Close Relationship between Ratios of “Golden Ratio Group” and Those of “Square-Root-of-2 Ratio Group”

Abstract

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 polyhedra, 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.