Consider the most ancient mathematical object  the wellknown, monotonically increasing 1Dseries of the common finite Natural Numbers:
1, 2, 3, 4, 5, 6, 7, . . . (I)
Rewrite the quite "boring" 1Dsequence (I) in the following 2Dform:
It is evident that the number of strings in such
the 2Dform of the series (I) is infinte.
Consider, further, any property of Natural Numbers,
say, a predicate P(n) given on the 1Dset (I) or, that is now the
same, on the 2Dtable. For example, consider the quite "trivial" set of
natural squares:
1, 4, 9, 16, 25, . . . (II)
and the predicat "to be or not to be a square of a natural number" or, more precisely:
It is natural that the colors are choiced arbitrary by our aesthetic liking.
So, we obtain the following colormusical 2Dimage (socalled PYTHOGRAM after Great Pythagoras) of a quite abstract mathematical property "to be or not to be a square of a natural number" (unfortunately, we are forced to leave hear a beautiful, mathematical pythograms' Music off screen ):
1. Quantity of numberscells in every string
of that 2Dtable is called the MODULUS
(further  mod) of the pythogram.
2. The pythogram, as a whole, visualizes
the twice abstract CONNECTION
between the additivity and multiplicativity properties of
Natural Numbers: BTW, the ADDITIVITY
of Natural Numbers is modelled by its COLOUR,
the MULTIPLICATIVITY of Natural
Numbers is modelled by their POSITION
in the pythogram, since all numbers n, for which n = k
x mod
+ j, j < mod, holds, are placed in the same
jth column of the pythogram.
3. According to the quite cogent argument
of known number theorists (Delone, Khinchin, etc.), main features of Number
Theory problems are defined just that twice abstract CONNECTION
between the additivity and multiplicativity properties of
Natural Numbers. Therefore, as a rule (but not always), we shall be interested
in not concrete properties of individual Natural Numbers, but just colormusical
pythograms of that twice abstract CONNECTION as a whole
("ornamental patterns").
1. We shall change the physical SIZE of pythogram cells ad liberum (by our wish). For example, we can decrease that size as follows :
Such the trivial transformation allows essentially to increase the length of the segment of the series (I) that is visualized: here from [1,55] to [1,745].
Then, we can change a MODULUS of pythograms ad liberum (by our wish). For example, we can increase a modulus as follows :
The modulus of pythograms is a unique degree of (mathematical) freedom, which essentially enlarges the segment length of the serties (I) under consideration (here from 745 to 12218). It allows to see absolutely new mathematical structures generated by common Natural Numbers which (such the structures) do not exist and can't be seen in the linear form of the 1Dseries (I):
The 11 stills only! From the trivial Fig 1 to the very nontrivial Fig 5,  and we have seen what a whole Mathematics was not seeing ever from the Pythagoras' time! Let us try to understand what we are seeing now.
In a far "bevorecomputer" epoch, great Gotfried Leibniz supposed that "figures are useful to awake a thought". The modern computer CCGtechnology open unique opportunities just for awaking a nontraditional and nonstandard mathematical thought. We suggest some not quite ordinary examples.
Let us try to understand the mathematical nature of some these visuallygraphical structures. Successively changing the modulus of the phythogram, we create a colormusical movie, which at once prompts us the following unique VISUAL CCGfact:
Fig. 6. The most beautiful and unexpected
CCGDiscovery: traditional (a) and nontraditional (b,c) forms of the visual
CCGrepresentation of the same mathematical object  of the wellknown
natural squares set, {1, 4, 9, 16, 25, 36, ...}. We have got a certain
new “paradox” in modern mathematics: the distance between a) and b) figures
is equal to ... about 2000 years! Indeed, even Pythagoras
himself could draw b). Moreover, in 1841, Meobius
drew very similar parabolas,  even just by the modulus 16 (!),  as a
nomogram for multiplication of natural numbers in his known nomographical
works. But only CCGtechnique has allowed us to see, in the first time,
this fantastic transformation! Indeed, the wellknown 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.
A quite not usual equation of a parabola in any nth local coordinate system is as follows:
The equation of the infinite family of the finite parabolas in the coordinate of the CCGspace (Fig. 8) is as follows:
EXAMPLE. Let n=4,
l = k  8.










































