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Appendix A.
Golden Isozonohedra: A6, O6, B12, F20, K30

The definitions of zonohedron and golden isozonohedron are given in the text. Five golden isozonohedra are A6, O6, B12, F20 and K30.

These are two golden rhombohedra A6 and O6. The angle of two edges at a principal vertex is acute in A6 and obtuse in O6.

Golden dodecahedron B12, discovered by Bilinski, can be regarded as the locus of an A6 or that of O6. It is composed of 2 A6's and 2 O6's. Among 4 quasi-lattice-points altogether, three are on the surface and the other one inside. The one inside is chosen of 2 equivalent possibilities. The whole configuration of B12 is four dimensional in the sense that only four kinds of nearest neighbour vectors are used among allowed six.

Golden icosahedron F20, discovered by Fedrov, can be regarded as the locus of a B12. It has a five-fold symmetry axis. It is composed of 5 A6's and 5 O6's. Among 10 quasi-lattice-points altogether, six are on the surface and the other four inside. The four inside are chosen of 10 equivalent possibilities. The whole configuration of F20 is five-dimensional 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 K30, discovered by Kepler, can be regarded as the locus of an F20. It has icosahedral symmetry. It is composed of 10 A6's and 10 O6's. Among 20 quasi-lattice-points 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 A6's and O6's are contained in three golden isozonohedra B12, F20, and K30.
 

Appendix B.
The Coordinates in the Six-Integer Representation

The configurations with bond orientational perfect long range order of icosahedral symmetry can be expressed in a form of six-integer-set. It is useful to introduce proper representation suitable for the purpose. For example, sometime trigonal and sometime pentagonal.
 

(1) The trigonal representation

The six quasi-bases, a-b-g-x-h-z 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 quasi-bases, 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 ].
 

(3) The coordinates

The coordinates of quasi-lattice-points ate listed in either of these representations.

In these three choices, if all of three formers or all of three latters are chosen, then the internal configuration has trigonal symmetry. There are twenty possibilities in the choice of the trigonal axis. In other six cases among 23 choices, there are three-times more possibilities for the direction of axis.

Therefore, there are altogether 23 × 20 = 160 configurations for ten inner quasi-lattice-points for a fixed surface configuration. It is useful for the following discussion to note that seven trigonal surface quasi-lattice-points among twenty are occupied by the principal vertices of the inner O6's and one among the seven can be on a trigonal axis.

In the case of a triacontahedral cage in the skeleton of an expanded A6, an O6, which separate the cage into two parts, is already fixed. Then, only nine inner quasi-lattice-points have freedom and the number of configurations is reduced to as 23 × (1 + 3 + 3 × 3) = 104.
 


 


 

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