## Abstract

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.

## 1. Introduction

Quasicrystals are aperiodic structures with long-range order that generate diffraction patterns with non-crystallographic symmetries. After earlier theoretical efforts for describing aperiodic structures, for instance, in the works of Penrose (1978), de Bruijn (1981) and Mackay (1982), the experimental discovery of quasicrystals in AlMn alloys by Shechtman *et al.* (1984) spurred great interest, also in the mathematical community. A standard way of generating mathematical models for quasicrystal structures is by means of the ‘cut-and-project’ method (de Bruijn 1981; Levitov & Rhyner 1988; Katz 1989; Kramer & Schlottmann 1989; Baake *et al.* 1990*a*,*b*; Senechal 1996): the points of the quasicrystal are obtained by projection, to a low-dimensional subspace, of suitably selected points of a higher dimensional lattice. If the projection subspace is invariant with respect to a subgroup of the point group of the lattice, then the resulting quasicrystal has the same symmetry.

Cut-and-project quasicrystals give rise to aperiodic tilings of space or the plane, which are obtained from the aperiodic point sets through a general method known as the dualization technique (Senechal 1996). This provides a general procedure for the construction of aperiodic tilings with non-crystallographic symmetry, such as icosahedral tilings of space.

We are interested in transformations of such tilings that preserve at least some symmetry, and to this end, we consider higher dimensional analogues of the Bain-like strains, i.e. lattice deformations such that the intermediate configurations during the transition maintain a common (perhaps maximal) symmetry subgroup. Such strains are considered, for instance, in the investigation of three-dimensional reconstructive martensitic phase transformations in crystalline substances (Bain 1924; Wayman 1964; Boyer 1989; Toledano & Dmitriev 1996; Christian 2002; Pitteri & Zanzotto 2002; Bhattacharya *et al.* 2004; Capillas *et al.* 2007). The basic observation is that, provided there exists some common symmetry subgroup of the higher dimensional lattices before and after transition, one can determine the corresponding higher dimensional Bain strains and induce, via projection, structural transformations in the associated quasicrystals and tilings, which preserve the intermediate symmetry if the projection subspace is also invariant under the common symmetry subgroup.

Figure 1 gives a low-dimensional sketch of the proposed procedure: a two-dimensional square lattice is deformed into a rhombic lattice by an affine deformation, the two-dimensional analogue of the Bain strain. The point sets (quasicrystals) change accordingly, and this in turn induces a transformation of the tilings in the projection subspace *E* through the formation of new tiles and the change of shape of the existing ones (cf. §3). Analogous effects occur in higher dimensions.

As a first application of our methods, we study the transformations of the Penrose tiling of the plane into the tilings induced by the five-dimensional face-centred cubic (FCC) and body-centred cubic (BCC) lattices, respectively, while keeping at least the fivefold symmetry at intermediate configurations. These quasicrystals are obtained by projection of the five-dimensional lattices onto a plane invariant with respect to a suitable integral representation of the cyclic group *C*_{5}. One of these lattice paths in the higher dimensional space is obtained through the compression of the five-dimensional hypercubic cell along a body diagonal, similarly to the rhombohedral strain relating simple cubic (SC) and BCC lattices in three dimensions (Pitteri & Zanzotto 2002).

We find that, in projection, the classical Penrose rhomb tiling of the plane transforms into triangle tilings through three basic mechanisms, involving the flipping, bisection and merger of tiles. These mechanisms result from, and indeed correspond to, changes in the geometry of the projection window as a consequence of the deformation of the five-dimensional lattice.

Figure 2 illustrates these effects by showing the external shape of the projection windows (see §3) and the corresponding plane tilings along a symmetry-preserving transition path between the SC lattice in five dimensions (Penrose tiling) and the BCC lattice in five dimensions (triangle tiling). The projection windows are three-dimensional, but it is indeed the structure of the projected three-dimensional facets (not shown here) of the full five-dimensional Voronoi cell that determines the shape and the structure of the tilings. Furthermore, the electronic supplementary material, movie (FCC_Movie.avi), shows how the projection window changes along a full transition path from an SC lattice to an FCC lattice.

