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ARTICLE 2
The Nature and Meaning of Fundamental Constants
Abstract
The description of the most general and many specific regularities of the
development of the surrounding world is seldom managed without the
presence of the so-called world constants — universal values entering into
the composition of the form-building means of such a description
(mathematical
formulas,
mathematical
dependences,
mathematical
expressions).
A great multitude of printed works is devoted to these values, and, despite
the abundance of investigations of their remarkable properties and
characteristics, the circumstance is universally recognized that until now the
true nature and the actual meaning of the fundamental world constants are
unknown to anyone.
The present work removes the veil of mystery from the very popular and
extraordinary numbers π, e, and φ, to the solving of the secret of which
numerous publications in various information sources are devoted.
On the example of these three most ancient and omnipresent numbers, the
article shows in what manner one should approach the establishment of the
nature and the revelation of the meaning of constants of this kind.
The work can be recommended to the attention of researchers as initial
material for the study of the background and the elucidation of the meaning
of the entire spectrum of existing fundamental constants, as well as as a
source of alternative views on the nature and essence of the surrounding
world.
Introduction
Many versions are known regarding the origin, purpose, and role of the
fundamental world constants. We are speaking, first of all, of the most
ancient constant "π" [2, 10, 11, 13, 17, 20, 22, 28, 34, 51, 52] — the
designation and numerical value of the ratio of the length of the
circumference to its diameter, as well as of the no less fundamental and
popular (although not so ancient as the first) constant "e" [2, 8, 10, 11, 17,
20, 26, 29, 31, 43, 51, 53] — the designation and numerical value of the
base of natural logarithms. In this same connection we shall not fail to recall
yet one more outstanding constant "φ" [1, 2, 3, 7, 14, 19, 23, 25, 35, 39,
40, 47, 50, 51] — the designation and numerical value of the so-called
golden ratio, or golden section, or golden proportion.

To the extraordinary and mysterious (not to say — mystical) numbers π (pi),
e, and φ (phi) a great multitude of printed works is devoted. Special
attention is traditionally paid to the enumeration of the astonishing
properties of these dimensionless constants. And at the same time, despite
the abundance of investigations and descriptions of the remarkable
characteristics of the indicated numbers, the circumstance is universally
recognized that until now the nature and the actual meaning of these values
are unknown to anyone.
In contrast to the mission of replenishing the list of enthusiastic descriptions
of the wonderful properties of the above-enumerated world constants, the
modest task of the present article consists in showing the source of their
extraordinary properties, in uncovering and laying bare the true nature and
essence of these values.
We shall solve the set task, relying on the concept of development described
in the previous article of this same collection, as well as in the works [5, 21,
54].
In investigating the essence of the mentioned constants, we proceed from
the fact that reality (the world surrounding us, nature) is always at least twosided (dual). Any object, process, phenomenon in the most general abstract
form represents a relation of two sides-constituents-components:
{content/form}. Then any concrete form (implying under itself the same
object, process, phenomenon) is determined by the correlation of the
indicated sides-constituents in each of their concrete relations. The
development of the forms themselves represents the change of the
correlations of the noted constituents within the framework of their relations.
In such a connection it becomes obvious that the forms determined by the
above-mentioned correlations can be both more and less contentful (by
virtue of the presence of different shares of the contentful and formal
components in their relations). The general tendency of the development and
change of forms (as the experience of human observations throughout
human history shows) is such that in the course of their (the forms')
development there takes place a shift of the correlation of the components
(in their relations) toward the "numerator," i.e., toward the contentful
component (what is meant is the principled orientation toward the general
movement-development of forms in the direction of the acquisition of the
highest contentfulness) [5, 21]. Such an installation-orientation is called the
principle of self-development, with the formulation of which the reader can
acquaint himself there — in the previous article of the present collection and
in the works [5, 21, 54]. This formulation means approximately the
following: everything that exists and develops has as the direction of its
development and existence the expression (coverage, absorption,
assimilation, comprehension...) by itself, existing and developing, of itself,

