful sets are also defined: OC (Λ) := {R ∈ O(n) | Σ(R) < ∞} , SOC (Λ) := {R ∈ OC (Λ) | det(R) = 1} . 4. CSL from reflections for the case of the square lattice It is well known that any isometry is the product of at most four reflections (see for instance Coxeter, 1973). In that sense, we can consider reflections as the primitive transformations1 and in this case, Clifford algebra is a natural tool for this problem. In Arag´on et al. (2001), we adopted this point of view and developed some preliminary results about CSL of the (hyper-)cubic lattice in arbitrary dimensions. The main results presented in that work, can be summarized as follows. Proposition 3 Let Λ = Zn = Ze1 ⊕ Ze2 ⊕ · · · ⊕ Zen , with {e1 , e2 , . . . , en } the canonical basis of Rn . If a ∈ Zn , then the reflection defined by Ta (x) = −axa−1 ,
is a coincidence reflection, that is Ta ∈ OC (Zn ). 1
We then proved that given an arbitrary coincidence reflection, we can always consider that the reflection is by a vector of the lattice. Thus coincidence reflections are identified with lattice vectors that generate the lattice Λ = Zn . We also conjectured that any orthogonal transformation T ∈ OC (Zn ) can be decomposed as a product of coincidence reflections in vectors of the lattice Zn . In what follows, we will prove that our conjecture is true for planar lattices, and explore some of its consequences. This result is presented in the following Proposition 5 Let T : R2 → R2 be an orthogonal transformation, different from the identity. If T ∈ OC Z2 , then there exists a ∈ Z2 such that T (x) = Ta (x) = −axa−1 T (x) = Ta Te2 (x) = ae2 xe2 a−1
if det(T ) = −1, if det(T ) = 1.
Proof. If T ∈ OC Z2 and det (T ) = −1 then T is a reflection by a line, say, l. Since T is different from the identity then T (ei ) 6= ei for any i = 1, 2. Now consider the vector b = T (e1 ) − e1 , which is orthogonal to the reflection line l and can be used to write the reflection as in Lemma 1. Since T (ei ) has rational components (as it is readily inferred from the fact that an orthogonal matrix with only rational entries is a coincidence isometry), then T (e1 ) − e1 has also rational entries and we can find λ ∈ Z such that λb ∈ Z2 . By defining a = λb, from (9) we get T (x) = −axa−1 . If T ∈ OC Z2 , different from the identity, and det (T ) = 1 then T is a rotation through an angle θ. Following the previous reasoning and using Proposition 2, we have that T (x) = be2 xe2 b−1 = Tb Te2 , where b = T (e1 )−e1 and there exists λ ∈ Z such that λb ∈ Z2 . By defining a = λb, we obtain Tb Te2 = Ta Te2 , T (x) = ae2 xe2 a−1 , that proves the proposition. It is important to remark that given the property (9), we can assume that in Proposition 5 the components of the vector a are relatively prime (they share no common positive divisors except 1). It is a consequence of the fact that if a = α1 e1 + α2 e2 ,
Contrary to crystallographers preference of considering translations, rotations and inversions as primitive transformations.
research papers and we define a0 =
α1 α2 1 e1 + e2 = a, (10) gcd (α1 , α2 ) gcd (α1 , α2 ) gcd (α1 , α2 )
where gcd(α1 , α2 ) stands for the greatest common divisor of α1 and α2 , then Ta = Ta0 . Notice that with all these results, we have characterized OC Z2 by providing a procedure to obtain an arbitrary ele 2 ment of OC Z . On the other hand, since Te2 (Z2 ) = Z2 , we have that Z2 ∩ (Ta Te2 ) Z2 = Z2 ∩ Ta Z2 , a ∈ Z2 , and an immediate consequence is the fact that every vector a ∈ Z2 defines the following two orthogonal transformations Ta (x) = −axa−1 ,
Ta Te2 (x) = Ra = ae2 xe2 a−1 .
