## research papers

## twins revisited: quinary twins and beyond

^{a}Laboratoire de Métallurgie de l'UMR 8247, IRCP Chimie-ParisTech, 11 rue Pierre et Marie Curie, F-75005 Paris, France^{*}Correspondence e-mail: denis.gratias@chimie-paristech.fr

A twin is defined as being an external operation between two identical crystals that share a fraction of the atomic structure with no discontinuity from one crystal to the other. This includes ). *The Painter's Manual: a Manual of Measurement of Lines, Areas and Solids by Means of Compass and Ruler*. Facsimile Edition (1977), translated with commentary by W. L. Strauss. New York: Abaris Books]. This paper will show that the Dürer twins, once defined in five-dimensional space, are simple twins, in the sense of Georges Friedel, leaving the five-dimensional invariant. This analysis will be generalized to some other higher-order -modules.

Keywords: merohedral twins; quinary twins; Dürer structure; polygonal tilings.

### 1. Introduction

From an historical point of view, twins have played a special role in mineralogy and crystallography as they are an aggregate of identical crystals oriented with respect to each other in a very specific and characteristic manner (see, for instance, Groth, 1906; Putnis, 1992). Several twin laws have been proposed that can all finally be summarized by the basic idea that the specific relative orientation of a twinned crystal is a special that keeps invariant – either exactly or approximately – a part of the atomic structure or of a specific property between the two twins. The idea is that the larger the common part is, the more stable is the twin and the more frequently it occurs in nature. Using this kind of intellectual guide, Friedel (1904, 1926, 1933) proposed a classification of twins based on the geometry of the sole crystal (Bravais, 1851; Mallard, 1885; Donnay, 1940):

(i)

twins where the crystals share the same (this can happen only for non-holohedral structures).(ii) Twins by reticular

where the crystals share only a fraction, a of the crystal this corresponds, in metals and alloys, to the so-called special boundaries between grains like the famous mirror twin along the (111) direction often observed in the f.c.c. (face-centred cubic) metals.(iii) Pseudo-merohedral twins or twins by reticular pseudo-merohedry where the previous definitions are satisfied only approximately.

In the present paper, we choose a general definition of ), Grimmer & Nespolo (2006), Marzouki *et al.* (2014 and references therein):

(i) Twinned crystals in mutual orientation by reticular

in three dimensions (two dimensions or one dimension) that share a common three-dimensional (two-dimensional or one-dimensional) sublattice.(ii) A twin by contact where only the habit plane is shared by the two adjacent crystals (epitaxy).

(iii) More generally, any

keeping invariant a fraction of the Wyckoff positions of the structure.### 2. Formalism

#### 2.1. Symmetry operations and space groups

A *g* and a translation part *t*, and noted , that transforms a point *r* in into a point in as

Designating by the set of the isometries of , we define the

of the crystal as the set defined byConsidering now the subset of the points that are invariant under the ), its symmetry group is defined

(see Fig. 1It is generally different to and is not necessarily a

of it. | Figure 1 of . |

Thus, the fundamental group–subgroup relation defining the geometry of the two processes implies the intersection group , that gathers the symmetry operations that belong to both and .

The group scheme is shown in Fig. 2. It defines two integers *n* and *m* that are the indices of in, respectively, and :

Their meaning is the following:

(i) *n*-1 is the number of different possible twinned crystals around one given crystal and all share with the central crystal the same subset of symmetry group :

(ii) *m* is the number of equivalent subsets in the same crystal of :

Each ^{1} *i.e.* an operation that relates two twinned crystals sharing the same and each represents an internal operation for the crystal that transforms the subset into one of its equivalents.

As a simple example, let us consider the standard so-called twin in cubic face-centred metals where the two individuals share a common *U*), with the intersection group with the same *U*, as shown in Fig. 3 on the right. This leads to *n* = 2, meaning that the connects two individuals and *m* = 4 ×4 ×3/4 = 12 different crystals – corresponding to the 4 ×3 families of (1, 1, 1) planes, *A*, *B* or *C* – can be formed around one single crystal. Finally, the that corresponds to the indices of the lattices only is , thus the term used to designate this kind of grain boundary.

| Figure 3 conservation between twinned crystals, whereas on the right, emphasis is put on the twinned crystals sharing a common |

This same twin can be as well defined through its *n* = 2 variants and *m* = 4 different individuals – the four orientation families of (1, 1, 1) planes – that can be formed around one given variant.