We also apply this technique to the study of the transformations of the three-dimensional icosahedral quasicrystals and their associated tilings of space, obtained by projection of the SC and FCC lattices in six dimensions (Papadopolos *et al.* 1993, 1997, 1998; Niizeki 2004), suggesting that the same mechanisms as above also occur in these three-dimensional aperiodic structure transformations.

The approach presented here complements the work modelling experimentally observed transformations between crystalline and quasicrystalline structures (Toledano & Dmitriev 1996; Tsai 2008), which have been studied from a six-dimensional viewpoint in the literature, see §4 for a discussion. In contrast, our main focus is the structural transformation of aperiodic structures, such as plane or space tilings, or of other icosahedral assemblies in three dimensions relevant in the investigation of viral capsids (Patera & Twarock 2002; Twarock 2004, 2006; Keef & Twarock 2009); our methods have indeed been applied in a recent study of the conformational changes in viruses with icosahedral symmetry (Indelicato *et al.* in press).

## 2. Transition paths for lattices

Let be the group of *n*×*n* unimodular integer matrices, the group of *n*×*n* invertible real matrices, *O*(*n*) and SO(*n*) the orthogonal and special orthogonal group of , and the set of *n*×*n* symmetric positive definite matrices with real coefficients. For a basis {*b*_{α}}_{α=1,…,n} of , we write for the matrix with column vectors *b*_{α}. We denote by
the lattice with basis {*b*_{α}}. All other lattice bases have the form , with . Moreover, we write
for the *lattice group* of , and
for its *point group*. The following notations are equivalent:
The point and lattice groups are related via the identity
2.1
and
for *R*∈*O*(*n*) and . We, therefore, will also write for the point group of the lattice .

A lattice basis is characterized (modulo rotations) by its *lattice metric*
and the lattice group is the subgroup of that fixes the metric (Pitteri & Zanzotto 2002)
2.2
Important examples of *n*-dimensional lattices are the SC, BCC and FCC lattices given by (Levitov & Rhyner 1988)
2.3
A basis for the above SC lattice is the canonical basis. The point group of the SC lattice is the so-called hyperoctahedral group
While the point groups of the three cubic lattices in (2.3) coincide, their lattice groups do not. Indeed, they are integral representations of the common point group, which are non-conjugate in (this is indeed the defining property for lattices with distinct Bravais types, see, for instance, Pitteri & Zanzotto 2002).

For a matrix group , we have the following standard definition.

### Definition 2.1

The centralizer of in is the group

The centralizer of can be obtained by solving the linear equations *G*_{i}*C*=*CG*_{i} in the unknown *C*, where *G*_{i} are generators of . Hence, the centralizers of a finitely generated group, in general, depend linearly on a finite list of real parameters.

We define a lattice transition as a continuous transformation between two lattices and , along which some symmetry is preserved, described by a common subgroup of the lattice groups of the intermediate lattices.

### Definition 2.2

Let and be two lattices, and . We say that there exists a transition between and with intermediate symmetry if there exist bases *B*_{0} and *B*_{1} of and , and a continuous path , with *B*(0)=*B*_{0} and *B*(1)=*B*_{1}, such that, for
one has
2.4
We call the linear mapping
2.5
the *transition*, while the curve is the *transition path*.

As mentioned earlier, we are mostly interested in transitions with maximal intermediate symmetry. Notice that, by continuity, and have the same sign, so that . The following equivalent statements characterize lattice transitions.

### Proposition 2.3

*Let* *and* *be two lattices, and* . *The following statements are equivalent*:

*there exists a transition between**and**with intermediate symmetry*;*there exist bases**B*_{0}*and**B*_{1}*of**and**such that for*, 2.6*there exist bases**B*_{0}*and**B*_{1}*of**and*,*such that*2.7*there exist a basis**B*_{0}of*and continuous paths**such that**R*(0)=*U*(0)=*I**and**R*(1)*U*(1)*B*_{0}=*B*_{1}*is a basis of*;*and**there exist bases**B*_{0}and*B*_{1}*of**and**and a continuous path**such that letting*,*and*,*then**C*(0)=*C*_{0},*C*(1)=*C*_{1}*and*2.8

