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Note added in proof
Two numerical values 104
appearing in Section 4 and 160 appearing in Section
4 and Appendix B (3) are the corrected ones
respectively from the original incorrect 164 and 400, which were reported
in the talk at the Seminar and written in Ref. .
In the text, the Penrose
transformation is described as an expansion or structuration of two rhombohedra
A6 and O6 so that the logical structure map be clearly seen. The obtained
structure may be more clearly seen by noting that a point is transformed
into a flower dodecahedron Fl60 consisting of 20 A6's (exactly speaking,
a concave 60-hedron which looks like a dodecahedron each of whose 12 faces
is a five-petalous flower) . In this point of view,
the tiling elements should not be regarded as A6 and O6 but as Fl60, F20,
C = 2K30-O6 and O6.
Though the concept of
self-similarity is useful, it may be better regarding it as a result of
some local rule. Self-similarity, as a basic principle of a growing process,
is a little curious: The situation may remind one a horoscope. If self-similarity
is really the basic rule, an atom may have to know the position of its
own star in deciding its position to attach. Therefore, it is expected
that the basic rule is replaced by some local rule which leads the equivalent
In the discussion, Bennema
gave a comment about the higher-dimensional crystallography. After that
a very similar three-dimensional structure to the present model has been
obtained as the projection of certain selected points of a six-dimensional
simple cubic lattice to the three dimensional space .
The relation between the two models have not been known until the author
figured it out .
Some two-dimensional Penrose-like
tilings are obtained by similar hierarchical transformations. Some of them
have pentagonal symmetry and some others have octahedral symmetry .