Some icosahedral patterns and their isometries Nicolae Cotfas 1 Faculty of Physics, University of Bucharest, PO Box 76-54, Postal Office 76, Bucharest, Romania Radu Slobodeanu 2 Faculty of Physics, University of Bucharest, PO Box 76-54, Bucharest, Romania Abstract For each of the icosahedral patterns obtained by translating the strip used in the construction of the three-dimensional Penrose tiling we determine the isometries which leave it invariant. The variation of the pattern determined by a translation of the strip may describe some phase transitions or phasons in quasicrystals. Using our results one can determine the corresponding variation of the symmetry group, and to choose an adequate translation of the strip. 1 E-mail address: ncotfas@yahoo.com 2 E-mail address: rslobi@hotmail.com
1 Introduction Some of the most important models in quasicrystal physics are based on the three-dimensional Penrose tiling (also called Ammann-Kramer-Neri tiling) [7-9]. It is believed that there are not non-trivial isometries leaving invariant this tiling. Our purpose is to give a rigorous proof for this assertion, and to study the existence of proper isometries in the case of the singular patterns obtained by a translation of the strip. The method we use is inspired by some mathematical techniques developed in the case of self-similarities [2-6]. We think that our results may have some applications in the description of phase transitions and phasons. 2 Some icosahedral patterns Let ε 1 = (1, 0, 0, 0, 0, 0), ε 2 = (0, 1, 0, 0, 0, 0),..., ε 6 = (0, 0, 0, 0, 0, 1) be the canonical basis of IE 6, and let τ = (1 + 5)/2. The rotations a(α, β, γ) = ( τ 1 2 α τ 2 β + 1 2 γ, τ 2 α + 1 2 β + τ 1 2 γ, 1 2 α + τ 1 2 β + τ 2 γ) b(α, β, γ) = ( α, β, γ) (1) generate the usual three-dimensional representation of the icosahedral group Y = 235 = a, b a 5 = b 2 = (ab) 3 = I. (2) The points of the Y -invariant set I = {e 1, e 2,..., e 6, e 1, e 2,..., e 6 }, where e 1 = (1, τ, 0) e 2 = ( 1, τ, 0) e 3 = ( τ, 0, 1) e 4 = (0, 1, τ) e 5 = (τ, 0, 1) e 6 = (0, 1, τ) (3) are the vertices of a regular icosahedron. The action of a and b on I is described by the signed permutations ( ) ( ) e1 e a = 2 e 3 e 4 e 5 e 6 e1 e b = 2 e 3 e 4 e 5 e 6 (4) e 2 e 3 e 4 e 5 e 1 e 6 e 1 e 2 e 5 e 6 e 3 e 4 and the corresponding transformations a, b : IE 6 IE 6 ( ) ( ) ε1 ε a = 2 ε 3 ε 4 ε 5 ε 6 ε1 ε b = 2 ε 3 ε 4 ε 5 ε 6 ε 2 ε 3 ε 4 ε 5 ε 1 ε 6 ε 1 ε 2 ε 5 ε 6 ε 3 ε 4 (5) define the orthogonal representation a(x 1, x 2, x 3, x 4, x 5, x 6 ) = (x 5, x 1, x 2, x 3, x 4, x 6 ) b(x 1, x 2, x 3, x 4, x 5, x 6 ) = ( x 1, x 2, x 5, x 6, x 3, x 4 ) (6) of Y in IE 6. The space IE 6 can be decomposed [3,8,9] into a sum of orthogonal Y -invariant spaces IE 6 = IE 6 IE 6, such that IE 6 = {(< r, e 1 >, < r, e 2 >,..., < r, e 6 >) r IE 3 }. The vectors u 1 = ϱ(1, 1, τ, 0, τ, 0) u 2 = ϱ(τ, τ, 0, 1, 0, 1) u 3 = ϱ(0, 0, 1, τ, 1, τ) where ϱ = 1/ 2τ + 4, form an orthonormal basis of IE 6, and the isomorphism Λ : IE 3 IE 6 Λr = ϱ(< r, e 1 >, < r, e 2 >,..., < r, e 6 >) (7) 2
