Platonic Solids as Structured Geometric Objects

Abstract

Andrey V. Voron

The possibility of constructing Platonic solids from structural elements is shown – Kepler triangles (ratio of legs 1: √1. 618..) and Fibonacci (ratio of legs 1:1.618...) – provided that the area of these elements remains unchanged. The number of elements (or pairs of elements) that make up the structure of the "tetrahedron", "octahedron", "cube" increases, thus, by two times, and the "icosahedron" – by five times in relation to the number of elements of the tetrahedron, while the indicator "area of all structural elements of the figure" and radius (r = 3) remain unchanged inscribed in the Platonic bodies of the sphere. In addition, the area of the structural elements of two dodecahedra (S = √959325) is equal to the area of the structural elements of any 5 Platonic solids, for example, 5 tetrahedra (or octahedra, cubes, icosahedra) (S = √38373). The possibility shown is in accordance with the text of Plato's work Timaeus, according to which Platonic bodies can "transform into each other."

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