Z2 ∩ Ta Z2 = Z 2 ∩ R a Z2 , Σ (Ta ) = Σ (Ra ) , a basis of Z2 ∩ Ra Z2 can be obtained if the basis of Z2 ∩ Ta Z2 is known. 4.1. The coincidence index
Besides Definition 5, the coincidence index has the following interpretation. If Λ0 = Za1 ⊕ Za2 ⊕ · · · ⊕ Zan is a coincidence lattice, which is a sublattice of Λ = Zb1 ⊕ Zb2 ⊕ · · · ⊕ Zbn , then the coincidence index [Λ : Λ0 ] is the ratio of the volume of the unit cell defined by the vectors ai and the volume of the unit cell defined by the vectors bi . In the two-dimensional case that we are considering, the 0 2 coincidence 2 lattice Λ = Za1 ⊕ Za2 is a subset of Z and 0 Z : Λ = |a1 ∧ a2 |. In what follows, we detail the procedure to find the coincidence index We have seen in Proposition 5 that given an arbitrary coincidence rotation R in the plane, we can always find a vector a ∈ Z2 , with relatively prime components, such that the transformation can be written as Ta Te2 . Even more, at the end of the previous section wehave shown that a basis for the coincidence lattice Z2 ∩ Ra Z2 can be obtained from the basis of the lattice Z2 ∩ Ta Z2 . We will then analyze a Ta ∈OC Z2 , where a = α1 e1 + α2 e2 ∈ Z2 and gcd (α1 , α2 ) = 1. By using (6) this transformation can be written as axa . a2
Notice that Ta (a) = −a and it can be verified that for any x ∈ Z2 we have a2 Ta (x) ∈ Z2 . Now consider the vector b ∈ Z2 defined by b = ae1 e2 = (α1 e1 + α2 e2 ) e1 e2 = −α2 e1 + α1 e2 .
Ta (b) = −
aba ba2 = 2 = b. 2 a a
It implies that the vectors c = Ta (a) and d = Ta (b) belong to Z2 ∩ Ta Z2 ; the lattice Λ1 = Zc ⊕ Zd is a sublattice of Z2 ∩ Ta Z2 and of Z2 . Even more: Λ1 < Z2 ∩ Ta Z2 < Z2 .
Now from a basic group theory result we have that 2 Z : Λ1 = Z2 ∩ Ta Z2 : Λ1 Z2 : Z2 ∩ Ta Z2 , and since Z2 : Λ1 = |c ∧ d| = α21 + α22 = a2 , by substituting in the previous equation we conclude that 2 Z : Z2 ∩ Ta Z2 divides a2 .
With this result, and the definition of coincidence index, we can state the following
Also, since
Ta (x) = −axa−1 = −
Since b is orthogonal to a, as can be easily inferred, then ab = −ba and
(11)
Lemma 3 If a = α1 e1 + α2 e2 ∈ Z2 , where gcd (α1 , α2 ) = 1, then Σ (Ta ) and Σ (Ra ) divides a2 . More can be said about the coincidence index. By considering a = α1 e1 + α2 e2 ∈ Z2 , where gcd (α1 , α2 ) = 1, we have that α22 − α21 e1 − 2α1 α2 e2 Ta (e1 ) = α21 + α22 −2α1 = a + e1 , α21 + α22 −2α1 α2 e1 − α22 − α21 α2 e2 Ta (e2 ) = α21 + α22 −2α2 = a + e2 . α21 + α22 If both α1 and α2 are odd numbers then α21 + α22 is even and, consequently a2 Ta (ei ) ∈ Z2 i = 1, 2. 2 It canbe proved that a2 /2 is the least positive integer such that a2 /2 Ta (ei ) ∈ Z2 and therefore a2 /2 divides Σ (Ta). Now let us consider the vectors c0 = a2 /2 Ta (e1 ) and 0 d = a2 /2 Ta (e2 ). The lattice spanned by these vectors is a sublattice of Ta Z2 ∩ Z2 and, by the above reasoning, we can 2 see that Σ (Ta ) divides a2 /2 . All the above results can be summarized as follows: a2 | Σ (Ta ) , 2 Σ (Ta ) | a2 , 2 2 a . Σ (Ta ) | 2
(12) (13) (14)
research papers 1. a2 is odd.
From (12) and (13) we get a2
= Σ (Ta ) q = pq.
2 =
a2 pq, 2
Since p and q are integers, we have two possibilities: 1. p = 2 and q = 1. In this case, we can conclude that a2 = Σ (Ta ), but from (14) we have
a2 2
2 a2
= 4k,
Za ⊕ Zb = Z2 ∩ Ta Z2 .
In this case, there exist vectors c, d ∈ Z2 ∩ Ta Z2 such that d2 = c 2 , dc = −cd.
= Σ (Ta ) 2, = Σ (Ta ) ,
Finally, let us consider that α1 and α2 are relatively prime and one of them is an even number. It can be proved that in this case a2 is the least positive integer such that a2 T (e1 ) ∈ Z2 . It implies that a2 | Σ (Ta ) and, from the above results, a2 = Σ (Ta ). We have now all the ingredients to state the following Proposition 6 If a = α1 e1 + α2 e2 ∈ Z2 and gcd (α1 , α2 ) = 1, then 1. If α1 and α2 are both odd numbers: Σ (Ta ) = Σ (Ra ) =
a2 . 2
2. If either α1 or α2 is even, then Σ (Ta ) = Σ (Ra ) = a2 . 4.2. Basis for the CSL
The following Theorem states that the CSL of Z2 is a square lattice and the basis vectors are obtained during the proof. Theorem 1 Let a = α1 e1 + α2 e2 ∈ Z2 be a vector such that gcd (α1 , α2 ) = 1, then Z2 ∩Ta Z2 and Z2 ∩Ra Z2 are square lattices. Proof. If either α1 or α2 is even, then a2 is odd and 2 Z : Z2 ∩ Ta Z2 = a2 , a2 odd. If both α1 and α2 are odd, then a2 is even and Z2 : Z2 ∩ Ta Z2
From the previous section we know that Ta (a) = −a, Tb (b) = b, and that the lattice spanned by a and b is a sublattice of Z2 ∩ Ta Z2 . Since |a ∧ b| = a2 , as can be easily verified, then
2. a2 is even.
