Platonic Solids

A man shows a concern to polyhedrons during all conscientious life from the two-years child playing with wood cubes up to mature mathematician. Some of regular and semi-regular solids meet in the nature in the form of crystals, other in the form of viruses, which can be considered with the help of a super-microscope. What is a polyhedron? For the answer this question we will remind that geometry is determined sometimes as science about space and spatial two- and three-dimensional figures. The two- dimensional figure can be defined as a set of line segments limiting a part of a plane. Such plane figure is called as a polygon. This implies, that the polyhedron can be defined as a set of polygons limiting a part of three-dimensional space. The polygons, which form polyhedron, are called as its faces.

A long time scientists were interested in "ideal" or "regular" polygons, that is, the polygons having the equal parts and equal angles. As the simplest regular polygon it is possible to consider an equilateral triangle because it has the least number of line segments, which can limit a part of a plane. General picture of regular polygons, interesting us, alongside with the equilateral triangle make: a square (four sides), pentagon (five sides), hexagon (six sides), octagon (eight sides), decagon (ten sides) and so on. Apparently, that theoretically there are no limitations on number of the sides of regular polygons, that is a number of regular polygons is infinite.

What is a regular polyhedron? A polyhedron is called regular one if all its faces are equal (or congruent) among themselves and they are regular polygons. How much exists of different regular polyhedrons? On the first glance the answer this question is very simple: as much how much there exists of regular polygons. However it is not so. In the "Euclidean Elements" we found a rigorous proof that there are only five regular polyhedrons, and only three kinds of regular polygons (triangles, squares and pentagons) can be faces of regular polyhedrons.

These regular polyhedrons are called "Platonic solids" (Fig.1). The first of them is a tetrahedron (Fig.1-a). Its faces are four equilateral triangles. The tetrahedron has the least number of faces among all Platonic solids and is three-dimensional analogy of a plane regular triangle, which has the least number of sides among regular polygons. The next Platonic solid is a hexahedron, called also as a cube (Fig. 1-c). The hexahedron has six faces, which are squares. The faces of the octahedron (Fig.1-b) are regular triangles and their number in the octahedron is equal to eight. A dodecahedron is the next Platonic solid. Its faces are regular pentagons and their number in the dodecahedron is equal to twelve. The regular icosahedron (Fig.1-d) closes a set of Platonic solids. Its faces are regular triangles and their number is equal to 20.

Figure 1. Platonic Solids.

The main numerical characteristics of Platonic solids are a number of faces F, a number of topics V and a number of plane angle E on the solid surface. These numeric characteristics are given in Table 1.

Table 1.

 Polyhedron F V E Form of faces Tetrahedron 4 4 6 Triangle Hexahedron 6 8 12 Square Octahedron 8 6 12 Triangle Icosahedron 20 12 30 Triangle Dodecahedron 12 20 30 Pentagon

"The theory of polyhedrons", in particular, of convex polyhedrons is one of the most fascinating chapters of geometry" - this statement belongs to the Russian mathematician Lusternak who made a great contribution to this area of mathematics.

First of all it is necessary to emphasize that the geometry of regular dodecahedron and icosahedron is connected to the golden proportion. Really, the faces of the dodecahedron are pentagons based on the golden proportion. If closely to look the regular icosahedron (Fig. 1-d), it is easy to see, that five triangles converge in each its topic; the exterior sides of these triangles form pentagon. Already it is enough of these facts to be convinced that the golden proportion plays an essential role in a construction of these two Platonic solids.

But there are more steep confirmations of a fundamental role played with the golden proportion in the regular icosahedron and dodecahedron. It is known, that these Platonic solids have three specific spheres. The first (internal) sphere is inscribed in the Platonic solid and concerns to its faces. Let's designate the radius of this internal sphere through Ri. The second (mean) sphere concerns to its edges. Let's designate the radius of this sphere through Rm. At last, the third (external) sphere is described around of the Platonic solid and passes through its topics. Let's designate its radius through Rc. It was proved in geometry, that the radius lengths of the indicated spheres for the regular dodecahedron and icosahedron having edges of length of 1 are expressed through the golden" proportion t (see Table 2).

Table 2.

 Polyhedron Rc Rm Ri Icosahedron Dodecahedron

Note that the ratio of the radiuses is identical both the icosahedron and for the dodecahedron. Thus, if the dodecahedron and the icosahedron have the same inscribed spheres then their described spheres also are equal among themselves. A proof of this mathematical result is given in the "Euclidian Elements".

It is well known in geometry and other relations for the dodecahedron and icosahedron verifying their connection to the golden proportion. For example, if we take the icosahedron and dodecahedron with edge length equal to 1 and to compute their external areas and volumes they will be expressed through the golden proportion (Table 3).

Table 3.

 Icosahedron Dodecahedron Outer area Volume

Thus, there is a huge number of relations obtained still by the antique mathematicians, verifying the remarkable fact what just the golden proportion is the main proportion of the dodecahedron and icosahedron, and this fact is specially interesting from the point of view of so-called "dodecahedron-icosahedronical doctrine", which will be considered at the next sections of our Museum.