with the property Λ(1, 0, 0) = u 1, Λ(0, 1, 0) = u 2, Λ(0, 0, 1) = u 3, is an isomorphism of representations of Y, that is, Λ g = g Λ, for all g Y. It allows us to identify the space IE 3 with IE 6. The orthogonal projectors corresponding to IE 6 and IE 6 expressed in terms of the basis {ε 1, ε 2,..., ε 6 } are [7,9] π = M(1/2, 5/10) π = M(1/2, 5/10) (8) where and they satisfy the relations M(α, β) = α β β β β β β α β β β β β β α β β β β β β α β β β β β β α β β β β β β α π π = π π π = π π g = g π π π = π π = 0 π + π = I π g = g π. (9) (10) For each x IE 6 there are x = π x IE 6 and x = π x IE 6 uniquely determined such that x = x + x. We denote A = π (A), A = π (A), for any A IE 6. The set L = κ Z 6, where κ = 1/ϱ, is a Y -invariant lattice. The vectors κε 1, κε 2,..., κε 6 form a basis of L, and taking into account the identification IE 6 IE 3 we have π (κε j ) = Λe j e j (11) for any j {1, 2,..., 6}. The mappings L L : x x and L L : x x are bijective. Let K = ([ κ/2, κ/2] 6 ), S = K + IE 6, and let us define the pattern Q w = (L (S + w)) (12) for all w IE 6. The pattern Q w is said to be regular if the boundary (S + w) of S + w does not contain any point of L, and singular in the opposite case. If the pattern Q w is regular then its points are the vertices of a tiling, called a Penrose tiling. 3 Isometries of icosahedral patterns Theorem 1. The pattern Q 0 is invariant under the representation (1) of Y as a group of rotations of IE 3. Proof. Since g(k) = K, g(l) = L, and g(ie 6 ) = IE 6 we get g(s) = g(k + IE 6 ) = g(k) + g(ie 6 ) = K + IE 6 = S whence g(q 0 ) = g( π (L S) ) = π (g(l S)) = π (L S) = Q 0, for any g Y. 3
Theorem 2. If w IE 6 and g Y are such that w gw L then the pattern Q w is invariant under the isometry IE 6 IE 6 : x gx + (w gw). (13) Proof. Let v = w gw. The transformation Γ g : IE 6 IE 6, Γ g x = gx + v leaves invariant the lattice L and the strip S + w = S + w. Indeed, Γ g (L) = g(l) + v = L + v = L and Γ g (S + w) = g(s + w) + v = S + gw + w gw = S + w. Since y L (S + w) = Γ g y = gy + v L (S + w) we get π y Q w = g(π y) + π v = π (gy) + π v = π (Γ g y) Q w that is, x Q w = gx + v Q w. Let c = ab, and let L = L (L + t) = κ 2 Z6, where t = κ( 1 2, 1 2, 1 2, 1 2, 1 2, 1 2 ). Since (α, α, α, α, α, β) a(α, α, α, α, α, β) = 0 (0, 0, α, β, α, β) b(0, 0, α, β, α, β) = 0 (α, α, α, β, β, β) c(α, α, α, β, β, β) = 0 (14) and t gt L, for any α, β IR, g Y, from theorem 2 we get the following results. Corollary 1. If w U + L, where U = { (α, α, α, α, α, β) α, β IR }, then the pattern Q w is invariant under the isometry IE 6 IE 6 : x ax + (w aw). (15) Corollary 2. If w V + L, where V = { (0, 0, α, β, α, β) Q w is invariant under the isometry α, β IR }, then the pattern IE 6 IE 6 : x bx + (w bw). (16) Corollary 3. If w W +L, where W = { (α, α, α, β, β, β) α, β IR }, then the pattern Q w is invariant under the isometry Let U = g(u) + L g Y IE 6 IE 6 : x cx + (w cw). (17) V = g(v ) + L Theorem 3. If the pattern Q w is invariant under a proper isometry g Y IE 6 IE 6 : x gx + u W = g(w ) + L. (18) then g Y, there is w IE 6 such that w = w, w gw L, and u = (w gw ). In addition: i) g is a five-fold rotation if and only if w U ii) g is a two-fold rotation if and only if w V iii) g is a three-fold rotation if and only if w W. g Y 4