So, here n is a parabola
number; but the "value" of the function Y*(n)
is not a usual number, but it is a sequence of the following NINE
squares of successive natural numbers:
(8n4)^{2},
(8n3)^{2}, (8n2)^{2},
(8n1)^{2}, (8n)^{2}
, (8n+1)^{2}, (8n+2)^{2},
(8n+3)^{2}, (8n+4)^{2},
generating the nth parabola for every n = 1, 2, 3, … .
Thus, a "point" (!) Y*(n) describes a whole GEOMETRIC object  the corresponding nth parabola. In other word, that infinite family is simply the following sequence of finite 9points parabolas:
Y*(0), Y*(1), Y*(2), . . . , Y*(n), . . . (III)
So, the sequence (III) is a sequence of NEW FINITE GEOMETRICAL OBJECTS that are VIRTUAL  in the common series (I), and are REAL  in the CCGSpace.
Let us look at the following pythogram of the predicate "to be or not to be a square" under a successive change of its modulus. For example, let the mod be equal to 8, 9, 10, . . . 20.
As it is easy to see (better, from right to left),
some parabolas are jumping, others
are moving down monotonically and dignifiedly.
Consider the last type of parabolas on the following
pythogram modulo mod=18:
The equation of such the running parabola in the shown coordinate system is such:
So, the function, Y*(X*) = Y*(mod), discribes and visually represents a REAL GEOMETRICAL object,  the finite parabola,  in every point X* = mod = 3, 4, 5, . . .
As is known, in Physics, an isolated, solitary wave
is called a SOLITON.
So, our isolated, solitary parabolic wave in Fig. 11, is a PARABOLIC
SOLITON, propagating along the integer axis Y' = 1, 2, 3, .
. .,
No mathematician, from the Pythagoras' time, ever
guessed that a lot of virtual geometrical objects of such the new type,
the parabolic solitons, are running along the usual series of the common
Natural Numbers during centures, and nobody saw them. Only CCGapproach
allowed really to reveal such the DYNAMIC GEOMETRICAL OBJECTS in the Cognitive
Reality of the CCGSpace.
On the jumping parabolas I shall tell you next time.
attracts attention of artists, sculptors, architects, and so on.
On the 23 October, 1999, the known Russian painter
Alexander Pankin, using CCGVisualization approach,
drew (without any computer!) some pythograms of initial segments of the
Natural Numbers series (I) with the predicate:
P(n) = " n is a Fibonacci Number", (1)
and HAVE SEEN the following (see Fig. 12)
VISUAL FACT 1. The threes of
Fibonacii's Numbers (21,13,5), (34,21,8), and (55,34,13) make
up straight lines, and the threes of Fibonacii's Numbers (3,8,13),
(5,13,21), and (8,21,34) are placed in one column.
Fig. 12. Pythograms of initial segments of Natural Numbers Series (I) with the predicate (1). The modules of these pythograms are equal to the Fibonacci's Numbers 5, 8, and 13, correspondingly.
Then he came out with a HYPOTHSIS that such the property
holds for ALL pythograms of such the kind. Our CCGSystem confirmed that
the Pankin's hypothesis is very plausible (see Fig. 13 and 14).
Consider ANY SEVEN of successive Fibonacci's Numbers {f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}, } and the corresponding pythogram modulo f_{4} with the predicate (1). The pythogram modulo 13, presented in Fig. 15, illustrates the situation for the third seven {3,5,8,13,21,34,55}.
Introduce the following designations.
Local coordinate system connected with the triangle
ABC:
Origin: C(0,0); Xaxis: the line CB; Yaxis: the
line CA.
In this local coordinate system, the points A  E will have the following coordinates, expressed via Fibonacci's Numbers:
PROOF.
a) THE EQUATION OF THE LINE AB. From
(Y  Y_{A} )/( Y_{B}  Y_{A} ) = (X  X_{A} )/( X_{B}  X_{A}) and (Y  4) / (4) = X / (f_{4 } f_{1}), we have
4 X + (f_{4 } f_{1}) Y  4 (f_{4 } f_{1}) = 0 (2)
b) From the definition itself of the Fibonacci's Numbers we have:
(f_{3 }= f_{4 } f_{2} ) & (f_{3 }= f_{2 }+ f_{1}) ® (2f_{3 }= f_{4 }+ f_{1}) ® (f_{3 }= (f_{4 }+ f_{1})/2).
f_{3 } f_{1} = (f_{4 }+ f_{1})/2  f_{1} = (f_{4 } f_{1})/2.
c) Since E(f_{3 } f_{1} , 2) = E((f_{4 } f_{1} )/2, 2), we have for the point E from (2) :
4 (f_{4 } f_{1} )/2 + (f_{4 } f_{1}) 2  4 (f_{4 } f_{1}) = 0.