2. p = 1 and q = 2. In this case we obtain a valid condition:
b = −α2 e1 + α1 e2 .
= a2 k, k ∈ Z,
that can not be fulfilled since a2 /4 has a remainder of 2.
a2 a2 2
Consider the vector defined in (11):
=
a2 , a2 even. 2
Both cases are now considered in detail.
It turns out that {c, d} is a square basis of Z2 ∩ Ta Z2 , as we will prove in what follows. Consider the following vectors in Z2 c = d =
1 2 1 2
(a − b) = Ta (c0 ), (a + b) = Ta d0 ,
(15)
where b is given in (11) and 1 c0 = − (a + b) , 2
d0 =
1 (b − a) . 2
The vectors defined in (15) fulfill c + d = a, c2
a2 , 2 a2 = Z2 : Z2 ∩ Ta Z2 , 2
= d2 =
|c ∧ d| =
so we can conclude that {c, d} is a square basis of Z2 ∩ Ta Z2 .
From the previous Theorem we have in summary that if a2 is odd, a basis for the CSL is {a, b}, where b is given by (11). Otherwise, if a2 is even, a basis of the CSL is {c, d}, where c and d are defined in (15). Corollary 1 If T : R2 → R2 is a transformation such that T ∈ OC Z2 , then Z2 ∩ T Z2 is a square lattice.
Proof. From Proposition 5, T can be written either as T (x) = −axa−1 or T (x) = ae2 xe2 a−1 , where a = α1 e1 + α2 e2 ∈ Z2 and gcd (α1 , α2 ) = 1. Thus, Theorem 1 applies.
research papers 4.3. Examples
1. As a simple example consider the problem of determining a basis of the CSL corresponding to Σ = 17. We know (see Ranganathan, 1966) that Σ = 17 = 12 + 42 and tan(θ/2) = 1/4, which gives a rotation angle θ = 28.07240. With respect to the canonical basis, the matrix associated with this orthogonal transformation is T =
15 17 8 17
8 − 17 5 17
.
Thus T (e1 ) − e1 = (−2/17, 8/17) and we can consider a = (−1, 4) ∈ Z2 . By using the geometric product, this transformation is written as Ta (x) = − (−e1 + 4e2 ) e2 xe2 (−e1 + 4e2 )
−1
.
From Proposition 6 we obtain the expected coincidence index Σ (Ta ) = a2 = 17. A basis for CSL is deduced from Theorem 1 (case aa odd). We have b = −4e1 − e2 and therefore a basis for Z2 ∩ Ta Z2 is {−e1 + 4e2 , −4e1 − e2 } . 2. Consider the set of orthogonal transformation in R2 that when applied to Z2 produces coincident lattices with coincidence index equal to 5, that is Σ5 = T ∈ OC Z2 | Σ (T ) = 5 .
With the procedure discussed in this Section, we can easily find the elements of Σ5 and basis for the coincidence lattice in each case. By using the simple facts that 12 + 22 = 5 and 12 + 32 = 10, the orthogonal transformations belonging to Σ5 are tabulated in Table ??. As explained above, a = α1 e1 + α2 e2 and in this case b is a vector orthogonal to a. Notice that the set is composed by 16 transformations: eight reflections and eight rotations. 5. The rectangular lattice Let Γ be a rectangular lattice in R2 , i.e., Γ = Za1 ⊕ Za2 , and a1 · a2 = 0. In this section we study and characterize the group OC(Γ). Here we can also associate coincidence reflections with reflections by vectors in Γ. This is stated in the following Proposition, which is valid for arbitrary lattices in R2 (see Proposition 4). Proposition 7 Let Γ = Za1 ⊕ Za2 be a lattice in R2 . If T ∈ OC (Γ) is a reflection such that T (x) = Tb (x) = −bxb−1 , b ∈ R2 , then we can always find c ∈ Γ such that T (x) = −bxb−1 = −cxc−1 .
