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.
TopFive Plato Bodies
Even in Ancient Greece, the existence of five regular convex polyhedrons - Platonic solids was established (Aleksandrov, 1950): tetrahedron, octahedron, cube, dodecahedron, icosahedron (Euler, 1736). In each of these polyhedrons, all the vertices are the vertices of polyhedral angles - gonohedrons, in Fedorov's terminology (Fedorov, 1885, 1889, 1891). Since the faces of the gonohedron are regular polygons, the plane angles at the vertices of the gonohedron satisfy equality𝛼 = (n – 2)𝜋/n,(1.1)n is the number of sides at face of gonohedron.
In addition, planar angles around the vertex of the gonohedron must satisfy three natural conditions necessary for the existence of a gonohedron
- 1.
Around the top of the gonohedron, there must be at least three flat corners.
- 2.
The sum of the planar angles located around the vertex of the gonohedron must be less than 2𝜋.
- 3.
The sum of the plane angles located around the vertex of the gonohedron, minus one angle, must be greater than this angle.
Taking into account these conditions and equality (1.1), only five different gonohedrons and the corresponding previously indicated regular convex polyhedrons can exist (see Table 1).
There cannot be other regular three-dimensional polyhedrons since there cannot be other gonohedrons with faces of regular polygons. Images of regular convex three-dimensional polyhedrons in projection onto a two-dimensional plane with infinitely distant center of the projection (while the projection retains the parallelism of the edges of the polyhedrons) are shown in Figure 1.
Figure 1. Regular convex polyhedrons
Table 1. The regular convex polyhedrons
The name of
polyhedron | N | K | M | l | r | {n, k} |
| tetrahedron | 3 | 3 | 4 | 4 | 6 | {3,3} |
| octahedron | 3 | 4 | 8 | 6 | 12 | {3,4} |
| cube | 4 | 3 | 6 | 8 | 12 | {4,3} |
| dodecahedron | 5 | 3 | 12 | 20 | 30 | {5,3} |
| icosahedron | 3 | 5 | 20 | 12 | 30 | {3,5} |
Key Terms in this Chapter
Semiregular Polyhedrons: Convex polyhedrons with faces of regular polygons of two or more types, if all gonohedrons of the polyhedron have the same composition and arrangement of faces.
Regular Polyhedron: Convex polyhedron with faces of regular polygons of one type.
Graph Theory: A graph G (V, E) is a pair of two sets, V and E, V=V(G) being a finite nonempty set and E=E(G) a binary relation defined on V.
Operation on Maps: Operation on maps are topological modifications of a parent graph.
Mechanical Operations on a Polyhedron: ?onceivable transformations of a polyhedron, including cutting a polyhedron (cutting off vertices), deforming a section of a cut to obtain a regular polygon in a section, rotating parts of a polyhedron after cutting relative to each other.
Gonohedron: A polyhedral corner whose faces are regular polygons.