2. Theory of the social fractals in tetrasociology

3. Mathematical theory of the social fractals in tetrasociology

4. Mathematical theory of the social harmony fractals in "Golden tetrasociology"

5. Index of sphere harmony in "Golden tetrasociology"

6. Conclusion

Notice:

At the given stage this project carries a raising character. Its task consists to formulate the substantial features of tetrasociology the most precisely for creation of the mathematical theories of its social fractals, fractals of social harmony and mathematical index of harmony. When these theories will be created by mathematics, they (theories and mathematics) will be included in this project and it will get the finishing theoretical form and collective authorship as the GHA project. Let's name mathematicians, agreed to take part in development of the sphere (tetrasociological) mathematics of harmony:

Alexey Stakhov, Professor, Canada

Peter Sergienko, Professor, Uzbekistan

Eduard Shultz, Dr., Russia

Eduard Soroko, Professor, Byelorussia

Oleg Bodnar, Professor, Ukraine

Vadim Trifanov,Dr., Russia

The GHA other members, having mathematical education, experience and abilities are kindly invited to participation in this project. You also could invite your friends - mathematicians at participation in this project: for this purpose their data and emails inform to me, please, that I could include their in mailing list of mathematical group for this project.

1.Historical and terminological information

A fractalis generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractusmeaning "broken" or "fractured." Prof. Benoit Mandelbrot is a founder of the fractals mathematical theory as indefinitely nested hierarchical recursive and self-similar sets [2]. The birth of the fractal theory is connected with publication in 1977 Mandelbrot’s book "The Fractal Geometry of Nature". The fractal theory gives the first mathematical expression to deep idea of the antique philosopher Anaxagoras "all in everything", i.e. the first quantitative representation of qualitative idea of self-similarity or mutual nesting of structures, which developed by many philosophers and scientists during 2.5 thousand years. The fractal theory found intensive development in theory of infinite hierarchical nesting of matter (or theory of infinite fractal distribution of matter; or theory infinite self-similar nesting of matter) [3].The basic results in this theory were received in the first five years of the 21^{st}century.

Fractals classifications. Basically fractals divide on geometrical, algebraic and stochastic. First two groups form determined fractals, and third - not determined. However there are also other classifications: man-made and natural. To man-made those fractals concern, which were thought up by the scientists: they at any scale have fractal properties. on natural fractals the restriction on area of existence, i.e. maximal and minimal size, is imposed, at which at object are observed the fractal properties. Multi-fractalis complex fractal, which can be determined not by an unique algorithm of construction but by the several consistently replacing each other algorithms. Fore-fractal is a self-similar geometrical figure, which each fragment repeats in the simplified kind at reduction of scale final number of times. Quantity of the scale levels, on which the similarity is observed, names as fore-fractal order. At the order aspiring to infinity fore-fractal passes in fractal. Quasi-fractal differs from ideal determined fractals by incompleteness and inexactitude of recurrences of structure. The majority meeting in a nature of the fractal-similar structures (borders of clouds, line of a coast, trees, leaves of plants, corals, …) are quasi-fractals, as on some small scale the fractal structure disappears. The natural structures can not be ideal fractals because of restrictions, imposed by the sizes of an alive cell and, at the end, by sizes of molecules [1, 2, 3, 4].

Tetrasociology is a term entered by Dr Leo Semashko in 1999 [5] for designation of a new direction of theoretical sociology developed by him since 1976, which originally titled as "Sphere Approach" [6]. Tetrasociology (i.e. four-dimensional sociology) unfolds a fractal nature of society at a qualitative level and opportunity of its harmony. Fractal hierarchical structure of a society is unfolded through mutual inclusion and self-similarity of the matrixes of sphere indices of resources, processes, structures and sphere classes of the population. The matrixes of sphere indices are social fractals of tetrasociology. At superposition on them Prof. Alexey Stakhov’s proportions of "Golden Section" [7] or other kinds of mathematics of harmony they turn out into social fractals of harmony or fractals of social harmony. Use the "Golden Section" proportions in tetrasociology allowed to Alexey Stakhov to enter the term "Golden tetrasociology", with which we quite agree as it reflects the more high level of development of this theoretical sociology.

This project is the first attempt of integration of the mathematical theory of fractals and social fractals of tetrasociology.

2. Theory of the social fractals in Tetrasociology

Fractal theory knows natural, biological, mathematical, cosmic [8] and that similar fractals but does not know social fractals. Tetrasociology offers their line, first of which is the sphere resources fractal. From five axioms of theoretical tetrasociology we shall be limited by the first axiom: for existence of a society and man in any place and time four necessary and sufficient resources are required: People (P), Information (I), Organizations (O) and Things (T). (Things make any material boons and services).These resources represent the extreme large complexes, which title as sphere resources. In a limiting case of modern humankind the global society (GS) in statics will represent a sum of the global, necessary and sufficient, resources:

GS = Ph + Ih + Oh + Th (1), where

Ph is modern number of humankind (planet population),

Ih is set of information (any knowledge) of humankind,

Oh is set of all organizations (juridical, political, financial etc.) of humankind,

Th is set of all things (material boons and services) of humankind

With reference to any society and not just global the formula (1) accepts the following generalized kind:

S = P + I + O + T (2).

This complex of four necessary and sufficient resources is designated as PIOT.

For permanent reproduction (to support existence) each of PIOT resources according to an axiom 1 the appropriate complex of PIOT parts is needed. Hence, each of PIOT will divide on four parts necessary and sufficient as resources for reproduction of PIOT. This will be expressed by the following matrix of dimension by 4х4, which titles as a base matrix:

P = P1 + P2 + P3 + P4, where P is population, and P1, P2, P3, P4 are by its sphere classes, necessary and sufficient for reproduction of the appropriate PIOT resources:

P1 – for P reproduction;

P2 - for I reproduction;

P3 – for O reproduction;

P4 – for T reproduction;

I = I1 + I2 + I3 + I4, where I is information, and I1, I2, I3, I4 are its complexes, necessary and sufficient for reproduction of the appropriate PIOT resources:

I1 – for P reproduction;

I2 - for I reproduction;

I3 – for O reproduction;

I4 – for T reproduction;

O = O1 + O2 + O3 + O4, where O is organizations, and O1, O2, O3, O4 are their complexes, necessary and sufficient for reproduction of the appropriate PIOT resources:

O1 – for P reproduction;

O2 - for I reproduction;

O3 – for O reproduction;

O4 – for T reproduction;

T = T1 + T2 + T3 + T4, where T is things, material goods, and T1, T2, T3, T4 are their complexes, necessary and sufficient for reproduction of the appropriate PIOT resources:

T1 – for P reproduction;

T2 - for I reproduction;

T3 – for O reproduction;

T4 – for T reproduction;

In the pure form the base matrix of PIOT resources looks like:

P = P1 + P2 + P3 + P4

I = I1 + I2 + I3 + I4

O = O1 + O2 + O3 + O4

T = T1 + T2 + T3 + T4

3. Mathematical theory of the social fractals in tetrasociology

4. Mathematical theory of the social harmony fractals in "Golden tetrasociology"

5. Index of sphere harmony in "Golden tetrasociology"

6. Conclusion

(These paragraphs will be translated with Russian on English and are put here to October 1, 2008)

References:

1. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company

2. Feder, J. (1988). Fractals. Plenum Press, New York

3. Theory of infinite hierarchical nesting of matter: http://en.wikiversity.org/wiki/Infinite_Hierarchical_Nesting_of_Matter