So, the point E(f_{3 } f_{1} , 2) really lies on the side AB of the triangle ABC.
THEOREM 2. FIBONACCI NUMBERS f_{3 }, f_{5 }, f_{6 } BELONG TO THE SAME COLUMN, f_{3 }, OF THE PYTHOGRAM MODULO f_{4 }, SO THAT THE NUMBERS f_{3 }, f_{5 }, f_{6 } ARE A MIGLINE OF THE TRIANGLE ABC.
PROOF. By the definition itself
of the Fibonacci's Numbers, f_{5} = f_{4} + f_{3},
f_{6}= f_{5 }+ f_{4} = 2 f_{4} + f_{3}.
So, f_{3 }, f_{5} , f_{6} = f_{3}
(mod f_{4}), i.e., the numbers f_{3 }, f_{5} ,
f_{6} belong to the same column, f_{3 } , of
the pythogram modulo f_{4 } and are placed in the strings
1,2,3, correspondingly. Since (f_{3 }  f_{1 }=
f_{2 }) & (f_{4 }  f_{3 }= f_{2
}) and the columns with numbers f_{1 } and f_{3 }are
always parallel, the line made up by the numbers f_{3 }, f_{5}
, f_{6} is a midline of the rectangular triangle ABC.
P(n) = " n is a Fibonacci Number", (1)
and HAVE SEEN the following (see Fig. 16)
VISUAL FACT 2. The pythograms
by modules 6, 10, 16, which are not Fibonacii's Numbers,
have topologically invariant structure like "metylcyclopropan".
Since Mathematics takes on trust nothing, this visual hypothesis was proved next day.
In this local coordinate system, the points A  F will have the following coordinates, expressed via Fibonacci's Numbers:
THEOREM 3. THE POINT E LIES ON THE SIDE AB OF THE TRIANGLE ABC.
PROOF. a) THE EQUATION OF THE LINE AB. From
(Y  Y_{A} )/( Y_{B}  Y_{A} ) = (X  X_{A} )/( X_{B}  X_{A}) and (Y  2) / (2) = X / (f_{4 } f_{1}), we have
2 X + (f_{4 } f_{1}) Y  2 (f_{4 } f_{1}) = 0 (3)
b) From the definition itself of the Fibonacci's Numbers we have:
(f_{3 }= f_{4 } f_{2} ) & (f_{3 }= f_{2 }+ f_{1}) ® (2f_{3 }= f_{4 }+ f_{1}) ® (f_{3 }= (f_{4 }+ f_{1})/2).
f_{3 } f_{1} = (f_{4 }+ f_{1})/2  f_{1} = (f_{4 } f_{1})/2.
c) Since E(f_{3 } f_{1} , 1) = E((f_{4 } f_{1} )/2, 1), we have for the point E from (3) :
2 (f_{4 } f_{1} )/2 + (f_{4 } f_{1}) 1  2 (f_{4 } f_{1}) = 0.
So, the point E(f_{3 } f_{1} , 1) really lies on the side AB of the triangle ABC.
THEOREM 4. FIBONACCI NUMBERS f_{3 } and f_{6 } BELONG TO THE SAME COLUMN, f_{3 }, OF THE PYTHOGRAM MODULO 2f_{4 }, SO THAT THE NUMBERS f_{3 } and f_{6 } ARE A MIGLINE OF THE TRIANGLE ABC.
PROOF. By the definition itself
of the Fibonacci's Numbers, f_{6}= f_{5 }+ f_{4}
= 2 f_{4} + f_{3}. So, f_{3 }, f_{6}
= f_{3} (mod 2f_{4}), i.e., the numbers f_{3
}and f_{6} belong to the same column, f_{3 }
, of the pythogram modulo 2f_{4 } and are placed in the strings
1 and 2, correspondingly.