All known types of twins enter the general group–subgroup tree of Fig. 2.

For instance,

twins are characterized by and having the same coincidence grain boundaries are twins by reticular with a grain boundary index being the order of the of in the of .### 3. Generalization

As we will show here, there are cases where the scheme of Fig. 2 leads to original results such as the twin structures first drawn by Albrecht Dürer and reproduced here in Fig. 4(*a*) from the original work *De symmetria partium in rectis formis humanorum corporum* (1532) and *Underweysung der Messung* (1538) (available on CD-ROM, Octavo Editions, CA, 2003).

#### 3.1. The Dürer structure

The basic structure invented by Albrecht Dürer is shown in Fig. 5(*a*). It is built with six adjacent regular pentagons and has the crystallographic two-dimensional *c*2*m**m*. Taking the radius of the elementary pentagon as the unit length (see Fig. 5*a* on the left), we find the parameters and where τ is the golden mean …. The whole structure is described by only two Wyckoff positions generated by the positions and drawn in green and blue in Fig. 5.

Twins of the Dürer structure can be generated in a very symmetrical tenfold symmetry according to various equivalent modes, either radiant central as in Dürer's original drawing, or spiral-like twins made of ten two-dimensional crystals along the ten directions of a regular decagon as seen in Fig. 4(*b*).

#### 3.2. The hidden symmetries of the Dürer structure

The very specific feature of the Dürer structure is that the atomic positions *x*_{j} are all vertices of interconnected identical regular pentagons so that they can all be defined as *integer* sums of the five vectors relating the centre to the five vertices of the elementary pentagon:

This structure is thus a periodic decoration of a -module^{2} of rank 5 (4 in fact, because the sum of the five unit vectors gives the zero vector) generated by the five vectors defined by the regular pentagon. In a more geometric view, the Dürer structure is a two-dimensional projection of a five-dimensional periodic structure in a four-dimensional hyperplane perpendicular to the five-dimensional main diagonal (1, 1, 1, 1, 1).

Embedding the Dürer structure in five-dimensional space is easily performed. We choose the origin in Ω as shown in Fig. 5(*b*) and find the defined by and , two vectors that are both perpendicular to the main diagonal (1,1,1,1,1) in five-dimensional space. The two Wyckoff positions are located at the nodes *w*_{1} = (0,1,0,0,0) for the blue one and *w*_{2} = (0,0,1,0,0) for the green one. The operations are 5 ×5 signed permutation matrices given by

and the corresponding space operations that generate the *c*2*m**m* are

These operations together with the translation group generated by (*A*,*B*) form a faithful representation of the group *c*2*m**m* in five-dimensional Euclidean space.

#### 3.3. Twin operation

Now, we choose the underlying five-dimensional *E*_{i}, *i* = 1,5 whose projections are the five basic vectors of the pentagon, as the geometric object that should be left invariant. The group of all operations that keep the five-dimensional invariant together with the two-dimensional cut space is *p*10 *m**m* generated by

Thus, the general group–subgroup tree (Fig. 2) is built with the point groups (order 20), (order 4) that is a of and thus qualifies this defect to be a * pure twin by merohedry* because , and thus it leaves the -module invariant.

The (*b*). These are the five different ways of constructing the Dürer structure using the *same pentagon* (and its inverse). For example, one among the possible cosets of equivalent twin operations is given by

and is the glide mirror shown in Fig. 6.

It can be easily verified that this twin is *perfectly coherent* although it has no two-dimensional coincidence The boundary is defined by a sinuous row of adjacent pentagons that belong to both structures. Moreover, irrespective of the centring *c*, the of the Dürer structure is the set of five-dimensional points *V* = *p* *A* + *q* *B* = (2*q*, *q*-*p*, -2*q*, -2*q*, *p*+*q*) where *p* and *q* are integers. This transforms into the set = and the common points are such that 2*q* = and = that has solutions only for , *i.e.* for the direction (1,1). This is the habit direction of the twin: we have a perfect with no (two-dimensional) coincident lattice.

