3. The primitive Projection Method 

Projection of a regular lattice of higher dimension into the real physical space of lower dimension is a very useful method to construct a model structure of general lattices unifying usual lattices and quasilattices; the well-known examples are one-dimensional quasilattice as a projection from two-dimension, 2D-PT from five-dimension and three-dimensional quasilattice from six-dimension. Calculations with computer are easily carried out by this method. Sometimes they are done without recognizing the features of the structure. It is, at the same time, a merit and demerit of this method. The procedure to construct a d-dimensional general lattices from a D-dimensional simple lattice can be summarized as follows. 

(1) Regard the D-dimensional original space W as direct sum of d-dimensional physical space W and d (= D - d ³ D/2)-dimensional pseudospace W = W₁ Å W₂. 

(2) Select the lattice points in W whose projection to the pseudospace lies in a certain region ^GC (the gazing crystal) which is defined as a projection of a D-dimensional hypercube. A general lattice G is the projection of them to the physical space. A quasilattice Q is a general lattice without periodicity. Let L be a D-dimensional vector from a lattice point to another lattice point. 

                                    L= L₁Å  L₂ ,   L_i   Î W_i   (i= 1 and 2)                                           (3.1) 

                                       G ={L₁; L₂  Π ^GC   Ì   W₂}  Π   W₁                                           (3.2) 

(3) Now, the projection from W to W₁can be regarded as a mapping from W₂ to W₁. When the mapping is an injection, the general lattice is a quasicrystal. 

Therefore, all the informations of an infinite quasiperiodic arrangement in the ^GC in the whole universe of physical space W₁ are concentrated in a small region ^GC in the pseudospace W₂. A whole network is collapsed into the small area ^GC, keeping its connectivity. Some fortune teller decodes she informations contained in the inside of ^GC into everything in physical space by gazing it. It is the reason why the area is referred to as the gazing crystal (^GC). In the projection from 2-dimension to one-­dimension, the point sequence {y_n} in ^GC is very similar to a chaotic behaviour in point sequences in a dynamical mapping or a Poincare mapping of a dynamical process. General cases can be regarded as chaotic motion in multi-dimensional or branching time from this point of view. 

From a conceptual point of view, the method is summarized as 
 
 

A d-dimensional general lattice is constructed by the projection of the D-dimensional lattice points whose Voronoi cells cross the d­dimensional space.

and 
 

Any information of the generalized crystal in the physical space, of which a quasicrystals included as special cases, contained in the gazing crystal in the pseudo space. A whole generalized lattice is collapsed into the gazing crystal keeping its connectivity.

It is noted that a gazing crystal is a compound of _DC_D-d  unit cells because it is a projection of  D-dimensional hypercube into (D-d)-dimensional space. It rigorously leads to an important conclusion that there are no simple repeat of any pair of the same unit cells in the primitive projection model. It is the most characteristic feature of the model. 

Some examples of the gazing crystals are listed in Table 7. 

Table 7: Examples of Gazing Crystals 
 

In Example 2, the primitive projection model, being (..LMLSMLMLSMLLMSLMLMSL..) has no self-similarity. There are several transformations leading to a self-similar arrangement. For example, L ® LM, M ® LS, S ® M (LMLSLMMLMLSLSLMLSLMM..) and L ® LMS, M ® LM, S ® L (LMSLMLLMSLMLMSLMSLML..). These sequences can be regarded as either of symbol sequence or point sequence of relative separation lengths L : M : S = a : b : g (in the foot note of Table 7) = a : b : 1 (in Appendix B). 

In Fig.1, the gazing crystal of the third example (4-dim. to 2-dim.) is shown. The closer the point in the gazing crystal, the higher the symmetry or the wider the ruling area of the local symmetry of the corresponding point in the physical space. It is noted that the primitive projection method exactly corresponds to the transformation method in this case. The corresponding Penrose transformation is shown in Fig.2.

