Fibonacci's pythograms Foundations of the natural pythography Usually we consider the Natural Numbers in the form of the monotonically increasing series, that is, in the one-dimensional (1D) form:
Rewrite the quite "boring" sequence (1) in the following bidimensional (2D) form:
We will call the table in Fig.1 representing the Natural Numbers series (1) as Note that we choose here five columns in the pythogram in Fig.1 for representation of the Natural Numbers. A number of columns in a pythogram is called CCG-method by Alexander Zenkin Great German mathematician Gotfried Leibnitz supposed that Applying CCG-method to investigate the pythograms he got a number of unexpected CCG-discoveries. One of them consists of the following. Let's consider now any property of Natural Numbers, for example
Let's represent the pythogram in Fig.1 in colored form when all natural squares (2) are marked by black color and all rest natural numbers by yellow color.
Investigating the pythograms for the natural squares (2) (see http://www.com2com.ru/alexzen/papers/vgeom/vgeom.html) Alexander Zenkin came to unexpected mathematical results. Increasing modules of pythograms and using computer simulation he showed that distribution of natural squares is reduced to new wave functions, called
Evaluating his mathematical discovery Alexander Zenkin writes: "But only CCG-technique has allowed us to see, for the first time, this fantastic transformation! Indeed, the well-known ONE, but Infinite, parabola is transformed into the INFINITE FAMILY but of FINITE parabolas! Such the transformation is not known in the modern Mathematics, and it brings to light new aspects of the eternal philosophical problem about a connection between the Finiteness and the Infiniteness". Mathematical discovery of the painter Alexander Pankin In 1999, the well-known Russian artist Alexander Pankin considering some simplest CCG-pythograms of the classical Fibonacci numbers,
discovered the following "The threes of Fibonacci's Numbers (5, 13, 21), (8, 21, 34), and (13, 34, 55) make up straight lines, and the threes of Fibonacci's Numbers (3, 8, 13), (5, 13, 21), and (8, 21, 34) are placed in one column".
Alexander Zenkin proved that this hypothesis is true for the arbitrary threes of the Fibonacci's Numbers (
For more information see: http://www.com2com.ru/alexzen/. Phythograms for the p-Fibonacci Numbers Recently, according to my request, Alexander Zenkin has constructed pythograms for For the case
If we take 2-Fibonacci's numbers 9, 13, 19 as modules of pythograms we get the following 2-Fibonacci's numbers pythograms (see Fig. 5)
Analysis of the pythograms in Fig. 5 shows that the nines consecutive 2-Fibonacci's numbers: {1, 2, 3, 4, 6, 9, 13, 19, 28} by the module 9, always have some regular arrangement at the corresponding pythograms (see the figures in the blue contours). For the case
If we take 3-Fibonacci's numbers 7, 10, 14, 19 as modules of the pythograms we get the following pythograms for 3-Fibonacci's numbers (see Fig. 6).
Analysis of the pythograms in Fig. 5 shows that the eights consecutive 3-Fibonacci's numbers: {3, 4, 5, 7, 10, 14, 19, 26} by the module 7, always have some regular arrangement at the corresponding pythograms (see the figures in the blue contours). Conclusion Thus, CCG-method (Zenkin, 1991) gives us a nice opportunity to discover new unexpected properties of the classical Fibonacci numbers (3) and its generalization, the "Die ganzen Zahlen hat der lieb Gott gemacht, alles andere ist Menshenwerk". And great Henry Poincare expressed this thought as the following: "All Mathematics can be produced from the concept of the Natural Number". |