Golden Proportion as the main mathematical constant of Harmony Mathematics

The Pythagorean mathematicians by means of discovery of "Incommensurable Line Segments" made the most essential contribution to development of the concept of Irrational Number, which is the next, after Natural Number, fundamental mathematical concept.

As is well known a number of irrational numbers is limitless. However, some of them occupy the special place in the history of mathematics, moreover in the history of material and spiritual culture. Their importance consists of the fact that they express some relations, which have a universal character and appear in the most unexpected applications.

The first of them is the irrational number , which equals to the ratio of the diagonal to the side of the square. The discovery of the incommensurable line segments and the history of the most dramatic period of the antique mathematics is immediately connected to this irrational number. Eventually this result brought into the elaboration of the irrational number theory and into the creation of modern "continues" mathematics.

The p-number and Euler's number of e are the other two important irrational numbers. The p-number, which expresses the ratio of the circle length to its diameter, entered mathematics in the ancient period along with the trigonometry, in particular the spherical trigonometry considered as the applied mathematical theory intended for calculation of the planet coordinates on the "celestial spheres" ("the cult of sphere").

The e-number entered mathematics much later than the p-number. Its discovery was immediately connected to the discovery of Natural Logarithms. As is well known the e-number expresses a number of the important geometric properties of the hyperbola. There exists the following mathematical expression connecting the p- and e-numbers:

where is the "imaginary unit", one more unexpected creation of mathematical thought.

The p- and e-numbers "generate" a variety of the fundamental functions called the Elementary Functions. The p-number number "generates" the trigonometric functions sin x and cos x, the e-number "generates" the exponential function ex, the logarithmic function logex and the hyperbolic functions namely the hyperbolic sine and the hyperbolic cosine:


The trigonometric functions are connected to the exponential function with the help of Euler's formulas:

The trigonometric functions and the exponential function play a special part in differential and integral calculus because they are invariants of integration and differentiation, i.e.


The trigonometric functions and the exponential function play a special part in differential and integral calculus because they are invariants of integration and differentiation, i.e. p- and e-numbers re the most widespread functions of calculus. That is why there appeared the saying: "The p- and e-numbers dominate over the calculus".

The "Golden Section" is one more fundamental irrational number. The latter entered science in the ancient period along with the p-number. Hence, dating back from the ancient Egyptian period in the mathematical science of nature there came into being two trends of the science progress based on different ideas as to the Universe harmony, viz. the trend of the p-number, basing on the idea regarding to the spherical character of planets' orbits, and the trend of the golden section, basing on the dodecahedron-icosahedronical idea about the Universe structure. The latter idea emerged from the analysis of cyclic processes within the Solar system and underlies the calendar systems and the time and geometric angle measurement systems, basing on the fundamental number parameters of the dodecahedron and icosahedron, i.e. on the numbers 12, 30, 60 and 360.

In the course of historical progress there occurred a separation of the said two trends of the mathematized nature science progress. The p-number trend added in the 16th century, after the discovery of logarithms, by the e-number, has became the heart of the simulation processes of inanimate nature. Newton's gravitation theory and differential and integral calculus are the highest accomplishments of such an idea. The "Golden Section" trend dating back to the antiquity and to the Renaissance epoch is associated more and more closely with the art and the biological processes. In the second half of the 20th century this idea of the Universe harmony was put into the forefront in the mathematical study of nature. Modern scientific discoveries concerning the Golden Section (quasi-crystals, resonance theory of the Solar system, new geometric theory of phyllotaxis, structural harmony of systems, etc.) are the enough convinced argument and the proof of the most astonishing scientific hypothesis, which appeared in the ancient Egypt and was realized in the Cheops pyramid. This hypothesis affirms that the golden section is the main proportion of the Universe.

But just as Kepler's investigations aimed at finding the principles of the Universe harmony (culminating in the discovery of planets' movements laws) eventually resulted in the emergence of Newton's gravitation theory and of the classical mathematical analysis (differential and integral calculus), adapted optimally for the description of the mechanical processes, the modern investigations in the field of the systems harmony can bring finally into the Harmony Mathematics optimally adapted for the description of harmonious processes in animate and inanimate nature. And just as the numbers of p and e, which "dominate over the calculus", are the basic constants of the classical mathematical analysis, so the golden section has to become the basic constant of the Harmony Mathematics.

In the recent years a number of new mathematical findings of fundamental importance have been obtained within the framework of the "harmonious" trend, which may well initiate the Harmony Mathematics development. Those are the Fibonacci number theory, the algorithmic measurement theory, the hyperbolic Fibonacci trigonometry, etc.

It follows from this consideration the idea to single out the mathematics part connected to Fibonacci numbers and the Golden Section under the common title of "Harmony Mathematics". First this idea was stated in the lecture "The Golden Section and Modern Harmony Mathematics" delivered by Prof. Stakhov on the 7th International Conference on Fibonacci Numbers and Their Applications (Austria, Graz, 1996).

'Applications of Fibonacci Numbers' (Volume 7)Stakhov's lecture 'The Golden Section and Modern Harmony Mathematics'

And we will tell about the contents of this lecture at the next pages of out Museum. Follow us!