Complementary Relationship between the Golden Ratio and the Silver Ratio in the Three-Dimensional Space & “Golden Transformation” and “Silver Transformation” applied to Duals of Semi-regular Convex Polyhedra September 1, 2010 Hiroaki Kimpara 1. Preface The Golden Ratio and the Silver Ratio 1 : serve as the fundamental ratio in the shape of objects and they have so far been considered independent of each other. However, the author has found out, through application of the “Golden Transformation” of rhombic dodecahedron and the “Silver Transformation” of rhombic triacontahedron, that these two ratios are closely related and complementary to each other in the 3-demensional space. Duals of semi-regular polyhedra are classified into 5 patterns. There exist 2 types of polyhedra in each of 5 patterns. In case of the rhombic polyhedra, they are the dodecahedron and the triacontahedron. In case of the trapezoidal polyhedral, they are the icosidodecahedron and the hexecontahedron. In case of the pentagonal polyhedra, they are icositetrahedron and the hexecontahedron. Of these 2 types, the one with the lower number of faces is based on the Silver Ratio and the one with the higher number of faces is based on the Golden Ratio. The “Golden Transformation” practically means to replace each face of: (1) the rhombic dodecahedron with the face of the rhombic triacontahedron, (2) trapezoidal icosidodecahedron with the face of the trapezoidal hexecontahedron, and (3) the pentagonal icositetrahedron with the face of the pentagonal hexecontahedron. These replacing faces are all partitioned into triangles. Similarly, the “Silver Transformation” means to replace each face of: (1) the rhombic triacontahedron with the face of the rhombic dodecahedron, (2) trapezoidal hexecontahedron with the face of the trapezoidal icosidodecahedron, and (3) the pentagonal hexecontahedron with the face of the pentagonal icositetrahedron. Again, the replacing faces are all divided into triangles. While the aforementioned rhombic polyhedra both belong to Pattern Ⅱ, the trapezoidal polyhedra belong to Pattern Ⅳ. On the other hand, the pentagonal polyhedra belong to Pattern Ⅴ. Accordingly, the author conducted “Golden Transformation” and “Silver Transformation” of the trapezoidal polyhedra and the pentagonal polyhedra, too, and looked into their results. It is to be noted that the “Golden Transformation” and the “Silver Transformation” cannot be applied to duals belonging to Pattern Ⅰ and Ⅲ because faces of these polyhedra are all triangles. The outcome of this study is presented in this paper. 1 2. Golden Ratio and Silver Ratio found in the Natural and Human Worlds The Golden Ratio and Silver Ratio are found in various aspects of the natural and human worlds. Their actual examples are shown in Table 1. (See References [1] and [2].) This comparative table suggests that the Silver Ratio serves as the basic and primordial ratio not only in the visible macro world but also in the microscopic realms invisible to the naked eye. 3. Polyhedra based on the Silver Ratio and Those based on the Golden Ratio Among regular polyhedra, the tetrahedron, cube, and octahedron are based on the Silver Ratio, whereas the dodecahedron and icosahedron are predicated on the Golden Ratio. Also, the cub-octahedron and rhombic dodecahedron mentioned in Table 1 are based on the Silver Ratio. The reasons are mentioned in Table 2 below. Reasons Regular The dihedral angle is 70°31’43”. This is the same as the acute angle of the silver tetrahedron rhombus with a ratio of the short diagonal to the long diagonal being equal to the Silver Ratio. The obtuse angle of the isosceles triangle made by connecting the center of the gravity with any two vertices is 109°28’. This is the same as the obtuse angle of the silver rhombus. Cube Any two faces in parallel with each other are considered. If 2 ends of the upper edge of one face are connected with 2 ends of the lower edge of the other face, a polygon is created. This polygon is the “Silver Rectangle” with a ratio of its 2 different edges being equal to the Silver Ratio. Regular Six vertices are the vertices of three squares that intersect one another at the right octahedron angle at the center. The ratio of any edge of the square to its diagonal is equal to the Silver Ratio. The dihedral angle is 109°28’16”. This is the same as the obtuse angle of the silver rhombus with a ratio of the short diagonal to the long diagonal being equal to the Silver Ratio. Regular Twelve centers of the 12 pentagons are the vertices of 3 rectangles that intersect one dodecahedron another at the right angle at the center. These 3 congruent rectangles are all “Golden Rectangle” with a ratio of 2 different edges being equal to the Golden ratio. The dihedral angle is 116°33’54”. This is the same as the obtuse angle of the golden rhombus with a ratio of the short diagonal to the long diagonal being equal to the Golden Ratio. Cub- The cub-octahedron is made by connecting the middle point of each edge of either octahedron the cube or the regular octahedron. These solids are both based on the Silver Ratio. 