Appendix
A.
The definitions of zonohedron and golden isozonohedron are given in the text. Five golden isozonohedra are A_{6}, O_{6}, B_{12}, F_{20} and K_{30}. These are two golden rhombohedra A_{6} and O_{6}. The angle of two edges at a principal vertex is acute in A_{6} and obtuse in O_{6}. Golden dodecahedron B_{12}, discovered by Bilinski, can be regarded as the locus of an A_{6} or that of O_{6}. It is composed of 2 A_{6}'s and 2 O_{6}'s. Among 4 quasilatticepoints altogether, three are on the surface and the other one inside. The one inside is chosen of 2 equivalent possibilities. The whole configuration of B_{12} is four dimensional in the sense that only four kinds of nearest neighbour vectors are used among allowed six. Golden icosahedron F_{20}, discovered by Fedrov, can be regarded as the locus of a B_{12}. It has a fivefold symmetry axis. It is composed of 5 A_{6}'s and 5 O_{6}'s. Among 10 quasilatticepoints altogether, six are on the surface and the other four inside. The four inside are chosen of 10 equivalent possibilities. The whole configuration of F_{20} is fivedimensional in the sense that only five kinds of nearest neighbour vectors are used among allowed six and the one parallel to the pentagonal axis is excluded. Golden triacontahedron K_{30}, discovered by Kepler, can be regarded as the locus of an F_{20}. It has icosahedral symmetry. It is composed of 10 A_{6}'s and 10 O_{6}'s. Among 20 quasilatticepoints altogether, ten are on the surface and th'e other ten inside. The ten inside are chosen of 400 equivalent possibilities. It is the minimum convex region where all of the allowed six nearest neighbour vectors appear. It
is noted that the same number of A_{6}'s and O_{6}'s are
contained in three golden isozonohedra B_{12}, F_{20},
and K_{30}.
Appendix
B.
The
configurations with bond orientational perfect long range order of icosahedral
symmetry can be expressed in a form of sixintegerset. It is useful to
introduce proper representation suitable for the purpose. For example,
sometime trigonal and sometime pentagonal.
(1) The trigonal representation The six quasibases, abgxhz are taken so that where t = cos q =1 /Ö5. For example, they can be taken as follows; where p = Ö((1 + t )/2 and q = Ö((1 t )/2). There are following relations among them, A
representation by these bases is expressed as ( i, j, k,
l,
m,
n
).
(2) The pentagonal representation The six quasibases, A B C D E Z are taken so that ( A, B )=( B, C )=( C, D )=( D, E )=( E, A )=( A, Z )=( B, Z )=( C, Z )=( D, Z )=( E, Z ) = t ( A, C )=( B, D )=( C, E )=( D, A )=( E, B ) = t For example, they can be taken as follows; A = a, B = b, C = x, D = z, E = h, and Z = g. A
representation by these bases is expressed as [ i, j, k,
l,
m,
n
].
The coordinates of quasilatticepoints ate listed in either of these representations.