4. Three-Dimensional Tiling Obtained by the Primitive Projection 

Suppose a 6-dimensional simple hypercubic lattice where the lattice vectors are {u_k; k = 1,2,...,6}. Introduce another orthonormal set (x, y, z, X, Y, Z) in the six­dimensional original space W so that they are expressed in terms of u_k's as. 

and 

The space is regarded as the direct sum of three-dimensional physical space W₁ spanned by (x, y, z) and three-dimensional pseudospace W₂ spanned by (X, Y, Z). Six renormalized quasilattice vectors are the renormalized physical-space­components of six bases of the original space. They are 

in the Cartesian coordinate system (x, y, z) above. Here a , b, g, x, h and z are given in Eq.(2.1). The arrangement of {±V_k; k = 1,2,...,6} has icosahedral symmetry. The renormalized pseudospace-components of six bases of the original space are three-dimensional vectors. 

in the Cartesian coordinate system (X, Y, Z) above. The arrangement of twelve {±W_k; k = 1,2,...,6} has icosahedral symmeetry too. The inner products among {V_k; k = 1,2,...,6} and among {W_k; k = 1,2,...,6} are summarized respectively in a matrix form.

and 

Being a three-dimensional projection of a six-dimensional hypercubic domain, ^GC is a rhombic triacontahedron K₃₀ whose edges are either of six unit vectors W_k's (see Chapter 2). Having icosahedral symmetry, K₃₀ is rather spherical. The radii along three kinds of symmetry axes and to the edges are tabulated in Table 8. 

Table 8 Several Radii r of a Kepler 30-hedron K₃₀ of Edge length 1 

For a while, a tiling is regarded as a network composed of only unit vectors and its connectivity is mainly investigated. Never mind here that the body-diagonal length of O among extremely short ~0.963. A neighbour in this aspects may be referred to as a vectorial neighbour. 

The variety of the vectorial environment of a point in physical space is nine types so long as the network structure concerns; the number of vectorial neighbours is either of 4, 5, 6 (two types), 7, 8, 9, 10 or 12. It is convenient to classify the 12-neighbour case into two cases as to be mentioned below. Then there are altogether ten types. These ten local structures as vectorial environment of a point have their associate domains in ^GC which are respectively denoted by 4, 5, 6₃, 6₅, 7, 8, 9, 10, 12₀ and l2* in Fig.3. 

It is a remarkable fact from which a kind of self-similarity of factor t³ follows. Suppose Q and Q + W_k are both in ^GC and their associate points P and P + V_k are both quasilattice points in the physical space. Let Q*= -t³Q, regarding Q as a vector from the centre of ^GC. Then Q* lies in the region 12* and Q* + W_k* = -t³(Q + W_k) also lies in 12*. Correspondingly P* and P* + V_k* are both quasilattice points. Both points Q* + W_k and Q* + W_k* -W_k lie in domain 6₅, for any k if Q* lies in 12*. The vector S_l^(k)W_k corresponds to the symmetry axis of F₂₀ in physical space. Then the tiling thus obtained by the primitive projection method has the same skeleton structure as that by the Penrose transformation mentioned in Chapter 2. Note that the primitive projection method is deterministic and there are some freedom in the transformation method. This difference is discussed in the next subsection. 
 
 

INSIDE OUT AND OUTSIDE IN 

As mentioned above, the points in 12* sketch a rough structure in physical space in a scale of t³. More detailed structure in physical space is reflected in other region of  ^GC. Generally speaking, a rough structure are reflected in the central parts and the details or their internal structure are in the outer parts of  ^GC. 

The correspondence between a point in physical space and a point in pseudospace can be regarded as a mapping, since two spaces are correlated as the projection of a 6-dimensioiial vectors. A golden rhombus specified by two V's is mapped into a golden rhombus specified by the corresponding two W's where the angle of two V's and that of two W's are supplementary. 
 

A₆ « O₆

An acute rhombohedron A₆ specified by three V's is mapped into an O₆ specified by the corresponding three W's where the principal diagonal is multiplied by t^-3. An obtuse rhombohedron O₆ specified by three V's is mapped into an A₆ specified by the corresponding three W's where the principal diagonal is is multiplied by  -t³. 

Though each of rhombohedra is mapped into a rhombohedron, thus mapped rhombohedra are mutually overlapping in pseudospace. Then the mapping of a compound of some rhombohedra is complicated. Let us investigate the mapping of an F₂₀ and K₃₀.
 

