SYMMETRY OF THREE-DIMENSIONAL QUASICRYSTALS

Tohru Ogawa

Institute of Applied Physics, University of Tsukuba Sakura, Ibaraki 305, Japan,

ogawa@bk.tsukuba.ac.jp
 
 

The relation between the three-dimensional quasicrystal models, the one by Penrose transformation method which was found by the author in 1985 and the other by the so-called projection method, is clarified. Primitive projection model is formulated as a mapping from a region in the pseudospace to the physical space by introducing the concept of the gazing crystal. It makes it possible to study the structure of the so­called projection method in detail. In conclusion, the primitive projection model, being deterministic, is included in the Penrose transformation model in the widest sense, which, having some freedom, is more flexible to realize some hierarchical local symmetries.
 
 

1. Introduction

Penrose tiling [1] is the prototype of the concept of the quasicrystal [2]. However, two-dimensional case, where the Penrose transformation is uniquely defined and coincides with the so-called projection method [3], is rather exceptional when compared with three-dimensional case. Most people seem to believe the Penrose transformation and the projection method (hereafter, to be referred to as the primitive projection) lead always to the same results. In this paper, it is shown to be false and some relations are figured out.

It is useful to regard the projection of a regular lattice in a D-dimensional original space to d-dimensional physical space as a mapping from a finite region referred to as the gazing crystal in (D-d)-dimensional pseudo space to d­dimensional physical space. Reminisce that some fortune tellers gaze a quartz and decode the information contained in it as that in a whole physical world.

From another point of view, there are infinite local five-fold centres in the two­dimensional Penrose tiling. Their variety of symmetry range form a kind of hierarchy. On the other hand, the range of icosahedral symmetry in the most strict sense is restricted to the only one case; to be referred to as flower dodecahedron Fl₆₀. Of course, three dimensional configuration can be icosahedrally symmetrical in looser senses, for example in a macroscopic sense that all of the six vectors are evenly appear.

In  Chapter 2, three-dimensional Penrose transformation, which was proposed by the author [4] in 1985, is given in a revised form. In Chapter 3, the primitive projection method is formulated. In Chapter 4, the three-dimensional configuration by the primitive projection method is studied in detail. Possible symmetry in the projection model is investigated in Chapter 5. The value t³ of the similarity factor and some similar numbers to golden number are discussed respectively in Appendix A and B.
 

2. Three-Dimensional Penrose Transformation

The interpretation of three-dimensional Penrose Transformation, which was discovered before by the author [4], is revised below.

The quasibases are six vectors arranging together with their six mirror images with icosahedral symmetry.

where

and

There are 15 = ₆C₂ rhombi whose edges are these six. They are all congruent as can be seen from the fact that

They are referred to as Golden rhombus the ratio of whose two diagonals is the golden number t=(Ö5+1)/2» 1.618. Some properties of a golden rhombus are listed in Table 1. Two kinds of rhombic Penrose tiles in two dimension are the projection of the rhombus along other quasibases. There are 20 = ₆C₃ rhombohedra whose edges are these six vectors. They are either of two golden rhombohedra, acute A₆ and obtuse O₆. (Tables 2, 3 and 4). They are the quasi-unit-cells of the present model. Three kinds of zonohedra, B₁₂,  F₂₀ and K₃₀, are constructed with A₆ and O₆ (Table 5). Twenty A₆'s organize a concave 60-hedron Fl₆₀ with icosahedral symmetry so that their principal vertices meet at a point. In order to give a gross image, this polyhedron is referred to as flower dodecahedron each of whose 12 faces looks like a five-petalous flower (see Table 6).
 

Table 1: Properties of a Golden Rhombus

Table 2: Properties of an Acute Golden Rhombohedron A
 
 

Table 3: Properties of an Obtuse Golden Rhombohedron O₆

Table 4: Some Useful Relations

Table 5: Number of Elements and Symmetry of Golden Zonohedra

Table 6: Number of Elements and Symmetry of Two Compounds of A₆ and O₆

Let us construct an A₆* and an O₆* (respectively a big A₆ and a big O₆ of edge length t³) only with A₆'s and O₆'s of edge length 1. If it can be done, an A** and an O₆** of edge lenght t⁹ can be constructed in a similar way and so on. Here, a Penrose transformation is defined as a transformation from a set of space-filling figures to their enlarged figures consisting only of the formers.