### Proof.

The implications (i) ⇒ (ii) and (i) ⇔ (v) are immediate. To prove that (ii) ⇒ (iii), we notice that, by (ii), letting , then . Hence, and are conjugate in : , with . By the polar decomposition theorem, writing *T*=*RU* with *R*∈SO(*n*) and , it follows that, for all , there exists , such that *G*=*U*^{−1}*R*^{−1}*HRU*, i.e. *UG*=*R*^{−1}*HRU*. By the uniqueness of the polar decomposition, since *G* and *R*^{−1}*HR*∈*O*(*n*), one has
and *U* belongs to the centralizer of in . This proves (iii). Notice that, as a by-product of the above argument, and are conjugate in SO(*n*), indeed
To prove that (iii) ⇒ (iv), notice first that SO(*n*) is arcwise connected, so that there exist paths connecting *R* to the identity. Furthermore, since *U* is positive definite, we can write *U*=*Q*^{⊤}*DQ*, with *Q*∈SO (*n*), *D*=diag (*λ*_{1},…,*λ*_{n}) and *λ*_{i}>0. Then, a path connecting *U* to the identity is, for instance, *U*(*t*)=*Q*^{⊤}*D*(*t*)*Q*, with *D*(*t*)=diag ((*λ*_{1}−1)*t*+1,…,(*λ*_{n}−1)*t*+1), which is still in the centralizer. In fact, if , then , which means that *DH*=*HD* for every . This is equivalent to *λ*_{i}*h*_{ij}=*λ*_{j}*h*_{ij} for all *i*,*j*=1,…,*n*, where *h*_{ij} are the entries of *H*. On the other hand, this identity is true if and only if [(*λ*_{i}−1)*t*+1]*h*_{ij}=[(*λ*_{j}−1)*t*+1]*h*_{ij} for all *t*∈[0,1], which implies that , so that for all *t*∈[0,1].

Finally, (iv) ⇒ (i): in fact, letting *B*(*t*)=*R*(*t*)*U*(*t*)*B*_{0}, one has
Hence, for every *t*∈[0,1], which proves the claim. ■

If *T*(*t*)=*R*(*t*)*U*(*t*) is a transition path with symmetry between the lattices and , the symmetry of the intermediate phase is also described by the group of orthogonal transformations
2.9
which is a subgroup of the point group of the intermediate phase.

The following result is an immediate consequence of the above characterization of lattice transitions, and shows that any centralizer of , not necessarily symmetric, defines a transition with that symmetry.

### Corollary 2.4

*Any continuous path*
*with B*_{0} *and B*_{1} *lattice bases for* *and* *defines a transition between* *and* *with intermediate symmetry **.*

## 3. Cut-and-project quasicrystals and canonical tilings

Let be an *n*-dimensional lattice with point group . Consider a subgroup of , and assume that there exists a *k*-dimensional subspace invariant under . Denote by *E*^{⊥} the orthogonal complement of *E*, so that
with and the corresponding projection operators. Also, denote by a fundamental domain of acting on as a translation group, such as, for instance, the Voronoi cell at (cf. Senechal 1996),
The Voronoi cell at the origin is invariant under the point group of the lattice (Senechal 1996), and hence also under . Notice that the collection of all the Voronoi cells of the *n*-dimensional lattice defines a cell complex in . We assume that from now on.

Fix now a regular shift vector in , i.e. a vector ** γ** (possibly

**0**) such that for every

*d*-dimensional facet of , with

*d*<

*n*−

*k*, and for all . Defining the projection window as a cut-and-project quasicrystal is the point set given by 3.1 When

*E*is totally irrational, i.e. , with the dual lattice, the set is dense in

*E*; otherwise it is a -module, possibly a lattice, in

*E*. We refer to the point set as

*quasicrystal*.

There exists a canonical method for constructing aperiodic tilings of the space *E* using the points of a cut-and-project quasicrystal as vertices (Katz 1989; Kramer & Schlottmann 1989). Indeed, the complex of the Voronoi cells of the *n*-dimensional lattice defines a periodic tiling of ; its dual complex is also a periodic tiling of , called the Delone tiling, constructed as follows: if is a *p*-facet of a Voronoi cell, i.e. a *p*-cell of the Voronoi complex, its dual cell has dimension *n*−*p* and is the convex hull of the centres of the Voronoi cells that intersect in , i.e. letting
then
where is the convex hull of the lattice points in .

