SYMMETRY OF THREEDIMENSIONAL QUASICRYSTALS Tohru Ogawa Institute of Applied Physics, University
of Tsukuba Sakura, Ibaraki 305 Japan
The relation between the threedimensional
quasicrystal models, the one by Penrose transformation method which
was found by the author in 1985 and the other by the socalled 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, twodimensional case, where the Penrose transformation is uniquely defined and coincides with the socalled projection method [3], is rather exceptional when compared with threedimensional 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 Ddimensional original space to ddimensional physical space as a mapping from a finite region referred to as the gazing crystal in (Dd)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 fivefold 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 Fl_{60}. 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,
threedimensional 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 threedimensional configuration by the primitive projection
method is studied in detail. Possible symmetry in the projection model
is investigated in Chapter 5. The value t^{3}
of the similarity factor and some similar numbers to golden number are
discussed respectively in Appendix A and B.
The interpretation of threedimensional 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 = _{6}C_{2} 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 = _{6}C_{3}
rhombohedra whose edges are these six vectors. They are either of two golden
rhombohedra, acute A_{6} and obtuse O_{6}. (Tables 2,
3
and 4). They are the quasiunitcells of the present
model. Three kinds of zonohedra, B_{12}, F_{20} and
K_{30}, are constructed with A_{6} and O_{6} (Table
5). Twenty A_{6}'s organize a concave 60hedron Fl_{60}
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 fivepetalous flower (see
Table
6).
Let us construct an A_{6}* and an O_{6}* (respectively a big A_{6} and a big O_{6} of edge length t^{3}) only with A_{6}'s and O_{6}'s of edge length 1. If it can be done, an A** and an O_{6}** of edge lenght t^{9} can be constructed in a similar way and so on. Here, a Penrose transformation is defined as a transformation from a set of spacefilling figures to their enlarged figures consisting only of the formers. Threedimensional Penrose Transformation
in the most general version is simply stated as follows:
More exactly, into its centre and of course the Fl_{60} should be placed so that directions of all edges coincide with the quasilattice axes. It is noted that two Fl_{60}'s at both principal vertices of an O_{6}* are connected with each other sharing an A_{6} in common and six O_{6}'s are placed between the two. Though there are many unifilled spaces, in an A_{6}* and an O_{6}*, 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 F_{20}, so long as the tiles are restricted to A_{6} and O_{6}. The inside of F_{20} is left undecided in the skeleton structure. Now, the skeleton of an O_{6}* is completely decided. In an A_{6}*, there is a space C, which looks like a cage, can contain 20 A_{6}'s and 19 O_{6}'s (see Table 6). The shape of the cage is regarded as two Kepler tetrakaidecahedron K_{30} sharing an O_{6} in common. The O_{6} is inevitably placed there, dividing the cage into two parts. Leaving the structure of other parts of the cage undecided, the skeleton of an A_{6}* is thus completely determined. The A_{6}* and O_{6}* are numerically related with A_{6} and O_{6} as
A_{6}* = C+20A_{6 }+ [6×(1/5)+6×(3/10)] F_{20
}=
55A_{6} + 34O_{6} ,
(2.3)
where 20 A_{6} and 19 A_{6} correspond to an Fl_{60} respectively in A_{6}* and O_{6}*. There, the facts
C = 20A_{6} + 19O_{6} ,
(2.5)
are also used. The matrix which composed of the coefficients of this equation has t^{9} 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 selfsimilarity of factor t^{3} in linear dimension and t^{9} 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 F_{20} and C. As F_{20} has a fivefold axis in its outlook. However, it contains only four points in its inside and it can not have real fivefold 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 A_{6} and O_{6} are both five in 10 ways. As mentioned above, a C is a compound of two K_{30}'s sharing an O_{6} in common. A K_{30}, 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 threefold axis among ten may be regarded as the main axis of the internal structure. On the main axis, an A_{6} and an O_{6} are placed so that their principal axes coincide. Now, seven points are already fixed. The part near the A_{6}side end of the main axis is uniquely filled with three A_{6}'s and three O_{6}'s. There are eight (=2^{3}) 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 threefold 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 = 2^{3} × 20, by taking the possible choice of the threefold axis for the main axis into account. The numbers of A_{6} and O_{6} are both ten as K_{30} = 10 A_{6} + 10 O_{6} (in 160 ways) (2.8) An O_{6} is fixed in a cage as a partition and seven O_{6}'s are on the surface of a K_{30}. Any of the seven can be chosen as the partition. Therefore the number of the internal structures of a C is 10816 = [2^{3} × (1 + 3 + 3^{2})]^{2} = 104^{2} (The value 164^{2} in the original paper is incorrect), by taking the possible choice of the threefold axis for the main axis into account. Note that Penrose transformation can be
defined for any choice of these possibilities for F_{20} and C.
Further more, the highest symmetry is realized by giving threefold symmetry
to the internal structure of C in A_{6}*. 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 K_{30}'s of C. As for the choice for F_{20}'s and the
chirality, some aspects from higher point of view is necessary as shown
later in Chapter 5.