Proof. In the space IE 6 IE 3 the first neighbours of each point x Q w belong to the set x + I. The isometry Q w Q w : x gx + u must transform the first neighbours of x into the first neighbours of gx + u. This is possible only if g(x + I) + u = gx + u + I, that is, only if g Y and u L. Let v L be the element satisfying the relation u = v. Since π (κε j ) = e j, π g = g π, and L (S + w) Q w : y y is a bijection, our isometry Q w Q w : x gx + v corresponds to the transformation L (S + w) L (S + w) : x gx + v (19) where x gx is the restriction of the transformation g : IE 6 IE 6 belonging to the representation (6). The transformation (19) can exist only if g(s + w) + v = S + w, that is, only if there is w IE 6 such that w = w, w gw L and v = w gw. The last part follows from the proof of the corollaries. Theorem 4. If w L then the pattern Q w of Y in IE 3 IE 6. is invariant under the affine representation T g : IE 6 IE 6 T g x = gx + (w gw) (20) Proof. The relation (20) defines a representation T h (T g x) = h(gx + (w gw) ) + (w hw) = (hg)x + (w (hg)w) = T hg x and w gw L, for any g Y. Theorem 5. The pattern Q w is singular if and only if w K + L. Proof. Evidently, 0 (w + S) if and only if w K. Since K = K, for any x L we get x (w + S) x w + K w x + K. Theorem 6. Each of the sets U, V, W is a countable union of straight lines dense in IE 6, and L U V W K + L. (21) Proof. The spaces U, V, W admit the bases {ε 1 + ε 2 + ε 3 + ε 4 + ε 5, ε 6 }, {ε 3 + ε 5, ε 4 + ε 6 }, and {ε 1 ε 2 + ε 3, ε 4 + ε 5 + ε 6 }, respectively. Since (ε 1 + ε 2 + ε 3 + ε 4 + ε 5 ) = (1/ 5)ε 6, (ε 3 + ε 5 ) = τ(ε 4 + ε 6 ), (ε 1 ε 2 + ε 3 ) = τ 3 (ε 4 + ε 5 + ε 6 ), the spaces U, V, W are one-dimensional, namely, U = IRε 6 V = IR(ε 4 + ε 6 ) W = IR(ε 4 + ε 5 + ε 6 ). It is known [7] that {αε 1 + βε 2 α, β [0, κ]} = {αε 1 + βε 2 α, β [0, κ]} is a face of the triacontahedron ([0, κ] 6 ). From the relation [ κ/2, κ/2] 6 = [0, κ] 6 t it follows that { F 12 = αε 1 + βε 2 κ 2 ε 3 κ 2 ε 4 κ 2 ε 5 κ [ 2 ε 6 α, β κ 2, κ ]} 2 {( = α, β, κ ) [ 2, κ 2, κ 2, κ α, β κ 2 2, κ ]} 2 is one of the 30 faces forming K, namely, a face parallel to the plane Sp{ε 1, ε 2 } generated by ε 1 and ε 2. Since g(k) = K, for all g Y, it follows that K = g Y g(f 12), and hence { K + L = (x 1, x 2,..., x 6 ) at least four of x 1, x 2,..., x 6 belong to κ } 2 + κ Z. (22) 5
This means that K + L is a countable union of planes, each of them parallel to one of the 15 planes Sp{ε i, ε j }, where i, j {1, 2,..., 6}, i j. Since U, V Sp{ε 4, ε 6 } and W Sp{ε 2, ε 6 }, in order to prove the relation (21) it is sufficient to show that Sp{ε i, ε j } K + L and Sp{ε i, ε j } + t K + L for all i, j {1, 2,..., 6}, i j. The second relation follows directly from (22). ( κ 2, κ 2 + κτ, κ 2, κ 2, κ 2, ) κ 2 = 0, we have Since Sp{ε 1, ε 2 } = {( α, β, κ 2, κ 2, κ 2, κ 2 ) α, β IR }. From (22) we get Sp{ε 1, ε 2 } K + L, and taking into account the Y -invariance of the set K + L, we obtain Sp{ε i, ε j } K + L, for all i, j {1, 2,..., 6}, i j. The theorems 3-6 show that a non-trivial isometry can not leave invariant a regular pattern, and describe the isometries of the singular patterns. To any continuous mapping [0, 1] IE 6 : s w(s) we can associate a continuous translation of the strip s S + w(s), and the corresponding variation s Q w(s) of the pattern which involve some discontinuities (jumps of points). Since L is dense in IE 6 and K is a closed surface, the image of any non-constant continuous mapping [0, 1] IE 6 intersects K + L, and hence, in the case of any continuous translation of the strip we can not avoid the singular patterns. A translation of the strip along a straight line contained in U V W, generally, keeps unchanged the symmetry group of the pattern. If w L then the