The Delone tiling of induces a tiling of *E* by projection on *E* of those Delone *k*-facets dual to the (*n*−*k*)-facets of the Voronoi tiling that have non-empty intersection with ** γ**+

*E*, where . By construction, the tiling of

*E*thus obtained has vertices at the points of the corresponding quasicrystal. If

**is invariant under , and if we choose to be the projection on**

*γ**E*

^{⊥}of the Voronoi cell at the origin , the associated tiling is invariant under the representation of in

*E*. In fact, commutes with the projection

*π*, so that -orbits in project on -orbits in

*E*, with the representation of in

*E*.

For ** y**=

*π*(

**), a point of a cut-and-project quasicrystal, the vertex star at**

*x***is the set of all tiles of**

*y**E*that have

**as one of their vertices. It can be shown (Senechal 1996) that all the vertex stars of a tiling obtained by dualization are determined by the intersections of the projections on**

*y**E*

^{⊥}of the (

*n*−

*k*)-dimensional facets of the Voronoi cell . In fact,

**=**

*y**π*(

**) is the vertex of a tile if , where is an (**

*x**n*−

*k*)-dimensional facet of the Voronoi cell . Hence, for instance, the number of tiles of which

**is a vertex is just the number of projected facets to which**

*y**π*

^{⊥}(

**)−**

*γ**π*

^{⊥}(

**) belongs.**

*x*When ** γ** is not regular, and a lattice point is such that

*π*

^{⊥}(

**)−**

*γ**π*

^{⊥}(

**) belongs to the common boundary of the projection of two, or more, (**

*x**n*−

*k*)-dimensional facets of the Voronoi cell, then

*π*(

**) is a vertex of all tiles dual to Voronoi facets intersecting in , and these tiles may overlap. In this case, to avoid self-intersections, we use as tiles the projection on**

*x**E*of the dual of the low-dimensional facet : these are called glue tiles (Kramer 2002).

The techniques summarized above have been extensively applied in the study of quasicrystals and tilings of the plane and space (de Bruijn 1981; Kramer & Neri 1984; Rokhsar & Mermin 1987; Katz 1989; Janot 1993; Papadopolos *et al.* 1997, 1998, 1993; Reiter 2002).

## 4. Structural transformations of cut-and-project quasicrystals

We now indicate how structural transformations of cut-and-project quasicrystals can be induced by transitions between higher dimensional lattices. Consider two *n*-dimensional lattices and , with point groups and , and two subgroups and . Assume that and have invariant subspaces *E*_{0} and *E*_{1}, respectively, with dim*E*_{0}=dim*E*_{1}=*k*. Without loss of generality, assume that
and consider the cut-and-project quasicrystals and .

### Definition 4.1

We say that there exists a transition between the cut-and-project quasicrystals and with intermediate symmetry , if there exists a transition with intermediate symmetry between and , such that for *T*(*t*)=*R*(*t*)*U*(*t*), with *R*(*t*)∈SO(*n*) and , *E* is invariant under , i.e.
4.1
with
4.2

Any such transition defines a family of cut-and-project quasicrystals in the same projection space *E*, all of which have symmetry .

It is useful to discuss the relation of our method with other higher dimensional approaches used in the literature to study experimentally observed transitions between icosahedral quasicrystals and cubic phases in three dimensions (Kramer 1987; Litvin *et al.* 1987; Li *et al.* 1989*a*,*b*; Torres *et al.* 1989; Li & Cheng 1990; Aragón & Torres 1991; Mukhopadhya *et al.* 1991; Cheng *et al.* 1992; Sun 1993), and for deformations of Penrose tilings of the plane (cf., for instance, Ishii 1993; Sing & Welberry 2006; Welberry & Sing 2007). As mentioned in §6, icosahedral quasicrystals can be obtained by projection of an SC lattice in six dimensions on a three-dimensional subspace that carries an irreducible representation of the icosahedral group. The procedure used to obtain a transformation of the projected three-dimensional point set in these works is either (i) by rotating the projection plane with respect to the six-dimensional lattice or (ii) by deforming the six-dimensional lattice through certain linear transformations called phason strains. These two approaches can be shown to be related (Cheng *et al.* 1992). The technique presented in this paper generalizes (i) and (ii), since it allows for both deformations and changes of symmetry of the lattice, controlling the intermediate symmetry. In detail, our approach relates to the rotation-plane method (Kramer 1987) as follows. For a fixed six-dimensional cubic lattice , it is known that suitable choices of three-dimensional subspaces *E*_{0} and *E*_{1} in yield sets and that correspond to an FCC crystal or to an icosahedral quasicrystal in . Suppose that and are six-dimensional representations of the three-dimensional cubic group *O* and of the icosahedral group, respectively, so that *E*_{0} is invariant under and *E*_{1} is invariant under . Then, there exist orthonormal bases and of (not necessarily lattice bases) that block-diagonalize and as the direct sum of two three-dimensional blocks. These bases are adapted to the subspaces *E*_{0} and *E*_{1}.

