Two Fibonacci's problems

When we told about Fibonacci we emphasized that he proposed and solved many combinatorial problems in his famous book "Liber Abaci" published in 1202. The problem of "rabbit reproduction" is the most known among them. This one led him to the famous number series 1, 1, 2, 3, 5, 8, 13, 21, ... made famous his name and became the topic of the Fibonacci Association investigation.

Leonardo Pisano Fibonacci (1170-1228)

The next problem looked by Fibonacci is called the "problem of choosing the best system of standard weights for weighing on the balance". The history of the problem is the following. From Fibonacci's works it "migrated" to the works of the famous Italian mathematician Luca Pachioli who was "the friend and scientific adviser of Leonardo da Vince". Luca Pachioli placed it in his work "Summa de Arithmetica, Geomeytria, Proprtioni et Proportionalita" published in 1494. This one is considered rightfully as the mathematical encyclopedia of the Renaissance epoch.

Luca Pachioli (1454-1514)

Later the "weighing problem" appeared in the "Collection of the nice and entertaining problems" (1612) written by the French mathematician Bashe de Misiriaque. From the "Biographical directory of figures in mathematics field" by Borodin and Bugaj we know the following:

"Bashe de Misiriaque (9.10.1581 - 26.2.1638) is the French mathematician and poet. He wrote in French, in Italian and in Latin, knew Greek. He collected the ancient mathematical problems in the work "The nice and entertaining problems" (1612). He singled out interesting aspects in these problems and tried to generalize the partial problems. Bashe gave the integer solution of indeterminate equations of the first degree and was hire near to continues fractions. In 1621 he issued the "Diophantus' Arithmetic" in Greek and Latin and gave to it the supplements and remarks".

In the Russian mathematical literature the "weighing problem" is well-known under the title of "Bashe-Mendeleev's problem". We told above about Bashe de Misiriaque. But who is Mendeleev? Is it possible that he is the famous Russian chemist Dmitry Mendeleev, the author of the Periodical Law? Yes, it is true. But why the Russian chemist was interested by the "weighing problem". The answer this question gives an acquaintance with some little-known facts of Mendeleev's biography.

Dmitry Mendeleev (1834-1907)

In 1890 student's disturbances appeared in the Peterburg University where Mendeleev worked as Professor. Mendeleev advocated the students and protecting students he gave the petition about retirement. In 1982 Mendeleev was appointed as the Learned Keeper of the Standard Weights Bureau, which was transformed in 1892 into the Principal Russia House of Measures and Weights according to Mendeleev's initiative. Mendeleev was as the Director of this House until his death. Thus, the concluded stage of Mendeleev's life (1992-1907) was connected with development of measurement affairs. Just in this period Mendeleev is interested in the "weighing problem". According to his initiative investigations on this problem were realized and it was proved that the "ternary" system of standard weights is optimal.

But now we address to the computer history. In the beginning of the 60th of the 20th century the new area of measurement technology, analog-to-digit and digit-to-analog converters, originated. The primary goal of this engineering area consisted of elaboration of the devices ensuring automatic conversion of information from the analogue, continuous form to the discrete, digital form. Such devices were called as analog-to-digital converters. Thus, the main goal of these devices is ensuring of connection of computers with the real objects where the information is presented in the continuous, analog form.

Like to the "weighing problem", which can be solved by using the different sets of the standard weights, the problem of analog-to-digital conversion can be solved by using the different methods or algorithms of analog-to-digital conversion. However, the best ("optimal") algorithms of analog-to-digital conversion present the greatest interest. Such problem, that is, the problem "of synthesis of the optimal analog-to-digital conversion algorithms" was put in 1963 as the subject of the PhD's thesis before the young researcher Alexey Stakhov, the post-graduate student of the Kharkov University of Radio Electronics.

The results of the new mathematical theory of analog-to-digital conversion obtained hereinafter the title of the "algorithmic theory of measurement" were elaborated in Stakhov's PhD thesis defended in 1966 and then in his DrSci thesis "Synthesis of the optimal analog-to-digital conversion algorithms" defended in 1972.

The talented Ukrainian mathematician Igor Vitenko (1938-1974), the graduate of the Lvov University mathematical faculty, who worked during that period as a teacher of the Kharkov University of Radio Electronics, was involved in research of the problem of the "optimal" analog-to-digit converters.

Igor Vitenko (1938-1974)

Just thanks to Vitenko the high mathematical level of researches was ensured, and he rightfully together with Stakhov can be called as one of the creators of the new mathematical theory of measurement.

Vitenko's fate was developed tragically. His father and mother were the soldiers of Stephan Bandera army. The father was killed during the Great Patriotic War and his mother stayed many years in Stalin's concentration camps. During that period Stephan Bandera's soldiers and even their children were subjected to all possible prosecutions from the side of the communistic regime. Also Igor Vitenko became by the victim of these prosecutions. In the last years of his life he worked as the senior lecturer of the Mathematical Logic Theory Department of the Uggorod University. When he completed his DrSci's thesis according to the existed rules the scientific and teacher characteristic was needed to him for defending his DrSci's thesis. Because the Communistic Party Committee of the University actually refused to him in issuing such characteristic he decided to end his life and this sad event happened in September 1974. So the life of one of the most talented young Ukrainian mathematicians was broken.

The concept of the "algorithmic measurement theory" was entered by Stakhov in 1973 in the lecture "Algorithmic measurement theory" delivered made by him at the All-Union conference on information-measuring systems (the city of Ivano-Frankovsk).

In 1977 he published the book "Introduction into the Algorithmic Measurement Theory" and in 1979 the brochure "Algorithmic Measurement Theory".

Just this book and this brochure got a wide notoriety in the Soviet and world science proclaimed about birth of the algorithmic measurement theory, the new trend in the theoretical metrology.

The first his book "Introduction into the Algorithmic Measurement Theory" (1977) was devoted by Stakhov "to the memory of my friend, talented mathematician Igor Vitenko".

Thus, the ideas of the "great Fibonacci" got a worthy development in modern science. On the one hand, due to the American mathematician Verner Hoggat the interest in Fibonacci's "rabbit reproduction problem" and in Fibonacci numbers was revived and these remarkable numbers became as the research object of the Fibonacci Association established in 1963. On the other hand, the Ukrainian scientist Alexey Stakhov initiated in 1963 the investigation of the other Fibonacci's problem, the "weighing problem", and this Stakhov's activity, in the issue, resulted in development of the algorithmic measurement theory. And we will tell about this interesting theory at the next pages of our Museum. Follow us!