Three-dimensional Penrose Transformation in the most general version is simply stated as follows:
 

More exactly, into its centre and of course the Fl₆₀ should be placed so that directions of all edges coincide with the quasilattice axes. It is noted that two Fl₆₀'s at both principal vertices of an O₆* are connected with each other sharing an A₆ in common and six O₆'s are placed between the two. Though there are many unifilled spaces, in an A₆* and an O₆*, the skeleton of big rhombohedra have already been essentially decided.

The mid part of an edge is almost surrounded with two flowers and can only be filled with a Fedorov icosahedron F₂₀, so long as the tiles are restricted to A₆ and O₆. The inside of F₂₀ is left undecided in the skeleton structure. Now, the skeleton of an O₆* is completely decided. In an A₆*, there is a space C, which looks like a cage, can contain 20 A₆'s and 19 O₆'s (see Table 6). The shape of the cage is regarded as two Kepler tetrakaidecahedron K₃₀ sharing an O₆ in common. The O₆ is inevitably placed there, dividing the cage into two parts. Leaving the structure of other parts of the cage undecided, the skeleton of an A₆* is thus completely determined.

The A₆* and O₆* are numerically related with A₆ and O₆ as

                 A₆* = C+20A₆ + [6×(1/5)+6×(3/10)] F₂₀ =  55A₆ + 34O₆ ,                                    (2.3)
                 O₆* = 19A₆ + 6O₆ + [6×(1/5)+6×(3/10)] F₂₀ =  34A₆ + 21O₆ ,                              (2.4)

where 20 A₆ and 19 A₆ correspond to an Fl₆₀ respectively in A₆* and O₆*. There, the facts

                                                  C = 20A₆ + 19O₆ ,                                                                   (2.5)
                                               F₂₀ = 5A₆ + 5O₆ ,                                                                        (2.6)

are also used. The matrix which composed of the coefficients of this equation has t⁹ and (t^-1, t^-2) respectively as a set of eigenvalue and eigenvector

                                                                                                         (2.7)

It means that the limiting tiling obtained by an infinite iteration of this transformation has a self-similarity of factor t³ in linear dimension and t⁹ in volume and the composition of two tiles are t^-1 and t^-2.

The description should be given on the internal structure of the two undecided parts F₂₀ and C.

As F₂₀ has a five-fold axis in its outlook. However, it contains only four points in its inside and it can not have real five-fold symmetry. It can be placed in ten ways for a given axis. There are no principle in this stage to choose some among them. The numbers of A₆ and O₆ are both five in 10 ways.

As mentioned above, a C is a compound of two K₃₀'s sharing an O₆ in common. A K₃₀, having icosahedral symmetry in its outlook, contains ten points in its inside. Their arrangement is altogether 160 (The value 400 in the previous papers is incorrect). In any of them, a three-fold axis among ten may be regarded as the main axis of the internal structure. On the main axis, an A₆ and an O₆ are placed so that their principal axes coincide. Now, seven points are already fixed.

The part near the A₆-side end of the main axis is uniquely filled with three A₆'s and three O₆'s. There are eight (=2³) possibilities about the position of the remaining three points, each of which has two seats available independently selectable. Six among eight have only reflection symmetry and other two have a three-fold axis. The two correspond to the chirality, which is the symmetry concerning the relation between a left hand and a right hand. Therefore the number of the internal structures is 160 = 2³ × 20, by taking the possible choice of the three-fold axis for the main axis into account. The numbers of A₆ and O₆ are both ten as

An O₆ is fixed in a cage as a partition and seven O₆'s are on the surface of a K₃₀. Any of the seven can be chosen as the partition. Therefore the number of the internal structures of a C is 10816 = [2³ × (1 + 3 + 3²)]² = 104² (The value 164² in the original paper is incorrect), by taking the possible choice of the three-fold axis for the main axis into account.

Note that Penrose transformation can be defined for any choice of these possibilities for F₂₀ and C. Further more, the highest symmetry is realized by giving three-fold symmetry to the internal structure of C in A₆*. There are four cases for such configurations, among which two have inversion symmetry at the centre of C and chirality and other two have reflection symmetry between two K₃₀'s of C. As for the choice for F₂₀'s and the chirality, some aspects from higher point of view is necessary as shown later in Chapter 5.