Consider now the tetrahedral group , which is a subgroup of both and : both and block-diagonalize , but since has just a single three-dimensional irreducible representation, we may choose and such that . Hence, there exists *C*∈SO(6) in the centralizer of such that . Actually, there exists a continuous path *C*(*t*) in the centralizer of that connects *C* to the identity (the ‘Schur rotation’). This in turn means that the path belongs to the centralizer of in SO(6), and is such that . To summarize, the Schur rotation defines a path in the centralizer of , connecting *E*_{0} to *E*_{1}, i.e. such that *E*_{1}=*R*(1)*E*_{0}, and *R*(*t*)*E*_{0} is invariant under for every *t*∈[0,1].

The Schur rotation can be used to define a transition with intermediate symmetry in the sense of the present paper because rotating the projection plane with respect to the lattice is equivalent to rotating the lattice with respect to a fixed projection plane: in fact, *E*_{0} is invariant under both and , and since , this means that the Schur rotation defines a family of rotated cubic six-dimensional lattices
with the property that -orbits in project on tetrahedral orbits in *E*_{0}. Consider now any path
Then, choosing any lattice basis *B*_{0} for , the path
defines a lattice transition with tetrahedral symmetry between and . Furthermore, since and have *E*_{0} as invariant subspace, this transition defines in turn a transition between the quasicrystals and with tetrahedral symmetry (cf. (4.1) and (4.2)).

## 5. Transformations between planar aperiodic tilings preserving the fivefold symmetry

In this section we present three examples of transformations that preserve the global fivefold symmetry between planar tilings of the same symmetry, in particular, the Penrose tiling. For the latter, we adopt here a five-dimensional approach instead of the usual one based on a four-dimensional minimal embedding (Baake *et al.* 1990*b*), because in this way it is simpler to describe the transitions in terms of deformations of the unit cubic cell in .

Consider the SC, BCC and FCC lattices, and the standard basis (*e*_{α}), *α*=1,…,5, in , together with the group of fivefold rotations about the body diagonal ** n** of the unit cube,
The group , which leaves all the three five-dimensional cubic lattices above invariant, has two mutually orthogonal invariant subspaces: the two-dimensional subspace

*E*and the three-dimensional subspace

*E*

^{⊥}with projections and Notice that

*E*

^{⊥}may in turn be split into the direct sum of two orthogonal invariant subspaces, a two-dimensional plane and a line parallel to

**.**

*n*With reference to §3, we choose ; projection of the SC lattice (2.3)_{1} and the related Delone tiling on *E* produces the well-known Penrose tiling of the plane, while projecting the FCC and BCC lattices in (2.3)_{2,3} gives more complex aperiodic planar tilings, see also Reiter (2002). All these aperiodic structures have a global fivefold symmetry about the origin, and our focus is on their structural transformation preserving such symmetry.

Proposition 2.3 guarantees that the *C*_{5}-preserving transition paths for the associated five-dimensional lattices are parametrized by the centralizers of *C*_{5} in , i.e. the group of matrices that has explicitly the form, for ,
We consider three specific transition paths: the first two are paths between the SC and the BCC lattices in five dimensions,
5.1
and
5.2
while the third one joins the SC to the FCC lattice,
5.3
Each intermediate lattice along these paths has global fivefold symmetry by construction. Since
the first two transition paths above involve a compression of the unit cube along a body diagonal ** n**. These paths also entail, through the dualization technique applied at each step, transformations between the Penrose and the BCC and FCC tilings. For a point of a high-dimensional lattice, the tiles that have

*π*(

**) as a vertex are the projections on**

*x**E*of the two-dimensional duals of the three-dimensional facets of , such that . To explain transformations in tilings in