Proof. Since T ∈ OC (Γ), for a given a ∈ Γ we can always find m ∈ Z such that mT (a) = T (ma) ∈ Γ. It is equivalent to say that there exists x ∈ Γ such that y = T (x) ∈ Γ, that is y = −bxb−1 ∈ Γ. Thus yb = −bx and, consequently y · b = −b · x, y ∧ b = −b ∧ x = x ∧ b.
The last equation implies that (y−x)∧b = 0 and since x, y ∈ Γ, there exists λ ∈ R such that y − x = λb ∈ Γ. By taking c = λb ∈ Γ, we get T (x) = −bxb−1 = −cxc−1 . The generalization to rectangular basis of Proposition 2, to decompose an arbitrary orthogonal transformation T into a product of reflections, reads as follows. Proposition 8 Let T : R2 → R2 be a nontrivial orthogonal transformation. If a1 , a2 ∈ R2 are vectors such that a1 · a2 = 0, then there exists b ∈ R2 such that T (x) = Tb (x) = −bxb−1 −1 T (x) = Tb Ta2 (x) = ba2 xa−1 2 b
if det (T ) = −1, if det (T ) = 1.
Proof. Assume T : R2 → R2 , different from the identity, and det (T ) = −1. In this case T is a reflection by a line, say, l. Since T is different from the identity and {a1 , a2 } is an orthogonal basis of R2 , then T (ai ) 6= ai for some i = 1, 2. Now consider the vector b = T (a1 ) − a1 , (16)
which is orthogonal to the reflection line l and can be used to write the reflection as in Lemma 1: T (x) = Tb (x) = −bxb−1 . Now assume det (T ) = 1. In this case T is a rotation through an angle θ and also T (ai ) 6= ai for i = 1, 2. Conidering the vector b defined in (16) and since Tb (a1 ) = T (a1 ), we have that Tb (T (a1 )) = Tb (b + a1 ) = −b + Tb (a1 ) = a1 .
Since det (Tb T ) = −1, it readily follows that Tb (T (a2 )) = −a2 and that Ta2 Tb (T (a2 )) = a2 . Therefore T = Tb Ta2 . As in the square lattice, here also any orthogonal transformation T ∈ OC (Γ) can be decomposed as a product of coincidence reflections by vectors of the lattice Γ, as follows from the following Proposition 9 Let Γ = Za1 ⊕ Za2 be a lattice in R2 , where a1 and a2 are orthogonal vectors. If T : R2 → R2 , is a nontrivial coincidence transformation, i.e. T ∈ OC (Γ), then there exists c ∈ Γ such that T (x) = Tc (x) = −cxc−1 −1 T (x) = Tc Ta2 (x) = ca2 xa−1 2 c
if det(T ) = −1, if det(T ) = 1.
Proof. Consider det (T ) = −1. Then T is a reflection and, by Proposition 8, there exists b ∈ R2 such that T (x) = Tb (x) = −bxb−1 . Since T ∈ OC (Γ), T (a1 ) has rational components with respect to the basis {a1 , a2 } and, consequently, b has also rational components. Thus, we can find λ ∈ Z such that λb ∈ Γ. By defining c = λb, and using (9), we have T (x) = −cxc−1 . Now assume det (T ) = 1. In this case T is a rotation through an angle θ and, by Proposition 8, there exists b ∈ R2 such
research papers −1 that T (x) = ba2 xa−1 2 b . By the above arguments, we can find λ ∈ Z such that λb ∈ Γ. By defining c = λb, we have −1 T (x) = ca2 xa−1 2 c . Notice that, contrary to the case of the square lattice, given a vector c ∈ Γ, the reflection Tc (x) = cxc−1 is not necessarily a coincidence reflection, that is, it may occur that Tc ∈/ OC(Γ). In Section 5.2. we provide conditions under which a reflection Tc is a coincidence reflection. As in the case of square lattices, we can consider that in Proposition 9 the components of the vector c, respect to the basis {a1 , a2 }, are relatively prime (see Equation 10). Also since Ta2 (Γ) = Γ, we have that
Γ ∩ (Tc Ta2 ) (Γ) = Γ ∩ Tc (Γ) , a ∈ Γ,
It can be proved by noticing that if y ∈ Γ ∩ Tc (Γ), then there exists x ∈ Γ such that y = Tc (x). From (17) we have y = Tc (αc + βd) = αTc (c) + βTc (d) = −αc + βTc (d), which leads to y + αc = βTc (d). Since m is the least natural number such that mTc (d) ∈ Γ ∩ Tc (Γ), then β = km, k ∈ Z, which proves our claim. The importance of the number m becomes evident if we evaluate the quantity c ∧ mTc (d) = −m(c ∧ cdc−1 ). By using (4) and (18) we have
and, consequently, every vector c ∈ Γ defines the following two orthogonal transformations
−m c ∧ cdc−1
Tc (x) = −cxc−1 , Tc Ta2 (x) = Rc (x) = ca2 xa2 c−1 .
m ccdc−1 − cdc−1 c 2 m = − (dc − cd) 2 = m (c ∧ d) = m (a1 ∧ a2 ) . = −
With this result, the coincidence index Σ(Tc ) turns out to be:
In particular, if Tc (x) ∈ OC (Γ), then Γ ∩ Tc (Γ) = Γ ∩ Rc (Γ) , Σ (Tc ) = Σ (Rc ) ,
Σ(Tc ) =
and a basis of Γ∩Rc (Γ) can be obtained if the basis of Γ∩Tc (Γ) is known.
