Modified strip projection method

Modified strip projection method arXiv:math-ph/0605046v4 31 Oct 2006 Nicolae Cotfas Faculty of Physics, University of Bucharest, PO Box 76-54, Post Office 76, Bucharest, Romania, E-mail address: [email protected] Abstract. The diffraction image of a quasicrystal admits a finite group G as a symmetry group, and the quasicrystal can be regarded as a quasiperiodic packing of copies of a G-cluster C, joined by glue atoms. The physical space E containing C can be embedded into a higher-dimensional space Rk such that, up to an inflation factor, C is the orthogonal projection of the set {(±1, 0, ..., 0), (0, ±1, 0, ..., 0), ... (0, ..., 0, ±1)}. The projections of the points of Zk lying in the strip S = E +[−1/2, 1/2]k+t obtained by shifting a hypercube [−1/2, 1/2]k+t along E is a quasiperiodic packing of partially occupied copies of C, but unfortunately, the occupation of clusters is very low. In our modified strip projection method we firstly determine for each point x ∈ Zk ∩ S the number n(x) of all the arithmetic neighbours of x lying in the strip S, and project the points of Zk ∩ S on E in the decreasing order of the occupation number n(x). In the case when n(x) represents more than p% of all the points of the cluster C we project all the arithmetic neighbours of x (lying inside or outside S). We choose p such that to avoid the superposition of the fully occupied clusters. The projection of a point x with n(x) less than p% of all the points of the cluster C is added to the pattern only if it is not too close to the already obtained points. Modified strip projection method 2 1. Introduction Quasicrystals are materials with perfect long-range order, but with no three-dimensional translational periodicity. The discovery of these solids in the early 1980’s and the challenge to describe their structure led to a great interest in quasiperiodic sets of points [11, 12]. The diffraction image of a quasicrystal contains a set of sharp Bragg peaks invariant under a finite non-crystallographic group of symmetries G, called the symmetry group of quasicrystal (in reciprocal space). In the case of quasicrystals with no translational periodicity this group is the icosahedral group Y and in the case of quasicrystals periodic along one direction (two-dimensional quasicrystals) G is one of the cyclic groups C8 (octagonal quasicrystals), C10 (decagonal quasicrystals) and C12 (dodecagonal quasicrystals). Real structure information obtained by high resolution transmission electron microscopy suggests us that a quasicrystal with symmetry group G can be regarded as a quasiperiodic packing of copies of a well-defined G-invariant cluster C, joined by glue atoms [7]. From a mathematical point of view, a G-cluster is a finite union of orbits of G, in a fixed linear representation of G. A mathematical algorithm for generating quasiperiodic packings of interpenetrating copies of G-clusters was obtained by author in collaboration with Verger-Gaugry several years ago [1]. This algorithm based on strip projection method [6, 8, 9, 10] works for any finite group G and any G-cluster, but in the case of a multi-shell cluster the dimension of the involved superspace is rather high, and the occupation of the clusters occurring in the generated pattern is too low. Some mathematical results recently obtained by author [3, 5] simplify the computer program and allow to use strip projection method in the superspaces required by this approach. Now, our aim is to present a way to increase the occupation of clusters occurring in the generated quasiperiodic set. 2. Two-dimensional packings of clusters Let G be one of the cyclic groups C8 , C10 , C12 . Each group Cn can be defined as Cn = h a | an = e i = { e, a, a2 , ..., an−1 } (1) and the formula   2π 2π 2π 2π − β sin , α sin + β cos a(α, β) = α cos n n n n (2) Cn (α, β) = {(α, β), a(α, β), a2 (α, β), ..., an−1 (α, β)} (3) define an R-irreducible representation in R2 . The orbit generated by (α, β) 6= (0, 0) contains n points (vertices of a regular polygon with n sides). Let C2 = {v1 , v2 , ..., vk , −v1 , −v2 , ..., −vk } (4) 3 Modified strip projection method where v1 = (v11 , v21 ), v2 = (v12 , v22 ),..., vk = (v1k , v2k ), be a fixed G-cluster, that is, a finite union of orbits of G. From the general theory [1] (a direct verification is also possible) it follows that the vectors w1 = (v11 , v12 , ..., v1k ) and w2 = (v21 , v22 , ..., v2k ) (5) from Rk are orthogonal and have the same norm hw1 , w2 i = v11 v21 + v12 v22 + ... + v1k v2k = 0 p 2 2 2 = ||w2||. + v12 + ... + v1k ||w1 || = v11 (6) We identify the physical space with the two-dimensional subspace E2 = { αw1 + βw2 | α, β ∈ R } (7) of the superspace Rk and denote by E⊥ 2 the orthogonal complement k E⊥ 2 = { x ∈ R | hx, yi = 0 for all y ∈ E2 }. The orthogonal projection on E2 of a vector x ∈ Rk is the vector D w Ew D w Ew 1 1 2 2 π2 x = x, + x, κ κ κ κ where κ = ||w1 || = ||w2 ||, and the orthogonal projector corresponding to E⊥ 2 is π2⊥ : Rk −→ E⊥ 2 π2⊥ x = x − π2 x. (8) (9) (10) We describe E2 by using the orthogonal basis {κ−2 w1 , κ−2 w2 }. Therefore, in view of (9) the expression in coordinates of π2 is π2 : Rk −→ R2 π2 x = (hx, w1 i, hx, w2i). The projection W2,k = π2⊥ (Λk ) of the unit hypercube   1 1 for all i ∈ {1, 2, ..., k} Λk = (x1 , x2 , ..., xk ) − ≤ xi ≤ 2 2 (11) (12) is a polyhedron (called the window of selection) in the (k−2)-dimensional subspace E⊥ 2, and each (k−3)-dimensional face of W2,k is the projection of a (k−3)-dimensional face of Λk . The vectors e1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), ..., ek = (0, 0, ..., 0, 1) from Rk form the canonical basis of Rk , and each (k−3)-face of Λk is parallel to (k−3) of these vectors and orthogonal to three of them. There exist eight (k−3)-faces of Λk orthogonal to the distinct vectors ei1 , ei2 , ei3 , and the set ) ( xi ∈ {−1/2, 1/2} if i ∈ {i1 , i2 , i3 } (13) x = (x1 , x2 , ..., xk ) xi = 0 if i 6∈ {i1 , i2 , i3 } contains one and only one point from each of them. There are ! k(k − 1)(k − 2) k = 6 3 (14) sets of 23 parallel (k−3)-faces of Λk , and we label them by using the elements of the set I2,k = {(i1 , i2 , i3 ) ∈ Z3 | 1 ≤ i1 ≤ k−2, i1 + 1 ≤ i2 ≤ k−1, i2 + 1 ≤ i3 ≤ k }. (15) 4 Modified strip projection method In R3 the cross-product of two vectors v = (vx , vy , vz ) and w = (wx , wy , wz ) is a vector orthogonal to v and w, and can be obtained by expanding the formal determinant v×w = i j k vx vy vz wx wy wz (16) where {i, j, k} is the canonical basis of R3 . For any vector u = (ux , uy , uz ), the scalar product of u and v × w is u(v × w) = ux uy uz vx vy vz . wx wy wz (17) In a very similar way, a vector y orthogonal to l = k−1 vectors ui = (ui1 , ui2, ui3 , ..., uik ) i ∈ {1, 2, 3, ..., l} (18) from Rk can be obtained by expanding the formal determinant y= e1 e2 e3 u11 u12 u13 u21 u22 u23 ... ... ... ul1 ul2 ul3 ... ek ... u1k ... u2k ... ... ... ulk (19) containing the vectors of the canonical basis in the first row. (x1 , x2 , ..., xk ) ∈ Rk , the scalar product of x and y is hx, yi = x1 x2 x3 u11 u12 u13 u21 u22 u23 ... ... ... ul1 ul2 ul3 ... xk ... u1k ... u2k ... ... ... ulk . For any x = (20) For example, y= e1 e2 e3 e4 e5 e6 ... ek 0 0 0 1 0 0 ... 0 0 0 0 0 1 0 ... 0 0 0 0 0 0 1 ... 0 ... ... ... ... ... ... ... ... 0 0 0 0 0 0 ... 1 v11 v12 v13 v14 v15 v16 ... v1k v21 v22 v23 v24 v25 v26 ... v2k = (−1) k−1 e1 e2 e3 v11 v12 v13 v21 v22 v23 (21) is a vector orthogonal to the vectors e4 , e5 , ..., ek , w1 , w2 , and hx, yi = (−1) k−1 x1 x2 x3 v11 v12 v13 v21 v22 v23 (22) 5 Modified strip projection method ⊥ ✏✏ E1 ✏✏ ✏✏ ❈ ✄ ❈ ✄ ❈ ✄ ✄ ❈ W1,3 ✄ ❈ ✄ ❈✏✏ ✏✏ ✏ ✏ ✄ ❈ ✄ ✄ ❈ ✏✏ ❈ ✄ ❈✏✏ ❈ S1,3 ✄ ❈ ✄ E1 ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ C2 ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣Ω Figure 1. Left: The strip S1,3 and the window W1,3 in the case of a 1D physical space E1 embedded into a three-dimensional superspace. Centre: A one-shell C8 -cluster C2 . Right: A fragment of a set Ω defined by C2 . The nearest neighbours of any point q of Ω belong to q + C2 , which is a copy of C2 with the center at point q. ⊥ for any x ∈ Rk . The vector y belongs to E⊥ 2 , and since ei − π2 ei is a linear combination of w1 and w2 , it is also orthogonal to π2⊥ e4 , π2⊥ e5 , ..., π2⊥ ek . Therefore, y is orthogonal to the (k−3)-faces of W2,k labelled by (1, 2, 3). Similar results can be obtained for any (i1 , i2 , i3 ) ∈ I2,k . Consider the strip corresponding to W2,k (see figure 1) S2,k = {x ∈ Rk | π2⊥ x ∈ W2,k } (23) and define for each (i1 , i2 , i3 ) ∈ I2,k the number d i1 i2 i3 = max αj ∈{−1/2, 1/2} α1 α2 α3 v1i1 v1i2 v1i3 . v2i1 v2i2 v2i3 (24) A point x ∈ Rk belongs to the strip S2,k if and only if − d i1 i2 i3 ≤ xi1 xi2 xi3 v1i1 v1i2 v1i3 v2i1 v2i2 v2i3 ≤ d i1 i2 i3 for any (i1 , i2 , i3 ) ∈ I2,k . (25) The set defined in terms of the strip projection method [6, 8, 9, 10] Ω = π2 (S2,k ∩ Zk ) = { π2 x | x ∈ S2,k ∩ Zk } (26) can be regarded as a packing of translated partially occupied copies of C2 . Since π2 ei = (hei , w1 i, hei , w2 i) = (v1i , v2i ) = vi (27) we get π2 ({ x ± e1 , x ± e2 , ..., x ± ek } ∩ S2,k ) ⊆ {π2 x ± v1 , π2 x ± v2 ..., π2 x ± vk } = π2 x + C2 (28) that is, the neighbours of any point π2 x ∈ Ω belong to the translated copy π2 x+C2 of C2 . A larger class of aperiodic pattens can be obtained by translating the strip S2,k . For each t ∈ Rk the set Ω = π2 ((t + S2,k ) ∩ Zk ) = { π2 x | x−t ∈ S2,k and x ∈ Zk } (29) 6 Modified strip projection method ♣ ♣♣♣♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣ ♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣ ♣♣♣ ♣♣♣♣♣ ♣ ♣ ♣♣♣ ♣♣♣♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣♣♣♣♣ ♣ ♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣ ♣ ♣♣♣♣ ♣♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣ ♣♣ ♣♣♣♣ ♣ ♣♣ ♣♣♣♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣♣ ♣ ♣♣ ♣♣♣♣ ♣ ♣♣ ♣♣♣ ♣♣♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣ ♣♣♣ ♣♣♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣ ♣♣♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣♣♣ ♣♣♣ ♣♣♣♣ ♣♣ ♣♣♣♣ ♣♣♣♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣♣♣ ♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣♣♣ ♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣ ♣ ♣♣♣ ♣♣♣ ♣ ♣ ♣♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣♣ ♣♣ ♣ ♣♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ Figure 2. Left: A one-shell C10 -cluster and a fragment of the corresponding quasiperiodic set. Right: A fragment of the quasiperiodic set defined by a two-shell C10 -cluster, obtained by using strip projection method in a ten-dimensional superspace. The starting cluster is a covering cluster, but for most of the points the occupation is extremely low. is a packings of partially occupied copies of C2 . Some particular examples can be seen in figures 1-3. 3. Modified strip projection method The algorithm based on the strip projection method presented in the previous section is very efficient. Hundreds of points of Ω can be obtained in only a few minutes for rather complicated G-clusters C2 , but, as one can remark in figures 1-3, for most of the points of Ω the occupation of the corresponding cluster is very low. On the other hand, the images concerning the quasicrystal structure obtained by high resolution transmission electron microscopy show the presence of a significant percentage of fully occupied clusters. The number n(x) of the neighbours of a point π2 x ∈ Ω occurring in Ω corresponds to the number of the points of the set {x ± e1 , x ± e2 , ..., x ± ek } belonging to the strip t + S2,k . In our modified strip projection method we firstly determine n(x) for all the points of Zk lying in the fragment of the strip we intend to project, and then we project the points in the decreasing order of the occupation number n(x). In the case when n(x) represents more than p% of all the points of the cluster C2 we project all the arithmetic neighbours of x (lying inside or outside the strip t + S2,k ). We choose p such that to avoid the superposition of the fully occupied clusters. The projection of a point x with n(x) less than p% of all the points of the cluster C2 is added to the pattern only if it is not too close to the already obtained points. We get in this way a discrete set Ω̃ containing fully occupied copies of the cluster C2 . In the structure analysis of quasicrystals, the experimental diffraction image is compared with the diffraction image of the mathematical model, regarded as a set of scatterers. In order to compute the diffraction image of a discrete set Ω one has to use P the Fourier transform and to identify Ω with the Dirac comb ω∈Ω δω , which will also be denoted by Ω . The set Ω defined in terms of the strip projection method is an infinite 7 Modified strip projection method ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ❛♣ ♣♣♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣♣ ❛♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ❛♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣♣ ❛♣ ♣ ♣♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ❛♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣♣ ♣ ❛♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ❛ ♣ ♣ ♣ ♣ ♣ ♣ ❛ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ❛ ♣ ♣ ♣ ♣ ♣ ♣ ❛ ♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣♣ ♣ ❛♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❛ ♣ ♣ ♣ ♣ ❛♣ ♣ ♣ ♣ ❛♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣♣ ❛♣ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ❛♣ ♣♣ ♣♣♣ ♣ ♣ ♣ ♣♣♣ ♣♣ ❛♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣❛ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣ ❛♣ ♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ❛♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❛♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ❛ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ❛ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ Ω0 ♣ ♣ ♣♣♣♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ For diffraction pattern please see [5] C2 Figure 3. Left: A set Ω0 containing 923 points defined by starting from a C12 -cluster C2 . Centre: The cluster C2 . Right: For the diffraction pattern Ω∗0 of Ω0 please see [5]. ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣♣ ♣ ♣ ♣ ♣♣♣ ♣♣♣ ♣ ♣♣ ♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ ♣ ♣ ♣♣♣ ♣♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣♣ ♣ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣♣♣ ♣ ♣♣♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣♣ ♣ ♣♣ ♣♣ ♣ ♣ ♣ Ω̃0 ♣ ♣ ♣♣♣♣♣ ♣ ♣ ♣♣♣ ♣♣ ♣♣♣ ♣♣ ♣♣♣ For diffraction pattern please see [5] C2 Figure 4. Left: The set Ω̃0 containing 1019 points corresponding to the set Ω0 from the previous figure, defined by using the modified strip projection method. Centre: The cluster C2 . Right: For the diffraction pattern Ω˜∗0 of Ω˜0 please see [5]. set, but the set Ω̃0 we can effectively generate is evidently a finite set, obtained by starting from a finite fragment Ω0 of Ω. This is not very bad since any quasicrystal has a finite number of atoms. Nevertheless, a fragment of Ω or Ω̃ can not be an acceptable model for a quasicrystal unless it is large enough. The diffraction pattern corresponding to Ω0 is related to the function 2 Ω∗0 : R2 −→ [0, ∞) 2 Ω∗0 (ξ) = |F [Ω0 ](ξ)| = X eihω,ξi (30) ω∈Ω0 P where F [Ω0 ] means the Fourier transform of the distribution Ω0 = ω∈Ω0 δω . In figure 3 we present a fragment Ω0 of the set Ω corresponding to a C12 -cluster and the set  1 Ω∗0 (0) in order to illustrate the shape and symmetry properties ξ ∈ R2 | Ω∗0 (ξ) > 1000 of the diffraction image of Ω0 . The case of the set Ω̃0 corresponding to Ω0 , obtained by using the modified strip projection method in R6 , is presented in figure 4. It is an open problem if the diffraction properties of the sets obtained by using the modified strip projection method are similar to those of sets obtained by the non- 8 Modified strip projection method modified version. A mathematical answer seems to be difficult, but some suggestions in this direction can be obtained by analysing larger fragments. By using our computer program the fragment containing 923 points presented in figure 3 and its diffraction pattern are obtained in one minute. In two hours one can obtain about 16000 points. The fragment generated by the modified method presented in figure 4 can be obtained in two minutes. 4. Quasiperiodic packings of icosahedral clusters The icosahedral group Y = 235 can be defined in terms of generators and relations as Y = ha, b | a5 = b2 = (ab)3 = ei and the rotations a, b : R3 −→ R3 a(α, β, γ) = τ −1 α 2 − τ2 β + 12 γ, τ α 2 + 21 β + τ −1 γ, 2 (31) − 21 α + τ −1 β 2 + τ2 γ
(32) b(α, β, γ) = (−α, −β, γ). √ where τ = (1 + 5)/2, generate an irreducible representation of Y in R3 . In the case of this representation there are the trivial orbit Y (0, 0, 0) = {(0, 0, 0)} of length 1, the orbits Y (α, ατ, 0) = {g(α, ατ, 0) | g ∈ Y } where α ∈ (0, ∞) (33) of length 12 (vertices of a regular icosahedron), the orbits Y (α, α, α) = {g(α, α, α) | g ∈ Y } where α ∈ (0, ∞) (34) of length 20 (vertices of a regular dodecahedron), the orbits Y (α, 0, 0) = {g(α, 0, 0) | g ∈ Y } where α ∈ (0, ∞) (35) of length 30 (vertices of an icosidodecahedron), and all the other orbits are of length 60. If the set symmetric with respect to the origin C3 = {v1 , v2 , ..., vk , −v1 , −v2 , ..., −vk } (36) where v1 = (v11 , v21 , v31 ), ..., vk = (v1k , v2k , v3k ), is a finite union of orbits of Y then the vectors w1 = (v11 , v12 , ..., v1k ) w2 = (v21 , v22 , ..., v2k ) w3 = (v31 , v32 , ..., v3k ) (37) from Rk are orthogonal hw1 , w2 i = hw2, w3 i = hw3 , w1 i = 0 (38) and have the same norm κ = ||w1|| = ||w2|| = ||w3 ||. They allow us to identify the physical space with the three-dimensional subspace E3 = { αw1 + βw2 + γw3 | α, β, γ ∈ R } (39) 9 Modified strip projection method of the superspace Rk . The orthogonal projection on E3 of a vector x ∈ Rk is the vector D w Ew D w Ew D w Ew 1 2 3 2 3 1 + x, + x, (40) π3 