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Front page > Geometry based on Silver Ratio and Golden Ratio > Sacred Geometry: Formulation of Rhombic Polyhedron with 132 Faces (Rhombhectotriadiohedron)
Geometry based on Silver Ratio and Golden Ratio

Sacred Geometry: Formulation of Rhombic Polyhedron with 132 Faces (Rhombhectotriadiohedron)

Figure 1 Rhombhectotriadio-hedron (T-type)

As mentioned in the body text, the rhombhectotriadiohedron (i.e. rhombic polyhedron with 132 faces) is of two types (T and C) and each one is composed of three kinds of rhombus.  Please Refer to Figure 1 and Figure 2.  My current task is to calculate the angles of these rhombuses by myself.

Mr. Yasunari Watanabe who is an excellent researcher on crystallography is well-versed in this geometric field.  His paper “Derivation of Some Equilateral Zonohedra and

Figure 2 Rhombhectotriadio-hedron (C-type)

Star Zonohedra” is considered to be highly instrumental in figuring out these angles.  It describes the way to build the equilateral-zonogon, star-zonogon, equilateral zonohedra and star zonohedra.  First of all, basic plane figures or solids to be used for such formulation are specifically selected.  While the basic plane figures or solids to be used vary depending on the equilateral-zonogon, star-zonogon, equilateral zonohedra and star zonohedra to be built, they are either regular polygon (2D), regular polyhedra, quasi-regular polyhedra, or compound of regular polyhedra and quasi-regular polyhedra (3D).  Then, base vectors are successively drawn from their center to their respective vertices.

The basic solid to be used to build the rhombhectotriadiohedron of T-type are either the truncated tetrahedron or the small rhombicuboctahedron.  On the other hand, the one for the C-type is the truncated cube.  In case of the truncated tetrahedron, a whole solid is to be used.  In case of the small rhombicuboctahedron and the  truncated cube, one half of solid is to be used.  Please refer to Figure 3.  The number of base vectors in these cases is 12, which is common to T-type and C-type.

Figure 3 Formulation of rhombhectotriadiohedron

Mr. Watanabe kindly provided me with the following additional advices.

The rhombohedra or equilateral hexahedra can be created from three base vectors selected.  The quantity of these rhombohedra or equilateral hexahedra thus created is equal to the number of combinations selecting 3 from 12.  They include duplications and, therefore, we select only those independent from each other.  Then, we make their paper models and put them together successively in accordance with the configuration of base vectors.

The foregoing is understood to be a method to formulate the rhombhectotriadiohedron (T-type and C-type).  If such formulation is successful, the angle of rhombuses in question will be easily calculated.  This is my current homework yet to be done.  If it is successfully conducted, a result of it will be uploaded shortly thereafter.

Reference:

Mathematical paper ”Derivation of Some Equilateral Zonohedra and Star Zonohedra”,  Y. Watanabe and T. Betsumiya, Research of Pattern Formation

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