The value of m is obtained by noticing that the components of the vector Tc (d), with respect to the basis {a1 , a2 }, are rational numbers. Let us assume that
5.1. Coincidence index and basis of the CSL
Let Γ = Za1 ⊕ Za2 be a lattice in R , where a1 · a2 = 0. From the previous results, we know that there exists c = α1 a1 + α2 a2 ∈ Γ, where gcd(α1 , α2 ) = 1, such that Tc ∈ OC(Γ). Given that α1 and α2 are relatively prime, there exist integers β1 and β2 such that α1 β1 + α2 β2 = 1.
Tc (d) =
2
Γ = Zc ⊕ Zd.
(17)
To see this, define d = β2 a1 − β1 a2 , and the previous statement follows from c ∧ d = (α1 a1 + α2 a2 ) ∧ (β2 a1 − β1 a2 ) = (α1 β1 + α2 β2 ) a1 ∧ a2 = a1 ∧ a2 .
Now, since Tc (c) = −c then Tc (c) ∈ Γ ∩ Tc (Γ) . By considering the least natural number m such that mTc (d) ∈ Γ ∩ Tc (Γ) , we claim that {c, mTc (d)} is a basis of Γ ∩ Tc (Γ).
then
p1 p2 a1 + a2 , q1 q2
Σ(Tc ) = m = lcm (q1 , q2 ) ,
where lcm stands for the least common multiple function. 5.2.
It guarantees the existence of a vector d ∈ Γ such that
m |a1 ∧ a2 | |c ∧ mTc (d)| = = m. |a1 ∧ a2 | |a1 ∧ a2 |
Characterization of OC(Γ)
We mentioned above that, in the case under study, it may occur that given a vector c ∈ Γ, Tc ∈/ OC(Γ). Here we provide conditions under which a reflection Tc is a coincidence reflection. As a general result, first notice that for any nonzero vectors x and v, we have that −vxv−1 = −2
(18)
x·v v + x. v2
which readily follows from vxv = v (x · v + x ∧ v)
= (x · v) v + v (x ∧ v) 1 = (x · v) v + v (xv − vx) 2 1 1 = (x · v) v + vxv − v2 x. 2 2
Now let Γ = Za1 ⊕ Za2 be a lattice in R2 (not necessarily rectangular). If c ∈ Γ, then Tc (x) = −cxc−1 ∈ OC(Γ) provided
research papers that there exist mi ∈ Z such that mi Tc (ai ) = mi (−cai c−1 ) ∈ Γ, for each i = 1, 2. That is 2 (c · ai ) mi −cai c−1 = −mi c + mi ai ∈ Γ, i = 1, 2. c2
i) It is then enough that −mi 2(c·a c2 c ∈ Γ or, equivalently
(c · ai ) ∈ Q, i = 1, 2. c2
(19)
Consider the rectangular lattice Γ = Za1 ⊕ Za2 , a1 · a2 = 0. Without lost of generality, suppose that a21 = 1 and a22 = λ. If c = α1 a1 + α2 a2 ∈ Γ, where gcd (α1 , α2 ) = 1, then c · a1
c · a2 c2
α1 + λα22 λα2 2 α1 + λα22
= =
α2 λ (α2 r − s)
Thus c, defined in Propo-
5 T (a1 ) − a1 = −8a1 + 9a2 , 4
which belongs to Γ. We now follow the procedure described in Section 5.1. to obtain the coincidence index and a basis of the CSL. Since −8(1) + 9(1) = 1, we define d = a1 − a2 to get Γ = Zc ⊕ Zd, T (x) = ca2 xa2 c−1 . Then after some calculation we have that 23 29 a1 − a2 , 25 25
and the least natural number m such that mTc (d) ∈ Γ ∩ Tc (Γ) is m = 25. With this results we obtain the coincidence index Σ(Tc ) = m = 25, and a basis of the CSL, Γ ∩ T (Γ): {c, 25Tc(d)} = {−8a1 + 9a2 , 23a1 − 29a2 } .
p , q r , s
6. The rhombic lattice
where p, q, r, s ∈ Z and gcd(p, q) = gcd(r, s) = 1. From the above expressions we obtain: pα22 λ
36 25 a2 .
