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 wellknown examples are onedimensional quasilattice as a projection from twodimension, 2DPT from fivedimension and threedimensional quasilattice from sixdimension. 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 ddimensional general lattices from a Ddimensional simple lattice can be summarized as follows. (1) Regard the Ddimensional original space W as direct sum of ddimensional physical space W and d (= D  d ³ D/2)dimensional pseudospace W = W_{1 }Å W_{2}. (2) Select the lattice points in W whose projection to the pseudospace lies in a certain region ^{G}C (the gazing crystal) which is defined as a projection of a Ddimensional 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 Ddimensional vector from a lattice point to another lattice point. L= L_{1}Å L_{2} , L_{i} Î W_{i} (i= 1 and 2) (3.1) G ={L_{1}; L_{2} Î ^{G}C Ì W_{2}} Î W_{1} (3.2) (3) Now, the projection from W to W_{1}can be regarded as a mapping from W_{2 }to W_{1}. When the mapping is an injection, the general lattice is a quasicrystal. Therefore, all the informations of an infinite quasiperiodic arrangement in the ^{G}C in the whole universe of physical space W_{1} are concentrated in a small region ^{G}C in the pseudospace W_{2}. A whole network is collapsed into the small area ^{G}C, keeping its connectivity. Some fortune teller decodes she informations contained in the inside of ^{G}C into everything in physical space by gazing it. It is the reason why the area is referred to as the gazing crystal (^{G}C). In the projection from 2dimension to onedimension, the point sequence {y_{n}} in ^{G}C 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 multidimensional or branching time from this point of view. From a conceptual point of view, the method
is summarized as
and
It is noted that a gazing crystal is a compound of _{D}C_{Dd} unit cells because it is a projection of Ddimensional hypercube into (Dd)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 selfsimilarity. There are several transformations leading to a selfsimilar 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 (4dim. to 2dim.) 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.
Fig.1 The Gazing Crystal
of Example 3 in Table7.
Fig.2 The corresponding
Penrose Transformation.
Suppose a 6dimensional 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 threedimensional physical space W_{1 }spanned by (x, y, z) and threedimensional pseudospace W_{2 }spanned by (X, Y, Z). Six renormalized quasilattice vectors are the renormalized physicalspacecomponents 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 pseudospacecomponents of six bases of the original space are threedimensional 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 threedimensional projection of a sixdimensional hypercubic domain, ^{G}C is a rhombic triacontahedron K_{30} whose edges are either of six unit vectors W_{k}'s (see Chapter 2). Having icosahedral symmetry, K_{30} 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 30hedron K_{30} 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 bodydiagonal 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 12neighbour 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 ^{G}C which are respectively denoted by 4, 5, 6_{3}, 6_{5}, 7, 8, 9, 10, 12_{0} and l2* in Fig.3.
Fig.3 The gazing crystal
K_{30} for 6dim, to 3dim. projection. Only a quater
SELFSIMILARITY The domain 12* corresponding to the central point of Fl_{60} is exactly similar of factor t^{}^{3 }to the whole ^{G}C with the centre as the centre ot similarity. Suppose a similar domain of factor t^{}^{2}. Domain 12_{0} is obtained by removing the domain 12* from it. The points in the domain correspond to the point in physical space with 12 vectorial neighbours but not the centre of Fl_{60}. In other words, not only A_{6}'s but some O_{6}'s share the point. The similar triacontahedral domain 12_{0 }of side length t^{}^{2 }outside of 12* is for the points with twelve neighbours, which do not correspond to Fl_{60} but the compound of 2n O_{6}'s and (20 n) A_{6}'s (n>0). Let S_{l }^{(k)}V_{l} be the summation over five unit vectors among twelve vectors {± V_{k}; k = 1,2,...,6} whose V_{k}component is t = 5^{1/2}. Then the following relations hold for the mutually associated two vectors in the physical space and the pseudospace It is a remarkable fact from which a kind
of selfsimilarity of factor t^{3} follows. Suppose Q and
Q
+ W_{k} are both in ^{G}C and their associate
points P and P + V_{k} are both quasilattice
points in the physical space. Let Q*= t^{3}Q,
regarding
Q as a vector from the centre of ^{G}C.
Then Q* lies in the region 12* and Q* + W_{k}*
= t^{3}(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 6_{5}, for any k if Q* lies in 12*.
The vector S_{l}^{(k)}W_{k
}corresponds
to the symmetry axis of F_{20} 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 t^{3}. More detailed structure in physical space is reflected in other region of ^{G}C. Generally speaking, a rough structure are reflected in the central parts and the details or their internal structure are in the outer parts of ^{G}C. 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 6dimensioiial 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.
A_{6 }« O_{6} An acute rhombohedron A_{6} specified by three V's is mapped into an O_{6} specified by the corresponding three W's where the principal diagonal is multiplied by t^{}^{3}. An obtuse rhombohedron O_{6} specified by three V's is mapped into an A_{6} specified by the corresponding three W's where the principal diagonal is is multiplied by t^{3}. 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 F_{20} and K_{30}.
F_{20} in physical space ® a 30hedron 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 20hedron with unit side length. As a three dimensional point configuration, they are regardes as 22 vertices of a 30hedron 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 F_{20} in physical space is
uniquely determined by the position of the polyhedron in tne gazing crystal
in pseudospace.
K_{20} in physical space ® a concave 60hedron 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 3fold axis instead of a 5fold axis. The 32 surface points of K_{30} 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 60hedron in Pseudospace
in the physical space. (1)(4) correspond to the internal points in the physical space. (b) A concave 60hedron in the pseudospace,which corresponds to F_{20} in the physical space. (1)  (10) correspond to the internal points in the physical space. The whole distribution of these 32 points
are regarded as a 60hedron which is constructed by putting a triangular
pyramid on each of 20 faces of an icosahedron whose circumradius is r^{5}.
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 60hedron (Fig.5) C is an overlapping of two K_{30}'s where an O_{6} ((2)(3)), whose points are all on the surface of eithier of two K_{60}'s is shared in common. Among the points of two A_{6}'s (1)(2) and (3)(4), those but (1), (2), (3), and (4) are internal of either of the K_{30}'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 60hedra ((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 K_{30} ((2)(4)) but not of K_{30} ((2)(4)). Points (3), (8), (9) and (10) are outside of K_{30} ((2)(4)) but not of K_{30} ((1)(3)). Therefore in less symmetrical cases, two O_{6}'s are not necessarily at the both ends of the main axis (2)(3) of the connected 60hedra 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 O_{6}'s and points (5)(10) is uniquely determined for any position of the connected 60hedron 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 A_{6}* never be trigonal, but it is rather uniform. From these investigations, the following
can be concluded
In the transformation model, the hierarchy
of symmetry can be designed by making use of its freedom. Start with a
flower dodecahedron Fl_{60} which is the only configuration with
full icosahedral symmetry among all the compound of A_{6}'s and
O_{6}'s. Apply the Penrose transformation to this. The original
Fl_{60} is kept unchanged and 20 F_{20}'s are put on the
20 flower faces. Fivefold symmetry and therefore icosahedral symmetry can
not be kept since a F_{20} does not have 5fold 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 Fl_{60}'s, the hirarchical
symmetry of
T_{h} or T is realizable.