x = x, κ κ κ κ κ κ The orthogonal projector corresponding to the orthogonal complement k E⊥ 3 = { x ∈ R | hx, yi = 0 for all y ∈ E3 } (41) ⊥ is π3⊥ : Rk −→ E⊥ 3 , π3 x = x − π3 x. If we describe E3 by using the orthogonal basis {κ−2 w1 , κ−2 w2 , κ−2 w3 } then the expression in coordinates of π3 is π3 : Rk −→ R3 π3 x = (hx, w1 i, hx, w2i, hx, w3 i). (42) The projection W3,k = π3⊥ (Λk ) of the unit hypercube Λk is a polyhedron in the (k − 3)-dimensional subspace E⊥ 3 , and each (k − 4)-dimensional face of W3,k is the projection of a (k − 4)-dimensional face of Λk . Each (k − 4)-face of Λk is parallel to (k −4) of the vectors e1 , e2 , ..., ek and orthogonal to four of them. There exist sixteen (k−4)-faces of Λk orthogonal to the distinct vectors ei1 , ei2 , ei3 , ei4 , and the set ) ( xi ∈ {−1/2, 1/2} if i ∈ {i1 , i2 , i3 , i4 } (43) x = (x1 , x2 , ..., xk ) xi = 0 if i 6∈ {i1 , i2 , i3 , i4 } contains one and only one point from each of them. There are ! k(k − 1)(k − 2)(k − 3) k = 24 4 (44) sets of 24 parallel (k−4)-faces of Λk , and we label them by using the elements of the set ) ( 1 ≤ i ≤ k − 3, i + 1 ≤ i ≤ k − 2, 1 1 2 . (45) I3,k = (i1 , i2 , i3 , i4 ) ∈ Z4 i2 + 1 ≤ i3 ≤ k − 1, i3 + 1 ≤ i4 ≤ k A point x = (x1 , x2 , ..., xk ) ∈ Rk belongs to the strip S3,k if and only if − d i1 i2 i3 i4 ≤ xi1 xi2 xi3 xi4 v1i1 v1i2 v1i3 v1i4 v2i1 v2i2 v2i3 v2i4 v3i1 v3i2 v3i3 v3i4 ≤ d i1 i2 i3 i4 (46) α1 α2 α3 α4 v1i1 v1i2 v1i3 v1i4 . v2i1 v2i2 v2i3 v2i4 v3i1 v3i2 v3i3 v3i4 (47) for each (i1 , i2 , i3 , i4 ) ∈ I3,k , where d i1 i2 i3 i4 = max αj ∈{−1/2, 1/2} For each t ∈ Rk , the pattern defined in terms of the strip projection method [6, 8, 9, 10] Ω = π3 ((t + S3,k ) ∩ Zk ) = {π3 x | x−t ∈ S3,k and x ∈ Zk } can be regarded as a quasiperiodic packing of copies of the starting cluster C3 . (48) Modified strip projection method 10 The algorithm based on the strip projection method presented above is very efficient. In the case of a three-shell Y -cluster formed by the vertices of a regular icosahedron, a regular dodecahedron and an icosidodecahedron we use a 31-dimensional superspace, W3,k is a polyhedron lying in the 28-dimensional subspace E⊥ 3 bounded by 31465 pairs of parallel 27-dimensional faces, but we obtain 400-500 points in less than 10 minutes [2]. With the modification indicated in the previous section we can obtain quasiperiodic packings of multi-shell icosahedral clusters containing a significant percentage of fully occupied clusters. 5. Concluding remarks Some of the most remarkable tilings and discrete quasiperiodic sets used in quasicrystal physics are obtained by using strip projection method in a superspace of dimension four, five or six, and the projection of a unit hypercube as a window of selection [4, 6, 9]. The mathematical results presented above allow one to use this very elegant method in superspaces of dimension much higher, and to generate discrete quasiperiodic sets with a more complicated structure by starting from the symmetry group G and the local structure described by a covering cluster C. In our approach the window (which, generally, is a polyhedron with hundreds or thousands faces) is described in a simple way and we have to compute only determinants of order three or four, independently of the dimension of the superspace we use. These mathematical results have allowed us to obtain some very efficient computer programs for our algorithm [5]. Hundreds of points of our mathematical models can be obtained in only a few minutes. The quasiperiodic set generated by starting from a G-cluster C is a packing of partially occupied copies of C, but for most of these copies the occupation is very low. The main purpose of the paper is to present a modified version of the strip projection method. We project certain points lying outside the strip and do not project certain points lying inside the strip in order to favour the apperance of fully occupied clusters. More exactly, we start from the pattern generated by the standard strip projection method and help the clusters with occupation above a certain threshold (p%) to complete their configuration up to a fully occupied clusters. We project a minimum number of points lying outside the strip, and avoid to project certain points lying inside the strip. The symmetry group G corresponding to a real quasicrystal can be deduced from the diffraction images, and the covering cluster C can be chosen by analyzing the real structure information obtained by high-resolution transmission electron