2 Regular Twelve vertices are the vertices of three rectangles that intersect one another at the icosahedron right angle at the center. These 3 congruent rectangles are all “Golden Rectangle” with a ratio of 2 different edges being equal to the Golden Ratio. Rhombic This solid is composed of 12 congruent rhombuses. The ratio of its short diagonal to dodecahedron long diagonal is equal to the Silver Ratio. Table 2: Reasons why selected polyhedra are based on either the Golden Ratio or the Silver Ratio 4. Application of “Golden Transformation” & “Silver Transformation” to Duals of Quasi-regular Polyhedra Duals of semi-regular polyhedra are classified into the following 5 patterns: Duals of Quasi-regular Polyhedra Pattern p-kis Dual of quasi-regular polyhedron [p,2q,2q], made of regular F-hedron (p,q). Ⅰ F-hedron Ⅱ Rhombic Dual of quasi-regular polyhedron [p,q,p,q] with E being the number of E-hedron edges of regular polyhedron (p,q). Ⅲ Hexakis Dual of quasi-regular polyhedron [4,2p,2q]. Faces are scalene triangles and F-hedron they alternately become mirrored images of adjacent ones, with (p,q) being a regular F-hedron and p being equal to 3. Ⅳ Trapezoidal Dual of quasi-regular polyhedron [3,4,4,4] or [3,4,5,4]. Faces are trapezoids. 2E-hedron This corresponds to a case of Pattern Ⅲ where triangles on both sides are on the same plane against a line segment which connects an original vertex with a point at a certain distance measured along a perpendicular line from the center of the face. Ⅴ Pentagonal Dual of quasi-regular polyhedron [3,3,3,3,4] or [3,3,3,3,5]. Faces are 2E-hedron irregular pentagons. Two of their edges holding the vertex with an acute angle between them are isometric and 3 other edges are also isometric. Four other interior angles at the other 4 vertices are also the same. Table 3: Five Patterns of Duals of Quasi-regular Polyhedra 4.1 Rhombic Polyhedra Among these polyhedral duals, the rhombic triacontahedron and the rhombic dodecahedron belong to Pattern Ⅱ. The former is composed of 30 golden rhombuses and the latter 12 silver rhombuses. New solids can be created by mutually replacing these rhombuses which are partitioned into two. There are two ways of halving the rhombuses. One is the partition by the long diagonal and the other by the short diagonal. 3 4.1.1 Polyhedra created by Silver Transformation First of all, each face of the rhombic triacontahedron is replaced with 2 isosceles triangles arising from the partition of the silver rhombus by the long diagonal. This process is the aforementioned Silver Transformation (abbreviated as S.T.) Then, a nonconvex polyhedron composed of 60 congruent isosceles triangles with obtuse angles is created. (Refer to Figure 1a.) This new polyhedron having its own circumsphere is considered to be of the same kind as the triakis icosahedron. For this reason, it is tentatively named “Triakis Icosahedron derived from S.T. [1st kind]”. This solid can be seen as an icosahedron with a trigonal pyramid covering each face and the dihedral angle of the pyramid against its base is 35˚15’52”. The length of three edges of this triangle is at a ratio of 2√2 : √3 : √3. This polyhedron represents stellation of the regular icosahedron. The dihedral angle created by the diagonal of the silver rhombus is 151°16’51”． Replacement of each face of the rhombic triacontahedron with 2 isosceles triangles arising from the partition of the silver rhombus by the short diagonal results in the creation of a convex polyhedron consisting of 60 congruent, isosceles triangles with acute angles. (Refer to Fig. 1b.) This new polyhedron having its own inscribed sphere is considered to be of the same kind as the pentakis dodecahedron. So, it is provisionally named “Pentakis Dodecahedron derived from S.T. [1st kind]”. It can be seen as a dodecahedron with a pentagonal pyramid covering each face, with its dihedral angle being 31˚19’03”. The length of 3 edges of this triangle is at a ratio of √3 : √3 : 2. Fig. 1a: Triakis Icosahedron derived from Fig. 1b: Pentakis Dodecahedron derived S.T. [1st kind] (top view) from S.T [1st kind] (top view) In case of the “Triakis Icosahedron derived from S.T. [1st kind]”, vertices of 20 triangular pyramids are all outward-directed. If they are directed inward, a new nonconvex polyhedron is created. (Refer to Fig. 1c). It represents further stellation of “Triakis Icosahedron derived from S.T.