Tc (d) = −cdc−1 =
= λα2 , = α21 + λα22 .
α21
= α1 (q − α2 p),
= rα21 .
Since λ ∈/ Q then necessarily α1 = 0 or α2 = 0, that is, c = a or c = b. Consequently, if λ = a22 ∈/ Q, then OC (Γ) = {I, Ta1 , Ta2 , Ta2 Ta1 }. The above results can be summarized in the following Theorem, which characterizes OC(Γ) by providing a way to generate it from vectors of Γ: Theorem 2 Let Γ = Za1 ⊕ Za2 be a lattice in R , where a1 · a2 = 0 and a21 = 1. Let c = α1 a1 + α2 a2 be a vector in Γ, where gcd (α1 , α2 ) = 1, and Tc (x) = −cxc−1 be a reflection by the vector c. We have the following possibilities 2
1. If
c=
= α1
Consequently, Tc ∈ OC(Γ) provided that λ ∈ Q. Now suppose λ ∈/ Q. It follows that
a22
is b = T (a1 ) − a1 = − 35 2 a1 + sition 9 can be
∈ Q then Tc ∈ OC(Γ).
2. If a22 ∈/ Q then OC (Γ) = {I, Ta1 , Ta2 , Ta2 Ta1 }. 5.3. Example
Let Γ be a rectangular lattice spanned by the vectors a1 = 3 4 8 2 e 5 1 + 5 e2 and a2 = − 15 e1 + 5 e2 . Now consider the orthogonal transformation T with matrix representation, with respect to the canonical basis, given by 24 7 − 25 − 25 , T = 24 7 − 25 25 which corresponds to a rotation through an angle θ = 106.26020, that belongs to OC (Γ). The vector b, defined in (16)
Let Γ be a rhombic lattice in R2 , i.e., Γ = Za1 ⊕ Za2 , where a1 and a2 are two linearly independent vectors such that a21 = a22 . In this section we study and characterize the group OC(Γ). First, notice that Proposition 7 is valid for arbitrary lattices R2 , so it remains to show that an arbitrary orthogonal transformation T can be decomposed into a product of reflections by using the basis vectors of a rhombic lattice. Proposition 10 Let T : R2 → R2 be a nontrivial orthogonal transformation. If a1 and a2 are linearly independent vectors such that a21 = a22 , then there exists b ∈ R2 such that T (x) = = T (x) = =
Tb (x) −bxb−1 Tb Ta1 −a2 (x) −1 b (a1 − a2 ) x (a1 − a2 ) b−1
if det (T ) = −1, if det (T ) = 1.
Proof. Let us first prove that a1 − a2 and a1 + a2 are orthogonal: 2 (a1 − a2 ) · (a1 + a2 ) = (a1 − a2 ) (a1 + a2 ) + (a2 + a1 ) (a1 − a2 ) =
a21 − a2 a1 + a1 a2 − a22
+ a2 a1 + a21 − a1 a2 − a22
= 0.
Therefore, {a1 + a2 , a1 − a2 } is an orthogonal basis of R2 . Let us define d1 = a1 + a2 , (20) d2 = a1 − a2 .
Assume T : R2 → R2 , different from the identity, and det (T ) = −1. In this case T is a reflection by a line, say, l.
research papers Since T is different from the identity and {d1 , d2 } is an orthogonal basis of R2 , then T (di ) 6= di for some i = 1, 2. Now consider the vector b = T (d1 ) − d1 , (21)
which is orthogonal to the reflection line l and can be used to write the reflection as in Lemma 1: T (x) = Tb (x) = −bxb−1 . Now assume det (T ) = 1. In this case T is a rotation through an angle θ and also T (di ) 6= di for some i = 1, 2. By considering the vector b defined in (21), we have that Tb (T (d1 )) = Tb (b + d1 ) = −b + T (d1 ) = d1 . Since det (Tb T ) = −1, it readily follows that Tb (T (d2 )) = −d2 and that Td2 Tb (T (d2 )) = d2 . Therefore T = Tb Td2 . Now we will see that any orthogonal transformation T ∈ OC (Γ) can be decomposed as a product of coincidence reflections in vectors of the lattice Γ. Proposition 11 Let Γ = Za1 ⊕ Za2 be a lattice in R2 , where a21 = a22 . If T : R2 → R2 , is a nontrivial coincidence transformation, i.e. T ∈ OC (Γ), then there exists c ∈ Γ such that T (x) T (x)
= Tc (x) = −cxc −1 = Tc Td2 (x) = −cd2 xd−1 2 c −1
if det(T ) = −1, if det(T ) = 1,
where d2 = a1 − a2 . Proof. Consider det (T ) = −1. Then T is a reflection and, by Proposition 10, there exists b ∈ R2 such that T (x) = Tb (x) = −bxb−1 . Since T ∈ OC (Γ), T (d1 ) has rational components with respect to the basis {d1 , d2 } and, consequently, b has also rational components. Thus, we can find λ ∈ Z such that λb ∈ Γ. By defining c = λb, and using (9), we have T (x) = −cxc−1 . Now assume det (T ) = 1. In this case T is a rotation through an angle θ and, by Proposition 10, there exists b ∈ R2 such −1 that T (x) = bd2 xd−1 2 b . By the above arguments, we can find λ ∈ Z such that λb ∈ Γ. By defining c = λb, we have −1 T (x) = cd2 xd−1 2 c . As in the case of the rectangular lattice, given a vector c ∈ Γ, it may occur that Tc ∈/ OC(Γ). In Section 6.2. we provide conditions under which a reflection Tc is a coincidence reflection. By following the same ideas applied to the case of the rectangular lattice, we have that every vector c ∈ Γ defines the following two orthogonal transformations Tc (x) = −cxc−1 ,
Tc Td2 (x) = Rc (x) = cd2 xd2 c−1 . In particular, if Tc (x) ∈ OC (Γ), since Td2 (Γ) = Γ then Γ ∩ Tc (Γ) = Γ ∩ Rc (Γ) , Σ (Tc ) = Σ (Rc ) , and a basis of Γ∩Rc (Γ) can be obtained if the basis of Γ∩Tc (Γ) is known.
