Regular icosahedron as the main geometrical object of mathematics

Among the five "Platonic Solids" the regular icosahedron and dodecahedron take the special place. In the Plato cosmology the regular icosahedron symbolized water, and dodecahedron - harmony of Universe. These two "Platonic Solids " are connected directly to "pentagram" and through it to the golden proportion. Dodecahedron and regular icosahedron form the basis of so-called "icosaedro-dodecahedral doctrine, which penetrates all history of all human culture, starting since Pythagor, Plato and up to now.

And probably, it is impossible to consider accidental that this doctrine got unexpected development in the works of the outstanding German mathematician Felix Klein.

Felix Klein (1849-1925)
Felix Klein (1849-1925)

Felix Klein was born in 1849 and died in 1925. He graduated from the University Bonn. Since 1875 he worked as a Professor of the Higher Technical School in Munich, since 1880 as a Professor of the University Leipzig. In 1886 he moved to Gettingen, where he headed the Mathematical Institute of the University Gettingen; during the first quarter of the 20th century this Mathematical Institute was recognized as the World mathematical center.

The main Klein's works were dedicated to the Non-Euclidean geometry, theory of continuous groups, theory of algebraic equations, theory of elliptic functions, theory of automorphic functions. His ideas in the field of geometry was stated by Klein in the work "Comparative consideration of new geometrical researches " (1872) known under the title "Erlangen Program".

According to Klein, each geometry is the invariant theory for the special group transformation. Dilating or narrowing down the group, it is possible to pass from one type of geometry to other. The Euclidean geometry is the science about invariants of the metric group, projective geometry about the projective group invariants, etc. The classification of transformation groups gives us the classification of the geometries. The proof of the Non-Euclidean geometry consistence is considered as the essential Klein's achievement.

Many Klein's books Κλεινΰ are translated to the Russian: "Higher geometry" (1939); "Elementary mathematics since the point of view of the Higher one" (1934-1935 γγ.); "The Non-Euclidean geometry " (1936); "The Lectures about development of mathematics in the 19th century" (1937).

Klein gave a considerable attention to problems of mathematics development. In the book "Elementary mathematics since the point of view of the Higher one" (1908) he wrote:

"Mathematics developed similarly to the tree, which grows not by finest forks, going from the roots, and scatters branches and leafs in breadth, diffusing them frequently downwards, to the roots ... In the main researches in the mathematics field there can not be a final completion, at the same time and finally established first beginning ...".

Klein's researches concerned also to regular polyhedrons. His book "The Lectures about a regular icosahedron and solution of the 5th degree equations", published in 1884 is dedicated to this problem.

Though the book is dedicated to the solution of the 5-th degree algebraic equations, but the main idea of the book is much deeper and is to show the role of the "Platonic Solids", in particular of the regular icosahedron, in development of mathematical science.

According to Klein, the tissue of mathematics runs up widely and freely by sheets of the different theories. But there are mathematical objects, in which some sheets converge. Their geometry binds the sheets and allows enveloping a general mathematical sense of the miscellaneous theories. The regular icosahedron, in Klein's opinion, is just similar mathematical object. Klein treats the regular icosahedron as the mathematical object, from which the branches of the five mathematical theories miss, namely geometry, Galois' theory, group theory, invariants theory and differential equations.

Thus, the main Klein's idea is extremely simple:

"Each unique geometrical object is somehow or other connected to properties of the regular icosahedron".

In what is the significance of Klein's ideas since the point of view of the harmony theory? First of all we can see that the regular icosahedron, one of the "Platonic Solids" is selected as the geometric object integrating the "main sheets" of mathematics. But the regular dodecahedron is based on the golden section! It follows from here that just the golden section is the main geometrical idea, which, following to Klein, can join all branches of mathematics.

Klein's contemporaries cannot understand and access properly a revolutionary nature of Klein's "icosahedral" idea. Its significance was accessed properly equally in 100 years, that is, only in 1984, when the Israel scientist Dan Shechtman published the article verifying an existence of special alloys (called quasi-crystals) having so-called "icosaedral" symmetry, that is, the 5-th order symmetry, which is strictly forbidden by the classic crystallography.

Thus, still in the 19th century the ingenious Klein's intuition resulted him in the thought that one of most ancient geometrical figures, the regular icosahedron, is the main geometrical figure of science, in particular, mathematics. Thereby Klein inhaled in 19th century the new life in development of the "icosaedro-dodecahedral" doctrine about the Universe structure; this doctrine was developed by the great scientists and philosophers: Plato who constructed his cosmology on the basis of the regular polyhedrons, Euclid who devoted his "Elements" to presentation of the "Platonic Solids" theory, Johannes Kepler who used "Platonic Solids" in the rather original geometrical model of the Solar System, and many others.