existing and developing. And such directionality or orientation of movementdevelopment is inherent in all developing forms without exception.
In the text of the article the reader will not encounter a single formulaic
expression for the determination (calculation) of the values investigated by
us, just as he will not see "juggling" with their numerical values, which at
present have the possibility of being reproduced with any preassigned
accuracy. The demonstration of such effects does not add, unfortunately,
clarity in the intention to find out the true nature and meaning of the
constants under consideration: for no matter how many times one says the
word "halva," it will not become sweeter in the mouth. For us in this case the
most important thing is to uncover and manifest the deep philosophy of
these phenomena, i.e., to reveal the essence, meaning, background, nature,
and the urgent necessity of the use by man of these parameters (values) in
his activity. The possibility of the solution of this task is determined by no
means by the skills of mastery of the craft, but by the level of mastery of the
art of "aerobatics," i.e., dialectics.
Without being distracted by the description of the astonishing properties and
characteristics of the dimensionless constants under consideration (these
properties are enumerated and described in abundance in the literary sources
by the references mentioned at the very beginning and in many, many
others), let us stop on the main, in our opinion, question: in what, strictly
speaking, consisted and consists for man the urgent need or necessity of the
introduction of these numbers into everyday use, into practice, into
circulation? In other words, we shall answer predominantly such a question:
what main problem for himself did man attempt to resolve (let it be
unconsciously, unthinkingly, intuitively) by way of the introduction of these
numbers into practical circulation? At that, let us note that, answering these
simple questions, we shall inevitably have to answer also the more general
question: what fundamental regularity of development leads to the necessity
of the seeking-out and use of such constants?
In two words and still very simplistically, the answer to these questions will
look approximately as follows.
When man does not know something, does not understand or underunderstands it (i.e., encounters something unexplored, both in the small and
in the large), he strives with all his might to express this unknown through
the known and cognized, — let it be not so smoothly and thoroughly, not
completely, but with some "appendage" of the not quite ordinary (say,
through some atypical or unknown — until the time — combinations of
known forms), but still mainly through the habitual and already familiar. And
this means that in such cases man is moved by the striving — to clothe the
new developing content (new knowledge) in a more or less known form, to
impart to this new content (new knowledge) a more or less already
meaningful and habitual appearance. Therefore, however strange this may
look, not only at the first steps, but always in such cases human thinking

solves the main philosophical problem for itself of the seeking-out and
clothing of the renewing and accumulating content of developing reality in a
form more or less (and in the limiting expression — in the most and
maximally) answering to this content and simultaneously "digestible" for
comprehension. In the same cross-section, the initial internal contentful
human activity or aspiration seeks embodiment or expression through the
external formal pragmatic execution; at that, the contentful side in this
tandem (relation of the contentful and formal components) always proves to
be the leading and moving side. Probably, for the majority of us this
assertion abides in a still insufficiently manifested and not fully realized form,
but the matter stands precisely so!
Now let us pass over to the detailed examination of what has just been said
with respect to the numerical constants of interest to us.
§ 1. Let us begin with the number "π" (pi)
As is known still from the school bench, the number "π" represents the ratio
of the length of the circumference to its diameter. This is the most distinctly
manifested relation of two sides-constituents of reality, for which are
characteristic in this case such basic relations of its properties as
{curvature/rectilinearity},
{continuity/discreteness},
{boundlessness/limitedness}, etc. In each of the noted relations there are
present and mutually linked two sides-constituents of reality. However,
notwithstanding the closest interconnection, neither the one nor the other
side is identical to the other and represents components not replaceable by
each other; at that, to such a degree not replaceable that between these
sides-components there always lies a naturally conditioned boundary of
division. Human thinking comparatively easily copes with the realization of
the formal side of such relations, i.e., of the "denominator," and matters
stand much worse with the perception of the contentful side — the
"numerator."
Proceeding from this, from the most ancient times everything still not
understood and incomprehensible man trivially strives to express through the
already understood and comprehensible. The relations adduced above in this
sense are no exception. In his attempts to express, to comprehend the
incomprehensible continuity and boundlessness of the curvature of the
circumference through the comprehensible limitedness and discreteness of its
rectilinear diameter, man inevitably comes to the sought relation, expressed
and designated by a certain from time to time approximately repeating
numerical value (subsequently named the constant "π"): in a rather obvious
and unpretentious way he manages to express the incomprehensible
irrational through the well-known and sufficiently studied rational.
Thereby, in the image of the number "π" we find the first productive attempt
of man to cope with quite serious difficulties, which are conditioned by the
objective presence of two objectively opposing and non-interchangeable

