Original Paper ________________________________________________________________________________________ Forma, 36, 1–14, 2021 Close Relationship between Ratios of “Golden Ratio Group” and Those of “Square-Root-of-2 Ratio Group” Hiroaki Kimpara Kawasaki Branch, Mathematical Association of Japan, 255 Ohkubo, Sakura-ku, Saitama 338-8570, Japan E-mail: [email protected] (Received February 3, 2020; Accepted September 22, 2020) While the golden ratio (1:φ) has been well known since the ancient Greece age and has already been studied in depth by many researchers, the square-root-of-2 ratio (1: 2 ) and other ratios of this group have gone largely unnoticed until now, despite the fact that the rhombic dodecahedron, one of the most important poly- hedra, is based on the square-root-of-2 ratio. According to a popular theory, these ratios are independent of one another. Against it, however, the author found out that there exists close relationship between the ratios of the golden ratio group and those of the square-root-of-2 ratio group through careful study on three kinds of rhombic polyhedron. Key words: Golden Ratio, Square-Root-of-2 Ratio, Rhombus, Rhombic Polyhedron, Dihedral 1. Introduction gon to a specific diagonal thereof. These ratios also rep- Until now, it has been the established fact that the golden resent the length ratios between two diagonals of spe- ratio is an independent ratio with no specific relationship cific rhombuses, and each of such rhombuses constitutes with other special ratios like the square-root-of-2 ratio. a specific group of rhombic polyhedron. These findings Against it, however, the author puts forth the following are explained in Sec. 3. The five kinds of regular polyhe- proposition based on careful research into three kinds of dron are classified into two groups; one is a square-root- rhombic polyhedron. of-2 ratio group and the other is a golden ratio group. Such classification is described in Sec. 4. Proposition. There exists a close relationship between Section 6 reveals that plane-filling is realized by com- the ratios of the golden ratio group and those of the bination of the golden rhombus and the “square-root-of- square-root-of-2 ratio group. (Note: An explanation for 2 square rhombus” and that another plane-filling becomes these two groups is given in Sec. 2.) possible with a combination of the “square-root-of-2 rhombus” and the new rhombus based on a new ratio. Good grounds for this proposition are shown in Sec. 5 The definition of these rhombuses is provided in Sec. 3. and Sec. 6. Section 5 deals with the agreement in the dihedral of 2. Relationship with the Diagonal of the Regular solids transformed from three kinds of rhombic polyhe- Polygon dron, which provides evidence for this proposition. They There exist several specific ratios of significance other are rhombic dodecahedron, rhombic triacontahedron, and than the golden ratio and the square-root-of-2 ratio. All rhombic enneacontahedron. A common property of these these ratios are shown in Table 1 below. Except the square- solids is that they can all be built up from the regular root-of-2 triplicate ratio, they are all equal to the length polyhedron. The rhombic triacontahedron and the rhombic ratio of one side of the regular polygon to a specific di- enneacontahedron are totally different from each other in agonal thereof. Yet, the golden ratio and the square-root- shape, face composition, etc. Nevertheless, transforma- of-2 ratio are considered exceptional in that they repre- tions of these solids based on a simple specific principle sent the sole such ratio inherent in the regular pentagon amazingly result in exactly the same shape. and the regular tetragon, respectively. This is because Section 2 indicates that there are several specific ratios there exists only one kind of diagonal in these polyhedra. of significance other than the golden ratio and the square- The ratio 1: 23 is derived from the dodecagon and, root-of-2 ratio and that they belong to either a golden ra- + tio group or a square-root-of-2 ratio group. Except the for this reason, it is tentatively named the “dodecagon “square-root-of-2 triplicate ratio”, these ratios are all ratio”. According to a common view, the ratio 1:1+ 2 equal to the length ratio of one side of the regular poly- represents the silver ratio. Note that the term “silver ra- tio” sometimes means the ratio 1: 2 . But, in this paper Copyright Society for Science on Form, Japan this definition is not used. Of these ratios, the golden ra- tio, square-root-of-2 ratio, and silver ratio are self-ex- doi:10.5047/forma.2021.001 1 2 H. Kimpara Table 