Ways of constructing regular polyhedra

Abstract

The paper deals with the construction of a 3D model of a dodecahedron using rectangles with aspect ratio based on the golden ratio.The construction of dual figures: icosahedron and dodecahedron, inscribed in a cube and circumscribed about it, and also their cascade using rectangles with sides of golden ratio in AutoCAD is given.The article uses the concept of "regular polyhedron". What is it? A regular polyhedron is a convex polyhedron, each face of which is a regular p-gon and at each of its vertices the same number q of such faces converges.An icosahedron is a geometric body with twenty faces, each of which is a right triangle. Dodecahedron is a geometric body with twelve faces, each of which is a regular pentagon.These polyhedrons belong to regular polyhedrons. There are various versions of their use: candlesticks, dice, a tool for calibrating water pipes (for this purpose round holes have different diameters), as a visual teaching aid for learning the basics of academic drawing.Numerous ways of constructing regular polyhedrons and the most complex of them icosahedron and dodecahedron are known. Among them the method of construction of icosahedron on the basis of golden rectangles with the ratio of sides of the golden ratio (ratio of the whole to the larger part) has aroused special interest.This method, characterized by simplicity, elegance and internal harmony is not studied enough, as it is not extended to the dodecahedron, which is built as a figure dual to the icosahedron.Purpose of work is to supplement the method of construction by means of rectangles with aspect ratio on the basis of golden section for dodecahedron. The constructions were performed in the graphic editor AutoCAD [1], but can be repeated in other known CAD systems.The algorithm of construction of icosahedron on the basis of golden rectangles is as follows: three equal rectangles are inserted one into another perpendicularly to each other along the middle parallel, it remains only to connect the vertices nearest to each other.To build a dodecahedron, we need to connect the centers of the faces of an icosahedron. There are many ways to construct regular polygons: one can inscribe them into a sphere or describe them about it; use consecutive cascade inscription-description relative to each other (the number of possible cascades is 5! = 120; construct figures on the basis of a cube (for example, a dodecahedron by the method of "roofs" proposed by Euclid); using proportions of golden sections; using formulas or only on the basis of geometrical constructions, etc. It is possible to create a flat image or volumetric model on the computer.