*E*, it is useful to look at the changes in how

**−**

*γ***is projected into these projected facets. As a general observation, we see that the transformations of the aperiodic structures proceed through a combination of three basic mechanisms.**

*x*—

*Splitting of a tile into two.*This occurs when a facet of the Voronoi cell splits into two. The projection on*E*^{⊥}of the facet before the split, and the two new facets, are shown in figure 3*a*. A lattice point, such that*x**π*^{⊥}(−*γ*) belongs to the region where the perpendicular projections overlap, will be a vertex of a single tile in the first step and two tiles in the subsequent step (figure 3*x**b*). Equivalently, two tiles join to become a single tile if the projection of−*γ*falls out of the intersection of two regions into a region covered by only one.*x*—

*Tile flips.*Rearrangements of tiles within limited areas, which we call tile flips, occur owing to points*π*^{⊥}(−*γ*) in*x**E*^{⊥}moving from one projected facet in*E*^{⊥}to another, such as going through the shaded area in figure 4*a*. In the upper half of the figure, the point−*γ*is projected into the projection of two Voronoi regions in*x**E*^{⊥}, and therefore*π*() is a vertex of two tiles, as shown on the left of figure 4*x**b*. When the projected point goes through the shaded face, it is now only within the projection of a single Voronoi facet and*π*() is thus a vertex of a single tile. Since the transitions are continuous, there is a time when a projected lattice point lies in the intersection of these regions producing overlapping tiles, and at this time we insert in the tiling a glue tile, in this case, a quadrilateral tile.*x*—

*Tile mergers.*During the lattice transition the Voronoi cells change, so that the projection on*E*^{⊥}may involve different sets of points, see, for instance, figure 5*a*. Pointssuch that*x**π*^{⊥}(−*γ*) fall outside the resulting acceptance window are deleted from the tiling, along with any tiles with these points as vertices. This produces tile mergers as seen, for instance, in figure 5*x**b*.

All the changes in the transforming quasicrystals and tilings can be described through the three mechanisms above. Figure 6 shows four steps in the transformations from SC to BCC along the paths defined by (5.1) and (5.2), as well as the transformation from SC to FCC defined by the path in (5.3). Figure 6*a* shows the tiling for *t*=0, obtained via projection from an SC lattice, which then branches into three pathways. For the path (5.1) shown in figure 6*b*,*c*,*f*, the first step in the transition is a splitting of all tiles along their long diagonal, followed by flips, deletions of vertices and further splits/recombinations. The second path (5.2) shown in figure 6*d*,*e*,*f* also displays at first a splitting of all the tiles, in this case, along the shorter diagonal, and then proceeds as above to further splits/recombinations. For the transition from SC to FCC (5.3), figure 6*g*,*h*,*i*, only the thin rhombs split at first, followed by further changes in the tiles, finally resulting in a much coarser tiling than the one produced by the BCC lattice.

## 6. Example of a transformation between icosahedral three-dimensional quasicrystals

We briefly discuss here a transformation between icosahedral quasicrystals in and their associated tilings. Related ideas are analysed in more detail in Indelicato *et al.* (in press), where a technique similar to the present one has been applied to study the configurational changes of icosahedral viral capsids.

The twofold and threefold rotations
6.1
generate a six-dimensional integral representation of the icosahedral group (cf. e.g. Katz 1989). The matrix group leaves the three six-dimensional cubic lattices (SC, FCC, BCC, cf. (2.3)) invariant and is a subgroup of the hyperoctahedral group *B*_{6} (Levitov & Rhyner 1988).

The representation of on is the sum of two non-equivalent irreducible representations of degree 3 (Katz 1989), and splits into the direct sum of two three-dimensional subspaces, *E* and *E*^{⊥}, invariant under the icosahedral group. In particular, following Katz (1989), we choose a matrix representation of the projection on *E*,
with . Notice that the vectors of the six-dimensional canonical basis project on vectors pointing to the vertices of an icosahedron.