microscopy. Our modified strip projection method allows one to generate a mathematical model, and to compute the corresponding diffraction image by using the Fourier transform. If the agreement with the experimental data is not acceptable one has to look for a more suitable covering cluster C. 11 Modified strip projection method The computer program in FORTRAN 90 used in the case of figure 3 ! PLEASE INDICATE HOW MANY POINTS DO YOU WANT TO ANALYSE INTEGER, PARAMETER :: N = 6000 ! PLEASE INDICATE THE DIMENSION M INTEGER, PARAMETER :: M = 6 OF THE SUPERSPACE ! PLEASE INDICATE THE RADIUS OF THE PATTERN REAL, PARAMETER :: R = 9.0 INTEGER I, J, K, L, I1, I2, I3, JJ, J1, J2 REAL D1, D2, D3, PR REAL, DIMENSION(M) :: V, W, TRANSLATION, WJ, EPSILON INTEGER, DIMENSION(N) :: CLUSTER REAL, DIMENSION(2,M) :: BASIS REAL, DIMENSION(2,2) :: C12 REAL, DIMENSION(1:M-2,2:M-1,3:M) :: STRIP REAL, DIMENSION(N,M) :: POINTS, STRIPOINTS REAL, DIMENSION(N + M) :: XPOINT, YPOINT REAL, DIMENSION(200,200) :: FOURIER REAL, DIMENSION(2,40000) :: PATTERN COMPLEX II II=(0,1) EPSILON = 0.0001 ! PLEASE INDICATE THE COORDINATES OF A POINT OF THE CLUSTER BASIS(1,1) = 1.0 BASIS(2,1) = 0.0 ! PLEASE INDICATE THE TRANSLATION OF THE STRIP YOU WANT TO USE TRANSLATION = 0.1 C12(1,1) = SQRT(3.0) / 2.0 C12(1,2) = -1.0 / 2.0 C12(2,1) = 1.0 / 2.0 C12(2,2) = SQRT(3.0) / 2.0 DO J = 2, 6 DO I = 1, 2 BASIS(I,J) = C12(I,1) * BASIS(1,J-1) & + C12(I,2) * BASIS(2,J-1) Modified strip projection method END DO END DO STRIP=0 DO I1 =1, M-2 DO I2 =I1+1, M-1 DO I3 =I2+1, M DO D1 =-0.5, 0.5 DO D2 =-0.5, 0.5 DO D3 =-0.5, 0.5 PR = D1 * BASIS(1,I2) * BASIS(2,I3) + & D3 * BASIS(1,I1) * BASIS(2,I2) + & D2 * BASIS(1,I3) * BASIS(2,I1) - & D3 * BASIS(1,I2) * BASIS(2,I1) - & D1 * BASIS(1,I3) * BASIS(2,I2) - & D2 * BASIS(1,I1) * BASIS(2,I3) IF ( PR > STRIP(I1,I2,I3) ) STRIP(I1,I2,I3) = PR END DO END DO END DO IF( STRIP(I1,I2,I3) .EQ. 0 ) STRIP(I1,I2,I3)=N * SUM( BASIS(1,:) ** 2) END DO END DO END DO PRINT*, ’COORDINATES OF THE POINTS OF THE ONE-SHELL C12-CLUSTER:’ DO J = 1, M PRINT*, J, BASIS(1,J), BASIS(2,J) END DO PRINT*, ’* STRIP TRANSLATED BY THE VECTOR WITH COORDINATES:’ PRINT*, TRANSLATION PRINT*, ’* PLEASE WAIT A FEW MINUTES OR MORE,& DEPENDING ON THE NUMBER OF ANALYSED POINTS’ POINTS = 0 STRIPOINTS = 0 POINTS(1,:) = ANINT( TRANSLATION) STRIPOINTS(1,:) = ANINT( TRANSLATION) K = 1 L = 0 DO I = 1, N V = POINTS(I, : ) - TRANSLATION JJ = 1 DO I1 =1, M-2 DO I2 =I1+1, M-1 12 Modified strip projection method DO I3 =I2+1, M PR = V(I1) * BASIS(1,I2) * BASIS(2,I3) + & V(I3) * BASIS(1,I1) * BASIS(2,I2) + & V(I2) * BASIS(1,I3) * BASIS(2,I1) - & V(I3) * BASIS(1,I2) * BASIS(2,I1) - & V(I1) * BASIS(1,I3) * BASIS(2,I2) - & V(I2) * BASIS(1,I1) * BASIS(2,I3) IF ( ABS(PR) > STRIP(I1,I2,I3) ) JJ = 0 END DO END DO END DO IF( JJ .EQ. 1 ) THEN I3 = 1 DO J = 1, L WJ = ABS(POINTS(I,:) - STRIPOINTS(J,:)) IF( ALL(WJ < EPSILON) ) I3 = 0 END DO IF( I3 == 1 .AND. SUM( V * V) < R **2 ) THEN L = L + 1 STRIPOINTS(L,:) = POINTS(I,:) ELSE END IF DO I1 = 1, M DO I2 = -1, 1, 2 W = POINTS(I, : ) W(I1) = W(I1) + I2 I3 = 0 DO J = 1, K WJ = ABS(W - POINTS(J,:)) IF( ALL( WJ < EPSILON)) I3 = 1 END DO IF ( I3 == 0 .AND. K < N ) THEN K = K + 1 POINTS(K, : ) = W ELSE END IF END DO END DO ELSE END IF END DO CLUSTER = 0 13 Modified strip projection method DO I = 1, L DO J1 = 1, M DO J2 = -1, 1, 2 W = STRIPOINTS(I,:) - TRANSLATION W(J1) = W(J1) + J2 JJ = 1 DO I1 =1, M-2 DO I2 =I1+1, M-1 DO I3 =I2+1, M PR = W(I1) * BASIS(1,I2) * BASIS(2,I3) + & W(I3) * BASIS(1,I1) * BASIS(2,I2) + & W(I2) * BASIS(1,I3) * BASIS(2,I1) - & W(I3) * BASIS(1,I2) * BASIS(2,I1) - & W(I1) * BASIS(1,I3) * BASIS(2,I2) - & W(I2) * BASIS(1,I1) * BASIS(2,I3) IF ( ABS(PR) > STRIP(I1,I2,I3) ) JJ = 0 END DO END DO END DO IF( JJ .EQ. 1 ) CLUSTER(I) = CLUSTER(I) + 1 END DO END DO END DO DO J = 1, L XPOINT(J) = SUM( STRIPOINTS(J,:) * BASIS(1,:) ) YPOINT(J) = SUM( STRIPOINTS(J,:) * BASIS(2,:) ) END DO PRINT*, ’NUMBER OF ANALYSED POINTS :’, K PRINT*, ’NUMBER OF OBTAINED POINTS :’, L DO I = 1, 100 DO J = 1, 100 D1=0.0 DO I1 = 1, L D1=D1+EXP( II * (-1.5+I*0.03)*XPOINT(I1)+ & II * (-1.5+J*0.03)*YPOINT(I1) ) END DO FOURIER(I,J)=(ABS(D1))**2 END DO END DO I2=0 DO I = 1, 100 DO J = 1, 100 14 Modified strip projection method IF(FOURIER(I,J)>0.001*L**2) THEN I2=I2+1 PATTERN(1,I2)=-1.5+I*0.03 PATTERN(2,I2)=-1.5+J*0.03 ELSE END IF END DO END DO PRINT*, ’INDICATE THE NAME OF A FILE WITH EXTENSION tex FOR RESULTS’ WRITE(4,60) 60 FORMAT(’\documentclass{article} & \begin{document} & \begin{figure} & \setlength{\unitlength}{1.5mm} & \begin{picture}(50,20)(0,0) ’ & ’\put(32.0,20.0){\circle*{0.2}} ’) DO J = 1, L IF( CLUSTER(J) < M+1 ) THEN WRITE(4,65) 10+XPOINT(J), 20+YPOINT(J) 65 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.2}} ’) ELSE END IF END DO DO J = 1, L IF( CLUSTER(J) > M ) THEN WRITE(4,70) 10+XPOINT(J), 20+YPOINT(J) 70 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle{0.4}} ’) ELSE END IF END DO DO J = 1, 6 WRITE(4,72) 32+BASIS(1,J), 20+BASIS(2,J) 72 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.2}} ’) WRITE(4,73) 32-BASIS(1,J), 20-BASIS(2,J) 73 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.2}} ’) END DO WRITE(4,75) 75 FORMAT(’ \setlength{\unitlength}{1.8mm}’) DO J = 1, I2 WRITE(4,80) 45+10*PATTERN(1,J), 17+10*PATTERN(2,J) 80 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.1}} ’) END DO 15 Modified strip projection method WRITE(4,90) 90 FORMAT( ’\end{picture} & \caption{Quasiperiodic set obtained by using & the strip projection method } & \end{figure} & \end{document}’) PRINT*, ’* COMPILE THE OBTAINED FILE AND SEE THE ".dvi" FILE’ END 16 17 Modified strip projection method The computer program in FORTRAN 90 used in the case of figure 4 ! PLEASE INDICATE HOW MANY POINTS DO YOU WANT TO ANALYSE INTEGER, PARAMETER :: N = 6000 ! PLEASE INDICATE THE DIMENSION M INTEGER, PARAMETER :: M = 6 OF THE SUPERSPACE ! PLEASE INDICATE THE RADIUS OF THE PATTERN REAL, PARAMETER :: R = 9.0 INTEGER I, J, K, L, I1, I2, I3, JJ, J1, J2, L1 REAL D1, D2, D3, PR, XP, YP REAL, DIMENSION(M) :: V, W, TRANSLATION, WJ, EPSILON INTEGER, DIMENSION(N) :: CLUSTER REAL, DIMENSION(2,M) :: BASIS REAL, DIMENSION(2,2) :: C12 REAL, DIMENSION(1:M-2,2:M-1,3:M) :: STRIP REAL, DIMENSION(N,M) :: POINTS, STRIPOINTS REAL, DIMENSION(N + M) :: XPOINT, YPOINT REAL, DIMENSION(200,200) :: FOURIER REAL, DIMENSION(2,40000) :: PATTERN COMPLEX II II=(0,1) EPSILON = 0.0001 ! PLEASE INDICATE THE COORDINATES OF A POINT OF THE CLUSTER BASIS(1,1) = 1.0 BASIS(2,1) = 0.0 ! PLEASE INDICATE THE TRANSLATION OF THE STRIP YOU WANT TO USE TRANSLATION = 0.1 C12(1,1) = SQRT(3.0) / 2.0 C12(1,2) = -1.0 / 2.0 C12(2,1) = 1.0 / 2.0 C12(2,2) = SQRT(3.0) / 2.0 DO J = 2, 6 DO I = 1, 2 BASIS(I,J) = C12(I,1) * BASIS(1,J-1) & + C12(I,2) * BASIS(2,J-1) Modified strip projection method ! ! BASIS(I,6+J) = C12(I,1) * BASIS(1,5+J) & + C12(I,2) * BASIS(2,5+J) END DO END DO STRIP=0 DO I1 =1, M-2 DO I2 =I1+1, M-1 DO I3 =I2+1, M DO D1 =-0.5, 0.5 DO D2 =-0.5, 0.5 DO D3 =-0.5, 0.5 PR = D1 * BASIS(1,I2) * BASIS(2,I3) + & D3 * BASIS(1,I1) * BASIS(2,I2) + & D2 * BASIS(1,I3) * BASIS(2,I1) - & D3 * BASIS(1,I2) * BASIS(2,I1) - & D1 * BASIS(1,I3) * BASIS(2,I2) - & D2 * BASIS(1,I1) * BASIS(2,I3) IF ( PR > STRIP(I1,I2,I3) ) STRIP(I1,I2,I3) = PR END DO END DO END DO IF( STRIP(I1,I2,I3) .EQ. 0 ) STRIP(I1,I2,I3)=N * SUM( BASIS(1,:) ** 2) END DO END DO END DO PRINT*, ’COORDINATES OF THE POINTS OF THE ONE-SHELL C12-CLUSTER:’ DO J = 1, M PRINT*, J, BASIS(1,J), BASIS(2,J) END DO PRINT*, ’* STRIP TRANSLATED BY THE VECTOR WITH COORDINATES:’ PRINT*, TRANSLATION PRINT*, ’* PLEASE WAIT A FEW MINUTES OR MORE,& DEPENDING ON THE NUMBER OF ANALYSED POINTS’ POINTS = 0 STRIPOINTS = 0 POINTS(1,:) = ANINT( TRANSLATION) STRIPOINTS(1,:) = ANINT( TRANSLATION) K = 1 L = 0 DO I = 1, N V = POINTS(I, : ) - TRANSLATION JJ = 1 18 Modified strip projection method DO I1 =1, M-2 DO I2 =I1+1, M-1 DO I3 =I2+1, M PR = V(I1) * BASIS(1,I2) * BASIS(2,I3) + & V(I3) * BASIS(1,I1) * BASIS(2,I2) + & V(I2) * BASIS(1,I3) * BASIS(2,I1) - & V(I3) * BASIS(1,I2) * BASIS(2,I1) - & V(I1) * BASIS(1,I3) * BASIS(2,I2) - & V(I2) * BASIS(1,I1) * BASIS(2,I3) IF ( ABS(PR) > STRIP(I1,I2,I3) ) JJ = 0 END DO END DO END DO IF( JJ .EQ. 1 ) THEN I3 = 1 DO J = 1, L WJ = ABS(POINTS(I,:) - STRIPOINTS(J,:)) IF( ALL(WJ < EPSILON) ) I3 = 0 END DO IF( I3 == 1 .AND. SUM( V * V) < R **2 ) THEN L = L + 1 STRIPOINTS(L,:) = POINTS(I,:) ELSE END IF DO I1 = 1, M DO I2 = -1, 1, 2 W = POINTS(I, : ) W(I1) = W(I1) + I2 I3 = 0 DO J = 1, K WJ = ABS(W - POINTS(J,:)) IF( ALL( WJ < EPSILON)) I3 = 1 END DO IF ( I3 == 0 .AND. K < N ) THEN K = K + 1 POINTS(K, : ) = W ELSE END IF END DO END DO ELSE END IF 19 Modified strip projection method END DO CLUSTER = 0 DO I = 1, L DO J1 = 1, M DO J2 = -1, 1, 2 W = STRIPOINTS(I,:) - TRANSLATION W(J1) = W(J1) + J2 JJ = 1 DO I1 =1, M-2 DO I2 =I1+1, M-1 DO I3 =I2+1, M PR = W(I1) * BASIS(1,I2) * BASIS(2,I3) + & W(I3) * BASIS(1,I1) * BASIS(2,I2) + & W(I2) * BASIS(1,I3) * BASIS(2,I1) - & W(I3) * BASIS(1,I2) * BASIS(2,I1) - & W(I1) * BASIS(1,I3) * BASIS(2,I2) - & W(I2) * BASIS(1,I1) * BASIS(2,I3) IF ( ABS(PR) > STRIP(I1,I2,I3) ) JJ = 0 END DO END DO END DO IF( JJ .EQ. 1 ) CLUSTER(I) = CLUSTER(I) + 1 END DO END DO END DO I1 = L DO I = 1, I1 IF ( CLUSTER(I) > M) THEN DO J1 = 1, M DO J2 = -1, 1, 2 W = STRIPOINTS(I,:) W(J1) = W(J1) + J2 I3 = 0 DO J = 1, L WJ = ABS(W - STRIPOINTS(J,:)) IF( ALL( WJ < EPSILON)) I3 = 1 END DO IF ( I3 == 0 ) THEN L = L + 1 STRIPOINTS(L, : ) = W ELSE END IF 20 Modified strip projection method 21 END DO END DO ELSE END IF END DO L1=0 DO I = 1, I1 IF ( CLUSTER(I) > M) THEN L1=L1+1 XPOINT(L1) = SUM( STRIPOINTS(I,:) * BASIS(1,:) ) YPOINT(L1) = SUM( STRIPOINTS(I,:) * BASIS(2,:) ) ELSE END IF END DO DO J = I1+1, L L1=L1+1 XPOINT(L1) = SUM( STRIPOINTS(J,:) * BASIS(1,:) ) YPOINT(L1) = SUM( STRIPOINTS(J,:) * BASIS(2,:) ) END DO D1=4.0 DO I =2, M IF(((BASIS(1,1)+BASIS(1,I))**2 + (BASIS(2,1)+BASIS(2,I))**2 ) < D1 ) & D1=(BASIS(1,1)+BASIS(1,I))**2 + (BASIS(2,1)+BASIS(2,I))**2 END DO DO I =2, M IF(((BASIS(1,1)-BASIS(1,I))**2 + (BASIS(2,1)-BASIS(2,I))**2 ) < D1 ) & D1=(BASIS(1,1)-BASIS(1,I))**2 + (BASIS(2,1)-BASIS(2,I))**2 END DO DO I = 1, I1 IF ( CLUSTER(I) <= M) THEN D2=4.0 XP = SUM( STRIPOINTS(I,:) * BASIS(1,:) ) YP = SUM( STRIPOINTS(I,:) * BASIS(2,:) ) DO J = 1, L1 IF( ((XP - XPOINT(J))**2 + (YP - YPOINT(J))**2 ) < D2 ) & D2=(XP - XPOINT(J))**2 + (YP - YPOINT(J))**2 END DO IF( D2 > 0.9*D1 ) THEN L1=L1+1 XPOINT(L1)=XP YPOINT(L1)=YP ELSE Modified strip projection method END IF ELSE END IF END DO DO I = 1, 100 DO J = 1, 100 D1=0.0 DO I1 = 1, L1 D1=D1+EXP( II * (-1.5+I*0.03)*XPOINT(I1)+ & II * (-1.5+J*0.03)*YPOINT(I1) ) END DO FOURIER(I,J)=(ABS(D1))**2 END DO END DO I2=0 DO I = 1, 100 DO J = 1, 100 IF(FOURIER(I,J)>0.0015*L**2) THEN I2=I2+1 PATTERN(1,I2)=-1.5+I*0.03 PATTERN(2,I2)=-1.5+J*0.03 ELSE END IF END DO END DO PRINT*, ’NUMBER OF ANALYSED POINTS :’, K PRINT*, ’NUMBER OF OBTAINED POINTS :’, L1 PRINT*, ’INDICATE THE NAME OF A FILE WITH EXTENSION tex FOR RESULTS’ WRITE(4,60) 60 FORMAT(’\documentclass{article} & \begin{document} & \begin{figure} & \setlength{\unitlength}{1.5mm} & \begin{picture}(50,20)(0,0) ’ & ’\put(32.0,20.0){\circle*{0.2}} ’) DO J = 1, L1 IF( CLUSTER(J) < M+1 ) THEN WRITE(4,65) 10+XPOINT(J), 20+YPOINT(J) 65 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.2}} ’) 22 Modified strip projection method ELSE END IF END DO DO J = 1, L1 IF( CLUSTER(J) > M ) THEN WRITE(4,70) 10+XPOINT(J), 20+YPOINT(J) 70 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.2}} ’) ELSE END IF END DO DO J = 1, 6 WRITE(4,72) 32+BASIS(1,J), 20+BASIS(2,J) 72 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.2}} ’) WRITE(4,73) 32-BASIS(1,J), 20-BASIS(2,J) 73 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.2}} ’) END DO WRITE(4,75) 75 FORMAT(’ \setlength{\unitlength}{1.8mm}’) DO J = 1, I2 WRITE(4,80) 45+10*PATTERN(1,J), 17+10*PATTERN(2,J) 80 FORMAT( ’\put( ’F10.5’,’F10.5,’){\circle*{0.1}} ’) END DO WRITE(4,90) 90 FORMAT( ’\end{picture} & \caption{Quasiperiodic set obtained by using & the modified strip projection method } & \end{figure} & \end{document}’) PRINT*, ’* COMPILE THE OBTAINED FILE AND SEE THE ".dvi" FILE’ END 23 Modified strip projection method 24 Acknowledgment This research was supported by the grants CERES 4-129 and CEx05-D11-03. References [1] Cotfas N and Verger-Gaugry J-L 1997 A mathematical construction of n-dimensional quasicrystals starting from G-clusters J. Phys. A: Math. Gen. 30 4283-91 [2] Cotfas N 2005 Preprint math-ph/0504044 [3] Cotfas N 2006 Discrete quasiperiodic sets with predefined covering cluster Phil. Mag. 86 895-900 [4] Cotfas N 2006 Discrete quasiperiodic sets with predefined local structure J. Geom. Phys. 56 24152428 [5] Cotfas N 2006 http://fpcm5.fizica.unibuc.ro/~ncotfas [6] Elser V 1986 The diffraction pattern of projected structures Acta Cryst. A 42 36-43 [7] Janot C and de Boissieu M 1994 Quasicrystals as a hierarchy of clusters Phys. Rev. Lett. 72 1674-7 [8] Kalugin P A, Kitayev A Y and Levitov L S 1985 6-dimensional properties of AlMn alloy J. Physique Lett. 46 L601-7 [9] Katz A and Duneau M 1986 Quasiperiodic patterns and icosahedral symmetry J. Phys. (France) 47 181-96 [10] Kramer P and Neri R 1984 On periodic and non-periodic space fillings of Em obtained by projection Acta Crystallogr. A 40 580-7 [11] Kramer P and Papadopolos Z (eds.) 2003 Coverings of Discrete Quasiperiodic Sets. Theory and Applications to Quasicrystals (Berlin: Springer) [12] Moody R V 1997 Meyer sets and their duals The Mathematics of Long-Range Aperiodic Order ed. R V Moody (Dordrecht: Kluwer) pp 403-41