sides of reality [5, p. 55]. This is an undoubted achievement and victory of
man, but not without subsequent disappointments.
Perhaps the simplest algorithm (way) of finding (computing) the constant "π"
reduces to the seeking of the limit of the ratio of the perimeters of the
regular polygons inscribed in the circumference and circumscribed (around it)
(with the unlimited increase of the number of sides of these polygons) to the
diameter of this circumference. In this way is realized the striving toward the
expression of the most contentful correlation between the contentful and
formal components in their relation {length of circumference/diameter
of circumference}, indirectly determining, in essence, also the correlation
between
such
fundamental
sides-components
of
reality
as
{curvature/rectilinearity},
{continuity/discreteness},
{boundlessness/limitedness}, etc.
The straight line and the circumference — these are the two most popular
lines on the plane of the times of the absolute leadership of geometry among
all the sciences then being born. Between the two remarkable lines — the
circumference and its diameter — is enclosed a whole family of broken lines
(polygons) inscribed in the circumference (and circumscribed around it) with
perimeters easily computable and arbitrarily closely approaching the length
of the circumference. An excursion into history shows in what unpretentious,
but quite laborious way rational man approached the comprehension of the
irrational: Archimedes, as is known, in his computations went as far as the
calculation of the perimeters of the 96-gons inscribed in the circumference
and circumscribed around it [42, p. 43; 52]. And, despite such laborious and
painstaking work, to express the length of the circumference wholly and
without remainder through its diameter still did not succeed [18, p. 291; 24,
p. 131; 51, pp. 33–34]. All subsequent attempts of new times only confirmed
the unsolvability of this task in an exhaustively exact expression. Today the
constant of interest to us has already been computed with an accuracy of up
to billions of significant digits after the decimal point [20], but in the final
form the task has still not been solved. What does this speak of? Only of the
fact that in the course of investigations of this kind man has come up closely
against the inexpressibility of the real (really existing) irrationality by a
simple rational way, even in the case of a deliberately limited, so-called flat
geometric space.
Alongside the testimony of the achieved victory, corresponding to some
smallish possibility of representing one through the other, the number "π" —
this is at the same time also an indicator of the futility of the attempts to
achieve the final and complete expressedness of the objective
boundlessness, continuity, curvature through the no less objective
limitedness, discreteness, rectilinearity.
The inexhaustibility and progressive productiveness of such attempts, but at
the same time also the principled unattainability of the expression of the
entire fullness of content in a final kind and form are interpreted and
substantiated by the above-mentioned formulation of the principle of self-

development, which proves to be the background of that very (sought)
regularity of development leading to the necessity of the seeking-out and use
of this constant.
In the indicated connection, the meaning of the constant "π" is seen in the
fact that it fixes a certain transitional ("borderline") numerical correlation
between the contentful and formal constituents in the relation {"curvilinear"
circumference/"rectilinear" diameter}. In this case human thinking contrived
to express in a quite definite manner — by a more or less concrete number
— the correlation between curvature, boundlessness, and continuity (the
personification of which is the curved line of the circumference), on the one
hand, and rectilinearity, limitedness, and discreteness (personified by the
rectilinear diameter of the circumference), on the other hand. And in this
same case we state by no means a final and exhaustive, but still a productive
attempt of thinking to throw a bridge from the rational reality (from the
standpoint of traditional human thinking) to a reality to some extent
irrational (the term "irrational" is used here in the meaning of something
not fully cognized, insufficiently investigated, and comparatively more
contentful, not yielding to realization in full measure, and not in the sense of
something mystical and otherworldly). In the striving to express the
irrational through the rational (i.e., to throw a "little bridge" from one to the
other) and appear the remarkable relations, similar to the number "π". By
reason of the objective unattainability of the full and exhaustive expression
of the irrational through the rational (the infinite through the finite, the
continuous through the discrete, the indefinite through the definite, the
abstract through the concrete, etc.), such relations are expressed and
displayed by irrational numbers (i.e., infinite non-periodic fractions). The
very irrationality of such numbers-relations (inherent in the dimensionless
world constants under consideration) always presupposes, as it were, a long
(infinite) "run-up" before the "jump" from the formal side-constituent of the
relation to its more contentful side-constituent (for example, from the
straight line or plane to the circumference or sphere in the case of the
determination of the number "π").
Just as any natural formation, man and his thinking are at least two-sided
(dual), and therefore contradictory. By virtue of the indicated peculiarities,
even such a powerful instrument as abstract human thinking with its
characteristic enormous intensity of the realization of content-form
transformations proves unable to solve such tasks, moving exclusively in
measured-gradual steps; only by a leap in thinking (i.e., at the level of the
forms-concepts and representations mediated by human thinking about the
contradictory components of reality in their actual relations of
interconnection and mutual transitions) is the barrier or boundary of division
between the opposing sides-components of reality overcome.
Whether we want it or not, dialectics is present here and further, as
they say, at full height.