### 4. Beyond the Dürer twin

Dürer-like structures can easily be found using identical regular polygons of order *n* (later on, designated as *n*-gons) connected by edges. All these structures have the basic property of being defined by Wyckoff positions that are all on the same -module and can thus be described as two-dimensional cut-and-projections of *n*-dimensional structures.

We discuss here some of the simplest of these kinds of polygonal tilings where the *n*-gons in the are all crystallographically equivalent. We shall designate these patterns as *monogeneous n*-gon patterns.

An efficient way of characterizing these patterns consists of reporting in a vector the sequence of the number of free edges between each connected edge around an *n*-gon as exemplified in Fig. 7 for *n* = 9. We call it the *vector of free edges*, the length of which is equal to the coordination of the *n*-gon. Under these notations, the Dürer structure of the previous section with *n* = 5 has coordination *p* = 3 and is characterized, up to a circular permutation, by the vector (0,1,1).

The search for possible periodic solutions is significantly simplified by observing that, for an *n*-gon surrounded by *p* identical *n*-gons, the vector of free edges is such that with .

Also, the maximum possible number *P*_{n} of non-overlapping *n*-gons sharing an edge of a central identical *n*-gon is given by

For the simple case *p* = 3, monogeneous non-overlapping *n*-gon periodic patterns are generated only if the centre of the central *n*-gon is *inside* the triangle formed by the centres of the three adjacent *n*-gons, in which case the triangle characterizes the of the structure (see Fig. 7).

The vector of free edges has the form . Assuming all index ranks are taken modulo *n*, the centres of the three *n*-gons are located at *V*_{1} = (1, 1, 0, …), *V*_{2} = (0, 0, …, 1, 0, …) and *V*_{3} = (0, 0, …, 1, 0, …), generating the (primitive) unit cells defined by *A* = *V*_{2}-*V*_{1} = 0, …, 1, 0, …) and *B* = *V*_{3}-*V*_{1} = 0, …, 1, 0, …).

All twins in these structures are *n*-gon). They are characterized by the symmetry elements of the *n*-dimensional that leave the projected two-dimensional space invariant and that do not belong to the symmetry group of the structure as dictated by the general group tree of Fig. 2. In two-dimensional space, these twin operations are symmetry operations of the central *n*-gon that are not symmetry elements of the two-dimensional periodic structure and that leave invariant a row of the structure to form a perfect plane of For *p* = 3, these elements are signed permutations of the *n* basic vectors generating the *n*-gon that transform into each other two of the other adjacent *n*-gons and put the third one in a new position. This translates in the vector of free edges in exchanging two symbols while keeping the third constant. For example, the vector of free edges (1,2,3) of the case *n* = 9 generates three possible coherent twinned crystals: (1,3,2), (3,2,1) and (2,1,3) as exemplified in Fig. 7, whereas the vector of free edges (1,2,2) of the case *n* = 8 generates only two, (2,2,1) and (2,1,2). The interface operations are glide mirrors oriented along the common row of *n*-gons shared between the two adjacent crystals, as shown in Figs. 8 and 9.

### 5. Conclusion

We propose here a formal extension of the notion of *merohedral* twins when they share the same -module as in the case of the Dürer twins.

Beyond this *n*-dimensional generalization, the interest of the present approach is the simplicity of its basic group–subgroup tree shown in Fig. 2 that can be used in all cases of actually known twins, once is identified.

### Footnotes

^{1}As discussed long ago (Guymont *et al.*, 1976; Gratias *et al.*, 1979; Portier & Gratias, 1982), interfaces (twins or grain boundaries) in homogeneous crystals are described by cosets of space groups. Indeed, consider two identical crystals related by the operation and with space groups, respectively, and . The operation relates any point *x* of the first crystal to an equivalent in the second crystal. Thus the is equivalently described by any operation that results in the product of any of by and finally any of , that is . This shows that a general interface operation between two identical crystals is not defined by a unique operation but by a of the form .

^{2}A -module of rank *p* in of dimension is the set of points *x* defined by where the basic vectors *e*_{i} are arithmetically independent (no non-zero *p* integers lead to a sum giving the null vector); it is isomorphic to an irrational projection in *d*-dimensional space of an *N*-dimensional with . For *d* = *p* the -module is a lattice.

### Acknowledgements

The authors are grateful to the French ANR for financial support of project ANR METADIS 13-BS04-0005.

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