The icosahedral group has three maximal subgroups: the tetrahedral group and the dihedral groups *D*_{5} and *D*_{3}. We are interested in transitions between the SC and the FCC, BCC lattices with maximal intermediate symmetry, described by one of these subgroups. However, there exist no tetrahedral transition paths among the cubic lattices in six dimensions. We show this for transitions between the SC and FCC lattices, a similar proof also being possible for transitions to a BCC lattice. Indeed, assume that a tetrahedral transition exists. Then, according to proposition 2.3(iii), the transition operators *T* can be decomposed as *T*=*RU*, with *R*∈SO(*n*) and such that
with *B*_{0} a basis of an SC lattice and *B*_{1} a basis of an FCC lattice. Any has the form (Indelicato *et al.* in press)
with . By (2.3), the metric of an FCC lattice has entries in , and since
and , also has entries in ,
with *a*=*z*^{2}+4*x*^{2}+*y*^{2}, *b*=2*xz*, *c*=2*yz* and .

Then, leads to the following equation:
which cannot be fulfilled for any choice of . In fact, for *a*=*α*/4, *b*=*β*/4, *c*=*γ*/4 with , the equation above reduces to
which cannot be solved by any integer *α*, *β*, *γ*. A similar argument shows that there cannot be tetrahedral transitions between the SC and any rescaled FCC and BCC lattices. The same holds also for the FCC–BCC lattice transitions in six dimensions.

We consider next the possibility of transitions with *D*_{5} symmetry (see more details in Indelicato *et al.* in press). As discussed above, such transitions belong to the centralizer of *D*_{5}, having the form
6.2
where *x*,*y*,*z*,*s*,*u*,*w* depend on *t*. As an example, we consider a path with intermediate symmetry *D*_{5} that deforms an SC lattice (at *t*=0) into an FCC lattice (at *t*=1),
6.3
Figure 7 shows three snapshots of a patch of the corresponding three-dimensional tiling around a fixed vertex, i.e. projections on *E* of a suitable portion of the Delone tiling of the lattices , for *t*=0,0.233 and 1. Throughout the transition, the tile arrangements have *D*_{5} symmetry, and we observe that they evolve through mechanisms similar to those discussed in detail for the planar case in §5.

## 7. Conclusions

In this work we have proposed a new approach for the investigation of structural transformations in cut-and-project quasicrystals, based on higher dimensional analogues of the classical three-dimensional Bain strains, i.e. deformation paths that maintain a given symmetry, possibly maximal, for the associated higher dimensional lattices during the transition. Our method is related to the phason-strain and rotation-plane techniques (see §4), but has the advantage of allowing for more general transitions between the higher dimensional lattices, possibly involving also different Bravais types, and takes explicitly into account all possible intermediate symmetries.

The main purpose of our work is to explore the effect these structural transformations have on the plane or space tilings associated with each quasi-crystal, for instance, by means of the dualization method. The local tile rearrangements can be understood in terms of the change of geometry of the Voronoi cell of the higher dimensional lattice during the transition, and we analyse the effect this has on the projected lattice points. Our results suggest that the possible ways in which an aperiodic tiling can change, while still conserving some intermediate symmetry, reduces to the three basic mechanisms of tile splitting, tile flipping and tile merger.

As case studies, we have examined how the Penrose tiling of the plane changes as the underlying five-dimensional cubic lattice undergoes a deformation to either a BCC or an FCC lattice through a transition path that preserves fivefold symmetry. We have also examined the transformation of an icosahedral Ammann tiling in three-dimensional space, obtained by projecting a suitable deformation path connecting an SC and an FCC lattice in six dimensions.

A rich landscape for structural transformations in quasicrystals emerges from our approach. The general patterns identified here may provide a basis for further analysis of structural transitions in quasi-lattices and for a possible classification of transitions in aperiodic structures. This is of interest both from a theoretical viewpoint and for applications, for instance, in virology for the study of the structural rearrangements of viral capsids important for infection.

## Acknowledgements

R.T. would like to thank the Leverhulme Trust for funding of her research team via a Research Leadership Award. P.C. and G.I. acknowledge partial funding by the research project ‘Modelli Matematici per la Scienza dei Materiali’ of the Università di Torino, Italy. P.C., G.I. and A.Z. acknowledge partial funding by the MATHMAT and MATHXPRE projects of the Università di Padova, Italy, and the Italian PRIN 2009 project ‘Mathematics and Mechanics of Biological Systems and Soft Tissues’. G.I. also acknowledges funding by the Marie Curie Project MATVIR.

- Received November 16, 2011.
- Accepted January 12, 2012.

- This journal is © 2012 The Royal Society