Chemistry "on Fibonacci"

The law of constant proportions of the chemical compounds is one of the fundamental chemical laws. This law became firmly established in the chemical science after researches of the French scientist Prust (1754-1826). Studying the chemical compounds, in particular, the oxides of metals, he came to the conclusion that the chemical compounds have a strictly constant structure, which is not dependent on conditions of their formation.

Due to the works of the English scientific Dalton (1766-1844) the atomic doctrine became firmly established in chemistry and the law of multiple ratios was formulated. According to this law there exist the simple integer ratios between atoms in the chemical structures. And now each schoolboy knows that the structure of the water is described by the formula H2O, the common salt - NaCl, the zinc oxide - ZnO. The chemistry became the precise science. The new branch of chemistry called stoichiometry, which learn the ratios of atoms in the chemical structures, was born.

The assertion of the law of multiple ratios is one of the remarkable achievements of world science: from the chaos of the atomic presentations the simple, orderly, beautiful system originated. The atoms of different elements can derivate infinite every possible combinations, connected by the forces of the chemical connection. But only some of them are steady and are saved, and other combinations perish by disintegrating on the steadier connections. And those combinations of atoms of different elements will be steady, which correspond to simple integer ratios of components. It is surprised simple, clearly, lucidly and corresponds completely to the Pythagorean doctrine about the dominating role of numbers in the Universe organization!

However such formulation of the main chemical law evokes some bewilderment. What means the "simple integer ratios"? It is not clear what to perceive under the "small" integers of atoms in the formulas of chemical compounds. While studied rather simple chemical compounds the atom ratio in them usually corresponded to "small" numbers, for example, H2O, Al2O3, Fe3O4, As2O5. But the range of the studied chemical compounds rapidly extends. There appeared the formulas of the chemical compounds with the stoichiometric factors of 7, 9, 15, 21 etc. And when the chemists begun to study the structure of the organic compounds it became inconveniently to speak about the "simple integer ratios". The chemical structure of the bacteriophage is the peculiar champion in the stoichiometry because it has the following formula: C5750H7227N2215O4131S590. What are the ratios of the "small" integers in this formula; here there figure the four-valued numbers.

We would not delve into the chemistry of the different compounds. We will be interested only in one problem: whether reveal themselves the Fibonacci numbers and the golden proportion in the formulas of chemical compounds?

The Ukrainian chemist Vasutinski attempted to give the answer to this problem. He found out the compounds based on the Fibonacci numbers at the analysis of the uranium and chromium oxides. At the uranium oxidation the structure of the generating oxides changes not continuously, and spasmodically, from one steady compounds with the integer atom ratio to other one. Between the uranium oxides of UO2 and UO3 a lot of the intermediate compounds are formed; their structure are described by the formulas U2O5, U3O8, U5O13, U8O21, U13O34. We can see that the atom ratios in these compounds are equal to the ratios of the Fibonacci numbers taken through one: 2/5, 3/8, 5/13, 8/21, 13/34. We already know that such ratios strives in the limit to the square of the golden proportion! But we also met the similar ratios at the analysis of the botanic phenomenon of phyllotaxis!

Each of the described uranium oxides can be represented as the sum of the two boundary oxides UO2 and UO3 taken in the different proportions, for example: U5O13 = 3UO3 + 2UO2; U8O21 = 5UO3 + 3UO2. Here the factors of the oxides UO3 and UO2 corresponds to the neighboring Fibonacci numbers! It means that the structures of the above-considered uranium oxides are subjected completely to the Fibonacci numbers regularity. Note that according to Vasutinski's assertion the chromium oxides Cr2O5, Cr3O8, Cr5O13, Cr8O2 have also the similar structure.

Considering the equation of the kind U5O13 = 3UO3 + 2UO2 there comes on the mind the algebraic equation of the golden proportion of the 4th degree x4 = 3x + 2 describing the butadiene structure. And comparing the equation U8O21 = 5UO3 + 3UO2 with the algebraic equation of the golden proportion of the 5th degree x5 = 5x + 3 we also can see that they also have the identical structure. Whether these analogies can be the beginning of rather interesting researches in the field of the stoichiometry?

It is generally accepted to determine the structure of the chemical compounds by the ratio of atoms of the elements included in this compound. But it is possible to esteem the chemical compounds as consisting of atoms (ions) of the different elements and mobile valence electrons, which "respond" for formation of chemical connections between atoms. So, for example, in the chromium oxide Cr2O5 the 10 valence electrons correspond to the 7 atoms of the chromium and oxygen. If to make similar calculations for all above-considered oxides we get the following ratios of the sums of the atoms to the sums of the valence electrons: 10/7; 16/11; 26/18; 42/29; 68/47. Note that the numerators of these fractions are connected by the Fibonacci ratios and the denominators represent the Lucas numbers. If now sequentially we decreased the numerators and the denominators of these fractions on the Fibonacci numbers corresponding to the metal atom quantity in compounds, that is, 2, 3, 5, 8, 13, in outcome we will get the series of the next Fibonacci numbers ratios: 8/5; 13/8; 21/13; 34/21; 55/34; in limit these ratios strive to the golden proportion.

Thus, Vasutinski rather convincingly demonstrated, that the chemical compounds organized "on Fibonacci" exist!

But as we saw earlier, there is an infinite number of the generalized Fibonacci numbers, the p-Fibonacci numbers, which directly follow from the Pascal triangle. Let's give these numerical sequences for the initial values of p = 1, 2, 3, 4, 5, 6.


It is clear, that the generalized Fibonacci numbers give considerably more of the stoichiometric factors for the chemical compounds, than the classic Fibonacci numbers and discover a broader field for the chemical researches.

And in conclusion once again we remind about the golden proportion equations of the higher degrees, which was revealed by the famous physicist Feynman at the research of the power condition of the butadiene. Whether are these algebraic equations, having the golden proportion as the radical, by the models of the power structures of other chemical compounds? If it will be affirmed, it will testify, that the golden proportion is the basis of the optimal organization of the chemical compounds.