6.1.
Coincidence index and basis of the CSL
The procedure to determine the coincidence index Σ(Tc ) and a basis of the CSL, Γ ∩ Tc (Γ), follows the same lines as in the case of the rectangular lattice. For that reason, here we only summarize the results without further details. Let Γ = Za1 ⊕ Za2 be a lattice in R2 , such that a21 = a22 . From the previous results, we know that there exists c = α1 a1 + α2 a2 ∈ Γ, where gcd(α1 , α2 ) = 1, such that Tc ∈ OC(Γ). Given that α1 and α2 are relatively prime, there exist integers β1 and β2 such that α1 β1 + α2 β2 = 1. It guarantees the existence of a vector d ∈ Γ such that Γ = Zc ⊕ Zd, where
d = β2 a1 − β1 a2 .
A basis of Γ ∩ Tc (Γ) is {c, mTc (d)}, where m is the least natural number such that mTc (d) ∈ Γ ∩ Tc (Γ) . The natural number m is also the coincidence index Σ(Tc ), and its value can be obtained by assuming that Tc (d) = then 6.2.
p1 p2 a1 + a2 , q1 q2
Σ(Tc ) = m = lcm (q1 , q2 ) . Characterization of OC(Γ)
In this section the conditions under which a reflection Tc is a coincidence reflection are deduced. We start from the condition (19) which is valid for arbitrary lattices. Let Γ = Za1 ⊕ Za2 , a21 = a22 , be a rhombic lattice. If c ∈ Γ, our purpose is to find conditions under which Tc (x) = −cxc−1 ∈ Γ. Without lost of generality, suppose that a21 = a22 = 1. If c = α1 a1 + α2 a2 ∈ Γ, where gcd (α1 , α2 ) = 1, then from (19) we have c · a1 c · a2
c2
= α1 + α2 a1 · a2 , = α2 + α1 a1 · a2 ,
= α21 + α22 + 2α1 α2 a1 · a2 .
Consequently, Tc ∈ OC(Γ) provided that a1 · a2 ∈ Q. Now suppose a1 · a2 ∈/ Q. It follows that α1 + α2 a1 · a2 + α22 + 2α1 α2 a1 · a2 α2 + α1 a1 · a2 α21 + α22 + 2α1 α2 a1 · a2 α21
= =
p , q r , s
where p, q, r, s ∈ Z and gcd(p, q) = gcd(r, s) = 1. From the above expressions we obtain: α2 (q − 2α1 p) a1 · a2 = p α21 + α22 − qα1 , α1 (s − 2α2 r) a1 · a2 = r α21 + α22 − sα2 .
research papers Since a1 ·a2 ∈/ Q then necessarily the coefficients of a1 ·a2 must vanish, that is α2 (q − 2α1 p) = α1 (s − 2α2 r) = p α21 + α22 − qα1 = r α21 + α22 − sα2 =
0, 0, 0, 0,
(22)
for some αi 6= 0, i = 1, 2. Suppose that α1 6= 0. In this case the following equation must be fulfilled: r 1 = , s 2α2 thus r = 1, s = 2α2 and (22) becomes r α21 + α22 − sα2 = 0, α21 + α22 − 2α22 = 0, α21 − α22 = 0, (α1 + α2 ) (α1 − α2 ) = 0.