F₂₀ in physical space ® a 30-hedron with bottle neeck in pseudospace (Fig.4(a)) 

In Fig.4 points are connected so that the whole configuration of these points may be clearly seen than their point to point correspondence. As a connectivity of unit vectors, the points which correspond to the surface points in physical space construct a collapsed rhmbohedral 20-hedron with unit side length. As a three- dimensional point configuration, they are regardes as 22 vertices of a 30-hedron with bottleneck, among whose 30 faces 10 are equilateral triangles, the other 10 are another kind of equilateral triangles and the last 10 are trapezia.

A set of four internal points in physical space is mapped into the outside of the polyhedron ((1)-(4)) in Fig.4(a)) They must be in the gazing crystal. This requirement is rather strong. Thus the internal configuration of F₂₀ in physical space is uniquely determined by the position of the polyhedron in tne gazing crystal in pseudospace.
 

K₂₀ in physical space ® a concave 60-hedron with icosahedral symmetry in pseudospace (Fig.4(b))

The attitude in drawing Fig.4(b) is the same as in Fig.4(a) except that along and perpendicular to a 3-fold axis instead of a 5-fold axis. The 32 surface points of K₃₀ in physical space are aranged with icosahedral symmetry. Then they are mapped so that the symmetry is kept. However their radii get smaller as given in Table 9. 

Table 9 Several Radii of a Concave 60-hedron in Pseudospace 

The whole distribution of these 32 points are regarded as a 60-hedron which is constructed by putting a triangular pyramid on each of 20 faces of an icosahedron whose circumradius is r⁵. As for the 10 internal points in physical space, they are mapped to the outside of the polygon as shown as (1)-(10) in Fig.4(b). Here an internal configuration with trigonal symmetry is chosen. 
 

C ® an overlapping compound of two concave 60-hedron  (Fig.5) 

C is an overlapping of two K₃₀'s where an O₆ ((2)-(3)), whose points are all on the surface of eithier of two K₆₀'s is shared in common. Among the points of two A₆'s (1)-(2) and (3)-(4), those but (1), (2), (3), and (4) are internal of either of the K₃₀'s. In the case of the maximum symmetry shown in Fig.5, the diameter of the sphere that contains all the mapped point is the maximum. In this case, some of the mapped points are inevitably on the edge of the gazing crystal. 

Though all of the points (1)-(10) are outside of both 60-hedra ((1)-(3) and (2)-(4)) in the case of the maximum symmetry, the following should be noted. Points (2), (5), (6) and (7) are the internal of K₃₀ ((2)-(4)) but not of K₃₀ ((2)-(4)). Points (3), (8), (9) and (10) are outside of K₃₀ ((2)-(4)) but not of K₃₀ ((1)-(3)). Therefore in less symmetrical cases, two O₆'s are not necessarily at the both ends of the main axis (2)-(3) of the connected 60-hedra and point (3), for example, can be inside of 60­hedron ((2)-(4)). By this, the diameter is smaller and all the mapped points can be inside the gazing crystal. 

Fig. 5

In the gazing crystal, an available arrangement of two O₆'s and points (5)-(10) is uniquely determined for any position of the connected 60-hedron as easily seen from that the projection is deterministic. The arrangement of the internal points in a cage is uniquely determined according to the circumstance of a cage so that no simple repeat of the same unit cells. This is the most striking feature of the model. Its local symmetry is not so high as can be seen from the fact that A₆* never be trigonal, but it is rather uniform. 

From these investigations, the following can be concluded 
 

In the primitive projection model, the internal points of all the cages arange in rather bad symmetry. A cage with the internal points can not have trigonal symmetry.

5. Symmetry in the Transformation Model 

In the transformation model, the hierarchy of symmetry can be designed by making use of its freedom. Start with a flower dodecahedron Fl₆₀ which is the only configuration with full icosahedral symmetry among all the compound of A₆'s and O₆'s. Apply the Penrose transformation to this. The original Fl₆₀ is kept unchanged and 20 F₂₀'s are put on the 20 flower faces. Fivefold symmetry and therefore icosahedral symmetry can not be kept since a F₂₀ does not have 5-fold symmetry as easily seen from the fact that it has four internal points. However, there are some configurations with symmetry belonging to the symmetry group T_h and T which are both subgroup of the group I_h. For the cage parts, trigonal symmetry can be kept. The same cage configuration can be always used. In such a case, system has a kind of chirality. By defining some rank for the centres of Fl₆₀'s, the hirarchical symmetry of T_h or T is realizable.