Since (f_{3 }  f_{1 }=
f_{2 })& (f_{4 }  f_{3 }= f_{2
}) and the columns with numbers f_{1 } and f_{3 }are
always parallel, the line made up by the numbers f_{3 } and
f_{6} is a midline of the rectangular triangle ABC.
IMPORTANT REMARK.
The calssical series of Fibonacci's Numbers is generated by the initial pair {0,1}:
{0, 1} 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . (IV)
All the Theorems proved above use the DEFINITION ONLY of the Fibonacci Numbers. Thefore they hold for the GENERALIZED FIBONACCI SERIES produced by any initial pair of integers {m,n}.
Now, let us return to the habitual, most ancient mathematical object that is investigated during about 5000 years,  to the usual series of the usual finite Natural Numbers:
1, 2, 3, 4, 5, 6, 7, . . . , n, . . . (I)
But nobody, from Pythagoras' time, even guessed that
there exists a stormy life of virtual FINITE GEOMETRICAL OBJECTS in this
series (I). And only the CCGtechnology of the cognitivesemantic visualization
of mathematical abstractions has allowed to see such the objects and to
understand some of their unexpected and wonderful mathematical properties.
It's wonderful that some of these CCGdiscoveries
was made by nonprofessional mathematicians, but just a painter and a chemist
(see above). That allows to state that our CCGTechnology is really an
effective way to scientific discoveries.
We believe that great Leopold
Kronecker was right, a thousand times, saying
Our today's CCGresults, concerning the Virual Geometry
of Natural Numbers, can be presented in the next "2Dform".




















1. A.A.Zenkin, Cognitive Computer Graphics. Applications in
Number Theory.  Moscow: "NAUKA", PhysMath. Literature, 1991,
191 pp.; 30 000 copies.
2. The WEBSite http://www.com2com.ru/alexzen/
3. A. A.Zenkin, Cognitive (Semantic) Visualization of the Continuum
Problem and MirrorSymmetric Proofs in the Transfinite Number Theory.
 International ejournal "VISUAL MATHEMATICS", Vol. 2, at the WEBSites:
http://www.mi.sanu.ac.yu/vismath/zen/index.html
http://members.tripod.com/vismath1/zen/index.html
4. A.A.Zenkin, Waring's problem from the standpoint of the
cognitive interactive computer graphics.  An International Journal
 "Mathematical and Computer Modelling", Vol.13, No. 11, pp. 9 
25, 1990.
5. A.A.Zenkin, Some picturesque generalizations of NechaevWaring's
problem obtained by means of cognitive interactive computer graphics.
 An International Journal "Mathematical and Computer Modelling",
Vol.13, No. 11, pp. 27  36, 1990.
6 A.A.Zenkin, Generalized Waring's Problem: On One New Additive
Property of Natural Numbers.  “Matem. Zametki”,
Ňîě 58, No.3, 372  378, 1995 {Translated in: "Mathematical Notes",
vol. 58, no. 3, pp.933937 (1995)}.
7. A.A.Zenkin, Superinduction: A New Method For Proving General
Mathematical Statements With A Computer.  Doklady Mathematics,
Vol.55, No.3, pp. 410413 (1997). Translated from Doklady Alademii Nauk,
Vol 354, No. 5, 1997, pp. 587  589.
8. A.A.Zenkin, The TimeSharing Principle and Analysis of One Class
of QuasiFinite Reliable Reasonings (with G.Cantor's Theorem on the Uncountability
as an Example)  Doklady Mathematics, vol 56, No. 2, pp. 763765
(1997). Translated from Doklady Akademii Nauk, Vol 356, No. 6, pp. 733
 735.(1997).
############################################
Prof. Alexander A. Zenkin,
Doctor of Physical and Mathematical Sciences,
Leading Research Scientist of the Computing Center
of the Russian Academy of Sciences,
Member of the AIAssociation and the Philosophical Society of the Russia,
FullMember of the Creative Union of the Russia Painters.
email: alexzen@com2com.ru
WEBSite http://www.com2com.ru/alexzen
"Artistic "PI"Number Gallery":
http://www.com2com.ru/alexzen/gallery/Gallery.html
############################################
"Infinitum Actu Non Datur"  Aristotle.
"Drawing is a very useful tool against the uncertainty
of words"  Leibniz.