From the set of information about all sorts of investigations of the number
"π" it is known that there exists a multitude of curves of different orders,
somehow correlating with the segments of straight lines connecting the ends
of these curves on the plane, but only the ratio of the length of the
circumference to its diameter is the most large-scale and "full-blooded" in
magnitude (i.e., the most contentful) of all such relations. The highest
contentfulness of the relation-form expressed and designated by the number
"π" (as the ratio of the length of the circumference to the length of its
diameter) is conditioned, first of all, by the remarkable property of the
"numerator" of the relation, i.e., of the circumference; as mathematics
asserts, of all closed curves of given length on the plane the circumference
bounds the region of maximal area [2]. From the outstanding contentful
fullness of the circumference and proceeds the intuitively-practical interest of
man in the calculation of the length of the circumference by means of its
diameter — in the striving and hope of the finding of a simple device for the
exhaustive expression of the irrational curvature through the rational
rectilinearity. Let us keep in memory this remark about the highest
contentfulness of this wonderful correlation of the two components under
consideration in their relation.
Thus, of all the correlations between the components of reality in the general
relation-form {curvature/rectilinearity}, the number "π" represents such a
special correlation of the components in their particular (and most kindred to
the
general)
relation-form
{length
of
circumference/diameter
of
circumference} which answers to the maximally possible contentful filling (or
the highest contentfulness) of this form-relation.
At that, the irrationality of the number "π" testifies to the principled
impossibility of the establishment in exhaustive form of the final numerical
correlation between the components of the relation-form {length of
circumference/diameter of circumference}, corresponding to its maximally
contentful filling, and thereby to the impossibility of the acquisition of the
highest contentfulness by this form-relation.
The impossibility of the exhaustive expression of the highest contentfulness
by the indicated form-relation is ascertained in the course of the modeling of
the process of the acquisition (achievement) of this highest contentfulness by
the procedure of the computation of the correlations between the perimeters
of the polygons inscribed in the circumference (or circumscribed around it)
and the diameter of this circumference with the unlimited growth of the
quantity of the sides of the polygons.
The productively realized attempt to link two independent and noninterchangeable sides of reality proves subsequently to be in demand in the
plan of the use of the found compromise number "π" in all other cases which
one way or another require the replication of similar "borderline" values for
the adequate description of more complex real situations. Thereby the
number "π" belongs to the series of remarkable and extraordinary numerical

correlations thanks also to the need for it for the subsequent application in
the composition of formal (formulaic) expressions and dependences
describing the development of diverse (including compromise) real complex
forms and processes. The presence of the constant "π" in such descriptions
indicates the belonging of the forms and processes described with its help to
the number of optimal (i.e., endowed with heightened contentfulness).
Examples of diverse tasks in the solution of which the number "π"
participates as a constant for the description of various kinds of optimal
forms and processes can be found, say, in such printed works [6, 10, 11, 16,
24, 27, 30, 32, 33, 38, 46, 48, 49].
§ 2. Let us pass over now to the investigation of the constant "φ"
(phi), responsible for the so-called golden ratio, or golden section, or
golden proportion
The main peculiarity and rule of the construction of the golden proportion are
more or less precisely, in our opinion, conveyed by the words of I. Kepler,
who called this proportion itself continuing itself: "It is arranged so that the
two junior members of this endless proportion in sum give the third member,
and any two last members, if one adds them, give the following member,
with the same proportion being preserved to infinity" [15]. This quotation
describes a special algorithm of the composition of the whole from the
aggregate of parts.
Continuing to answer the question posed at the beginning of the article —
what fundamental regularity of development leads to the necessity of the
seeking-out and use of such a constant and of the golden ratio displayed by
it, — we state that this regularity too is conditioned by the above-mentioned
principle of self-development.
In the course of the realization of this principle, the universal internal
mechanism (algorithm) creates ever more "living" (i.e., ever more
contentful) unique natural forms-formations. Submitting to the single
principle of the exhaustive expression of the whole integrity of the developing
content, nature by successive approximations solves the task of the
coverage, representation, and comprehension of this contentful integrity by a
totality of ever newer and newer forms. In this concrete case investigated by
us, the role of the contentful and formal constituent sides in the generalized
abstract relation {content/form} is played by the components
{whole/part}.
Nature, as a definite-level external formalization of the developing reality,
constantly gropes for the most optimal way of solving the indicated task. By
the empirical path (by way of the realization of numerous approximations)
there are produced forms which most correspond to the requirements of the
exhaustive in the final analysis solution of the mentioned task, i.e., the most
contentful forms. Forms-relations reproduced on the basis of the golden
correlation between the components of these relations stand in the row of