symmetry group of Q w is isomorphic to Y. 4 Further observations It is known [10] that the set K of all locally finite tilings of IR 3 has a structure of metric space, complete and totally disconnected. On K there is a natural action of the group of isometries. Restricted to the space of icosahedral tilings discused above, this action is free, acording to section 2 (and it s discontinuous, obviously). So the quotient space X of tilings generated by regular patterns Q w, modulo isometries has a natural topological structure. This space seems to be very rich: in [9] it was noticed that X can be labelled by an uncountable set. Unfortunately, Conway s local isomorphism property says that one cannot distinguish between two tilings from X only by inspection of its finite patterns. So the natural topology on X is trivial: classical tools fails trying to reveal us the space X. It was the idea of Connes [1] to study the space X by means of non-classical tools of Noncommutative Geometry. That is, instead of using the involutive algebra C(M) of continuous functions as we do in topological category (algebra of classical observables, which contains all the topological informations about M), to asociate to M a noncommutative C -algebra (algebra of quantum observables ) which can provide us - via a sort of K-theory - data about M. This method is appropriate to more general spaces. We have to mention here that, in our case, self-similarity properties of icosahedral tilings [9] (the fact that there are infinitely many ways of extracting a sub-tiling from a initial one) makes impossible the attempt to develop the construction of noncommutative structure on X in an analogous manner as in 2-dim. case [1]. So we are forced to use a more complicated construction [10]. The equivalence relation between tilings, R, given by isometries has structure of a grupoid. On this grupoid we have the C -algebra C c (R) of all continuous complex functions with compact support. The completion, A, of C c (R) with respect to the reduced norm [10] will be the C -algebra asociated with our space, 6
X. It will be of great interest to calculate the (invariance) groups K 0 (A), K 1 (A), in order to elucidate the structure of the space X. Acknowledgment. This research was supported by a CNCSIS grant. References [1] Connes A. 1994 Noncommutative Geometry Academic Press, Inc. [2] Cotfas N 1998 On the self-similarities of two icosahedral patterns Z. Kristallogr. 213 311-315 [3] Cotfas N 1998 On the affine self-similarities of the three-dimensional Penrose pattern J. Phys. A: Math. Gen. 31 7273-7277 [4] Cotfas N 1999 Permutation representations defined by G-clusters with applications to quasicrystals Lett. Math. Phys. 47 111-123 [5] Cotfas N 1999 G-model sets and their self-similarities J. Phys. A: Math. Gen. 32 8079-8093 [6] Cotfas N 1999 On the self-similarities of a model set J. Phys. A: Math. Gen. 32 L165-L168 [7] Duneau M 1988 Pavages, structures quasi-périodiques et modélisation des quasi-cristaux Du Cristal à l Amorphe ed C Godrèche (Paris: Les Editions de Physique) 157-197 [8] Elser V 1986 The diffraction pattern of projected structures Acta Cryst. A42 36-43 [9] Katz A and Duneau M 1986 Quasiperiodic patterns and icosahedral symmetry J. Physique 47 181-196 [10] Kellendonk J. 1994 Noncommutative Geometry of Tilings and Gap Labeling arxiv: cond-mat/9403065 7