These equations lead to c = a1 + a2 or c = a1 − a2 . The same conclusion is obtained if α2 6= 0. Consequently, if a1 · a2 ∈/ Q, then OC (Γ) = {I, Td1 , Td2 , Td2 Td1 }, where d1 and d2 are given by (20). The above results can be summarized in the following Theorem which characterizes OC(Γ) by providing a way to generate it from vectors of Γ: Theorem 3 Let Γ = Za1 ⊕ Za2 be a lattice in R2 , where a21 = a22 . Let c = α1 a1 + α2 a2 be a vector in Γ, where gcd (α1 , α2 ) = 1, and Tc (x) = −cxc−1 be a reflection by the vector c. We have the following possibilities 1. If a1 · a2 ∈ Q then Tc ∈ OC(Γ).
2. If a1 · a2 ∈/ Q then OC (Γ) = {I, Td1 , Td2 , Td2 Td1 }, where d1 = a1 + a2 and d2 = a1 − a2 .
6.3. Example
Consider the hexagonal Γ = Za1 ⊕ Za2 , where a1 = √ lattice (1, 0) and a2 = 1/2, 3/2 . The diagonals (20) are then given √ √ by d1 = 3/2, 3/2 and d2 = 1/2, − 3/2 . Now let T : R2 → R2 be an orthogonal transformation with matrix representation, with respect to the canonical basis, given by √ ! − 3143 −√13 14 . 3 3 13 − 14 14 With respect to the basis {a1 , a2 } the matrix is 8 − 7 − 73 . 3 − 75 7
Therefore (21) yields b = T (d1 ) − d1 = −(45/14)e1 − √ (9 3/14)e2 = −(18/7)a1 − (9/7)a2 , and the vector c (=λb) can be c = 2a1 + a2 .
Since the determinant of the matrix representation is 1, the transformation is a rotation given by T (x) = Tc Td2 (x). Now, since α1 = 2 and α2 = 1 then Γ is spanned by {c, d}, where d = a1 + a2 . As described at the end of Section 6.1, a basis of Γ ∩ Ta (Γ) is {c, mTc (d)} and the value of m is obtained from Tc (d) = −cdc−1 . By substituting the values of c and d, after some length but simple calculation we get: Tc (d) = −
11 2 a1 − a2 . 7 7
Thus m = Σ(T ) = 7 and a basis of the coincidence lattice is {2a1 + a2 , −11a1 − 2a2 }. 7. Conclusions In this work we solve the coincidence problem for planar lattices by using Clifford algebra. It allows us to algebraically characterize the group of coincidence isometries in terms of the lattice vectors (objects on which the group operates), and to provide explicit expressions for both, the coincidence index and the basis of the coincidence lattices. Clifford algebra has proved to be a useful language in many areas of physics, engineering and computer science and we hope that its power for expressing geometric ideas, used here to solve the coincidence problem in planar lattices, would motivate its use in other fields of geometric crystallography. The present approach can be extended to higher dimensions an this work is under way. The authors wish to thank Prof. Gerardo Arag´on for fruitful discussions and suggestions. This work has been supported by DGAPA-UNAM and CONACyT through grants IN-108502-3 and D40615-F, respectively. One of us (M.A.R.) wishes to thank support from COFAA-IPN. References Arag´on, G., Arag´on, J.L., D´avila, F., G´omez, A. & Rodr´ıguez, M.A. (2001). In Geometric Algebra with Applications in Science and Engineering, edited by E. Bayro-Corrochano & G. Sobczyk, pp. 371386. Boston: Birkha¨user Arag´on, J.L., Romeu, D., Beltr´an, L. & G´omez, A. (1997). Acta Crystallogr. A 53, 772-780. Baake, M. (1997). In The Mathematics of Aperiodic Order, edited by R.V. Moody, pp. 9-44. Dordretch: Kluwer Academic Publishers. Bayro-Corrochano, E. & Sobczyk, G. (2001). Editors. Geometric Algebra with Applications in Science and Engineering. Boston: Birkha¨user. Coxeter, H.S.M. (1973). Regular Polytopes. New York: Dover. Duneau, M., Oguey, C. & Tahal, A. (1992). Acta Crystallogr. A 48, 772-781. Fortes, M.A. (1983). Acta Crystallogr. A 39, 351-357. Hestenes, D. (2002). In Applications of Geometric Algebra with Applications in Computer Science and Engineering, edited by L. Doerst, C. Doran & J. Lasenby, p. 3-34. Boston: Birkha¨user. Hestenes, D. & Sobczyk, G. (1985). Clifford Algebra to Geometric Calculus. A unified Language for Mathematics and Physics. Holland: D. Reidel Publishing Co. Ranganathan, S. (1966). Acta Crystallogr. A 21, 179-199. Sutton, A.P. & Baluffi, R.W. (1995). Interfaces in crystalline materials. Oxford: Clarendon Press.
