Regular and Semi-Regular Three-Dimensional Polytopes

Abstract

The convex three-dimensional regular and semiregular polyhedrons were investigated using mental mechanical operations on polyhedrons. They include cutting polyhedrons (cutting off vertices), the necessary deformations of the section sections to shape the sections into regular polygons, and rotating parts of the polyhedrons relative to each other. There is proved the existence of 16 semiregular polyhedrons, that is, three more polyhedrons than in the study of “operations on maps.” It is shown that any regular or semiregular convex three-dimensional polytope can be passed to any other regular or semiregular polyhedron in a finite number of steps.