such most contentful forms for their levels. The developing reality is capable,
through human thinking as yet another link-form in the chain of forms of
its contentful development, of fixing and evaluating the degree of
contentfulness of its own (both already existing and newly arising) linksforms in the striving of the latter toward the possession of the highest
contentfulness on their levels.
What is characteristic of the heightened (and in the limiting case — the
highest) contentfulness, and in what does it express itself with respect to the
forms-relations endowed with it? At the level of human consciousness the
mentioned contentfulness of the correlation displayed by the number "φ" and
the effectiveness of the chosen path are evaluated and confirmed by the
maximal density of the "mastering" (i.e., the "best mastering") of free space
by separate forms [2, 15], as well as, by virtue of this circumstance, by the
establishment of the presence of the indicated correlation in a whole series of
natural formations-forms-relations [1, 3, 14, 19, 23, 25, 40, 50, 51].
As in the case with the remark (see earlier) kept in memory about the
highest contentfulness of the correlation designated by the number "π", with
respect to the number "φ" the speech likewise goes about forms with
correlations (between the components {whole/part} in their relation)
answering to the expression of heightened contentfulness. The golden
relation-form differs from other relations-forms by such a correlation of the
contentful and formal components which ensures the maximal filling with
content of this form-relation at the given (this or that) stage-level of its
development.
A similar remarkable correlation is fixed by a certain number "φ" in the row
of all other numerical values corresponding to all other correlations within the
framework of the relation under consideration. And the matter, perhaps, is
not even in some concrete numerical value of this correlation, but in the fact
that such a remarkable correlation exists in principle among the multitude of
correlations and of the numerical values corresponding to them.
The special correlation of the contentful and formal components, ensuring the
maximal contentfulness of the golden forms-relations, endows these forms
with heightened stability and viability. For the expression of the remaining
advantages of such outstanding forms-relations, and in fact for the
evaluation of the degree or measure of their internal contentfulness, man has
invented and uses in everyday life a great multitude of all sorts of epithets
and definitions (directly linked, among other things, with the aesthetic
principle manifesting itself at the level of external human emotions), such as,
for example: divine beauty, perfection of form, ideality, exceptionality,
uniqueness, harmony, optimality, special property, high quality, "pleases the
eye," and many, many others. The heightened (maximally attainable or
highest for its level and stage of development) internal contentfulness of
such forms-relations, expressing itself in a set of their remarkable external
characteristics and properties, is connected with the presence, as already
noted, of the above-mentioned concrete numerical correlation in a whole
series of nonliving and living forms known to man. Confirmation of this can

