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:

 1, 2, 3, 4, 5, 6, 7, ... . (1)

Rewrite the quite "boring" sequence (1) in the following bidimensional (2D) form:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ...

Fig. 1. 2D-notation of the Natural Numbers.

We will call the table in Fig.1 representing the Natural Numbers series (1) as "2D-notation of Natural Numbers" or "Pythogram". The latter name is used in honour of Great Pythagoras who first proclaimed that "The World is a Number, ... and Harmony of singing celestial spheres". It is evident that the number of strings in the 2D-notation (Fig.1) is infinite.

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 "module" of its 2D-representation. But such the choice is arbitrary, i.e., it is a unique degree of freedom which allows to animate the pythograms, changing their modules, and to see number-theoretical properties in dynamic.

CCG-method by Alexander Zenkin

Great German mathematician Gotfried Leibnitz supposed that "figures are useful to awake a thought". The modern computer technologies open unique opportunities just for awaking a non-traditional and non-standard mathematical thought. The Russian mathematician Alexander Zenkin developed new computer method of mathematical research called "Cognitive Computer Graphics"(CCG-method) (see the book "Cognitive Computer Graphics. Applications in Number Theory" by Alexander Zenkin, Moscow: Publisher "Nauka", 1991).

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 "to be square of natural numbers"; then we get the following numerical sequence:

 1, 4, 9, 16, 25, ... . (2)

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.

Fig. 2. Pythogram for squares of natural numbers.

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 "parabolic solitons" by analogy with solitons used widely in modern physics.

Fig. 3. Parabolic Solitons.

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,

 1, 2, 3, 5, 8, 13, 21, 34, ... , (3)

discovered the following visual fact (see Fig. 3):

"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".

Fig. 4. Pythograms of Fibonacci Numbers with the modules equaling to the Fibonacci's Numbers 5, 8, 13 respectively.

Alexander Zenkin proved that this hypothesis is true for the arbitrary threes of the Fibonacci's Numbers (Fn, Fn+2, Fn+3) if the module of the corresponding Fibonacci's pythogram is equal to Fn. In general, he proved the following quite unexpected statement.

Zenkin's theorem (1999). Any seven of successive Fibonacci Numbers {F1+k, F2+k, F3+k, F4+k, F5+k, F6+k, F7+k} generates (in the 2D-pythogram by the module F4+k) a rectangular triangle with the hypotenuse {F4+k, F6+k, F7+k}, the legs of the triangle {F1+k, F2+k, F3+k, F4+k} and {F1+k, F7+k}, and the median {F3+k, F5+k, F6+k}, where k is the given integer nonnegative number (k = 0, 1, 2, 3, ...).

Phythograms for the p-Fibonacci Numbers

Recently, according to my request, Alexander Zenkin has constructed pythograms for p-Fibonacci's numbers (see my definition of such the p-Fibonacci numbers at http://www.goldenmuseum.com/) and showed that the pythograms (see Fig. 5, 6) have a similar regularity. Let's consider two cases of p (p = 2 and p = 3).

For the case p = 2 we have the following numerical sequence called 2-Fibofonacci's numbers:

 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, ... (4)

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)

 p = 2 Fig. 5. Some pythograms for 2-Fibonacci's numbers with "modules" 9, 13, 19.

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,

{2, 3, 4, 6, 9, 13, 19, 28, 41} by the module 13, and

{3, 4, 6, 9, 13, 19, 28, 41, 60) by the module 19,

always have some regular arrangement at the corresponding pythograms (see the figures in the blue contours).

For the case p = 3 we have the following numerical sequence called 3-Fibonacci's numbers:

 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, ... (5)

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).

 p = 3 Fig. 6. Some pythograms for 3-Fibonacci's numbers by modules 7, 10, 14, and 19.

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,

{4, 5, 7, 10, 14, 19, 26, 36} by the module 10,

{5, 7, 10, 14, 19, 26, 36, 50} by the module 14, and

{7, 10, 14, 19, 26, 36, 50, 69} by the module 19,

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 p-Fibonacci's numbers, which are parts of the Natural Numbers (1). And we believe that great Leopold Kronecker was right, a thousand times, saying:

"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".