Appendix Appendix
A
An interpretation is given on the fact that the similarity factor t^{}^{3} is the minimum value for three dimensional Penrose tiling with icosahedral symmetry and selfsimilarity. Six quasibases are a, b, g, x, h, z introduced in Chapter 2. The vectors that can be expressed as a summation of these vectors are restricted as follows. Along a 5fold axis (parallel to a), it is in the form and the length is Along a 2fold axis (g + z ) whose length is Along a 3fold axis (x + h + z ) whose length is In Table 10, the ratio
of distance to the corresponding minimum value are given for some small
values of (m, n). The ratios 5t and t
can not be realized for 3fold axis. On contrary,
the ratio t^{3} can
be realized for all symmetry axes as shown in the last column in Table
10. In order to expand an O_{6}, therefore, the value is the
first candidate of the similarity factor from numerical point of view.
As a matter of fact, the suitable arrangement is actually found as shown
in Chapter 2.
Table 10 The possible length along
symmetry axis and the ratio to the corresponding minimum value
Let the lengths of two kinds of diagonals of a regular heptagon be a and b where a>b. They are given by where q = p/7. They are respectively the biggest root of cubic equations and satisfy the following relations and It is noted that the relations are natural extension of that for a regular pentagon Two sequences and associate with them as Fibonacci sequence
does with regular pentagon. Similar relations hold generally for the lengths
of diagonals of regular polygons. The matrices can be regarded as those
for onedimensional wave propagation.