1. Seven specific ratios of significance including the golden ratio and the square-root-of-2 ratio. The above calculation indicates that the dodecagon ra- tio is equal to the length ratio of one side of the regular dodecagon to its shortest diagonal. The last two ratios in Table 1 are derived by the use of Fig. 2 below, which is a regular octahedron. Shown in Fig. 2 below is a regular octahedron projected to a XY- plane with the Z-axis serving as 2-fold rotational sym- metry axis. With the tetragon AHBC being a square-root- of-2 rhombus, HO: AO = 1: 2 . Therefore, HO: AB = 1:2 2 . This means that the square-root-of-2 triplicate ratio is equal to the length ratio of one-half edge of the Fig. 1. (a) Regular decagon; (b) Regular dodecagon. regular octahedron to its space diagonal. 3. Relationship with the Diagonals of Rhombuses planatory. Therefore, explanations are given to the other Various ratios mentioned above also represent the length ratios in Table 1. ratios between two diagonals of the following special The second ratio in Table 1 is expressed by AB:AD rhombuses, as shown in Table 2. shown in Fig. 1(a), which shows a regular decagon. Its Out of these angles, the acute angle of the square-root- proof is given below. of-2 rhombus (≈70°31′43.606″), obtuse angle of the square-root-of-2 rhombus (≈109°28′16.394″), obtuse an- gle of the golden rhombus (≈116°33′54.184″), obtuse angle of the golden square rhombus (≈138°11′22.866″), and obtuse angle of the square-root-of-2 triplicate rhom- bus (≈141°03′27.212″) are respectively equal to the di- hedral angles (those of two neighboring faces) of the regu- lar tetrahedron, regular octahedron, regular dodecahedron, regular icosahedron, and triangular bipyramid. The tri- The foregoing calculation indicates that the golden square angular bipyramid is a type of hexahedron, being the first ratio is equal to the length ratio of one side of the regular in the infinite set of face-transitive bipyramids. It is the decagon to its 2nd shorter diagonal. dual of the triangular prism with 6 isosceles triangle faces. Shown in Fig. 1(b) is a regular dodecagon, which also Particularly, the obtuse angle of the square-root-of-2 includes a regular tetragon (i.e. square) and a regular hexa- gon. In fact, the right isosceles triangle ADG is equiva- lent to one-half of a regular tetragon and the trapezoid ACEG is equivalent to one-half of a regular hexagon. In this figure, the triangle ACO is a regular triangle. There- fore, AC = AO = AG /2 . AC : AG = 1:2. This means that the “square-root-of-2 square ratio” is equal to the length ratio of one side of the regular hexagon to its longer diagonal. Fig. 2. Regular octahedron projected to a XY-plane with the Z-axis serving as 2-fold rotational symmetry axis. Close Relationship between Ratios of “Golden Ratio Group” and Those of “Square-Root-of-2 Ratio Group” 3 Table 2. Length ratios between two diagonals of special rhombuses. Table 3. Special rhombuses constituting a specific rhombic polyhedron. Fig. 3. How to make the rhombic icosahedron and the rhombic dodecahedron of the 2nd kind from the rhombic triacontahedron. rhombus (≈109°28′16.394″) is called the regular tetrahe- Of these poyhedra, the rhombic dodecahedron of the dron angle (called Maraldi’s angle, also) in the field of 2nd kind and the rhombic icosahedron are created from crystallography. the rhombic triacontahedron. This process is illustrated Here, it is to be noted that twice the acute angle of the in [3]. For reference purpose, this illustration is excerpted golden rhombus (≈63°26′05.816″) is equal to the obtuse and shown in Fig. 3 below. This means that the rhombic angle of the “square-root-of-2 square rhombus” dodecahedron of the 2nd kind, rhombic icosahedron and (≈126°52′11.632″) and that twice the acute angle of the rhombic triacontahedron all fall under the same category. square-root-of-2 rhombus (≈70°31′43.606″) is equal to Figure 3 illustrates the process of making the rhombic the obtuse angle of the “square-root-of-2 triplicate rhom- icosahedron and the rhombic dodecahedron of the 2nd bus” (≈141°03′27.212″) and that twice the acute angle of kind from the rhombic triacontahedron, (a) → (b) → (c) the “dodecagon ratio rhombus” (≈54°44′08.197″) is equal → (d) → (e) → (f). The rhombic icosahedron is created to the obtuse angle of the square-root-of-2 rhombus by pulling out 10 rhombuses, i.e. shadowed portion of (≈109°28′08.197″). (b), from the triacontahedron and by combining the up- Furthermore, these rhombuses constitute a specific per portion with the lower portion. Further, the rhombic rhombic polyhedron as listed in Table 3 below.