be found in numerous literary sources (see the references above and below)
devoting excessive attention to the description of the wonderful properties of
the golden section. Coming to such a kind of optimal correlations in a series
of certain forms, nature from year to year continues by spiral-cyclical
movements to "polish" its creations, achieving results still more astonishing
and still more approximated to the expression of the highest contentfulness.
In such a way the developing reality itself, through the host of its changing
and newly arising forms in the channel of the algorithm (formulation) of the
principle of self-development, by periodic reciprocating movements
"refines" the value of this correlation in the relation {whole/part}.
Meanwhile, human admiration, deification, and exaltation of the known
properties represent a reaction of man only to the external signs of the
manifestation of the internal contentful fullness of such special forms and
leave without due attention this very internal contentfulness. Not being
sophisticated in such questions, man is inclined to absolutize and mystify
everything that is above his understanding, taking the superficial
phenomenon for the innermost essence. And the golden section as an
external manifestation of the heightened internal contentfulness of definite
forms is not an exception in this case.
So, in what, in fact, does the divinity of the forms-relations reproduced by
the algorithm of the golden proportion consist? Only in the fact that precisely
such a correlation, such a proportion of the part and the whole ensures the
maximally possible contentful filling of this or that form at a definite stage of
its development, i.e., at this definite segment of its path the given form is
most contentful (stands out from the other forms by its heightened
contentfulness). The loss or forfeiture by a form of a part of its heightened
internal contentfulness at another segment (stage) of its path inevitably
leads also to the change (or disappearance) of the corresponding external
signs of this form.
As in the case with the number "π", the establishment of yet another
remarkable correlation "φ" between the contentful and formal components in
their relation is, undoubtedly, yet another victory of rational man. About the
special relations of the whole and the part man has reflected from time
immemorial. The ancient Greek philosophers (beginning with Plato) supposed
that the whole does not reduce to the simple sum of its parts. But all the
most important suppositions and conjectures in this connection to this very
day abide in the category of the theoretical (not to say scholastic) and have
neither the status of recognized knowledge nor practical application. The
official encyclopedic editions likewise limit themselves to very meager and
vague utterances on this score [45, p. 768; 37, p. 1500].
The whole — this is not a simple sum (or totality) of its parts, but plus yet
something else, obtained as a result of the interaction of the parts among
themselves and of the parts with the whole. When the whole grows by
parts in such a correlation with the whole as is prescribed by the

algorithm of the golden division, then the forming whole (under which
are implied both the newly arising and the changing forms-relations)
acquires the maximal contentful filling. Thus, the number "φ" is a
quite concrete quantitative correlation between the whole and the
part in certain relations-forms, answering to the maximally possible
contentful filling of these forms-relations in the course of their
growth by parts in the mentioned quantitative correlations with the
whole at certain definite stages-levels of the development of these
forms-relations. Precisely this correlation (this proportion) of the whole and
the part answers to the highest contentfulness of the whole growing by a
part in the indicated correlation.
The regularity of the construction of the golden proportion described by
Kepler in an astonishing way resembles the sequence of Fibonacci numbers.
Astonishing, however, only at first glance. The abstract and, it would seem,
unconnected with practical application numerical sequence of Fibonacci is
nothing other than a model of the optimal, most contentful growth of the
whole by parts, i.e., a model of the modification of existing and the formation
of new forms of heightened (in the striving toward the highest)
contentfulness.
[A passing remark. Mathematics is excessively full of surprises, sometimes
considered products of the fruitless play of a brilliant mind, but having,
however, a direct relation to the objective reality, although not always an
obvious one. The whole matter is in the excess, speaking simplistically, of the
intensity of the content-form transformations at the level of human thinking
over the analogous intensity at the level of the subject-practical activity of
man, in consequence of which many contentful developments of the "pure"
reason gather dust on the shelves, humbly awaiting their "binding" and
demand at the level of formal practical embodiment.]
It should be especially noted that under the term "golden section" and its
numerical value there hides a boundary correlation between the components
of the whole and the part with its (the correlation's) limit, guessed in the
perspective of the infinite growth of the totality of the constituent parts in the
whole, modeled (the speech is about the growth) by the ascending sequence
of Fibonacci numbers.
The analysis of the tendency of the change of the values of the relations of
the neighboring members of the Fibonacci sequence, lying in the basis of the
search for the most contentful correlation between the whole and the totality
of its constituent parts in concrete forms-relations built after the type of the
golden section, testifies to the fact that the increase of the number of parts
included in the whole leads to the refinement of such a correlation and to its
ever greater optimization, but the obtaining of an exhaustively exact
numerical value answering to the most optimal correlation, even with an
arbitrarily great (unlimited) growth of the quantity of parts in the
composition of the whole (as also with the proportional increase of the

geometric dimensions of the whole and the parts), is impossible neither
practically nor theoretically. In other words, the contentfully fulfilled whole is
uncountable (does not yield to counting) and inexpressible wholly and
without remainder by any number of limited parts and their totality; with the
unbounded increase of the number of parts entering into the whole, modeled
by the ascending Fibonacci sequence, the relation of the neighboring
members of the sequence arbitrarily closely approaches a certain limit, but
never attains it. Only in such an unfinished way and in an approximate
manner is expressed (and can be expressed) the contentful fullness of the
uncountable whole through the infinite succession of countable totalities of its
parts.
Thereby the analysis in the indicated connection of the unending sequence of
Fibonacci numbers allows coming to an important conclusion about the
principled unattainability of the exhaustive expression of the content of the
whole by any formal totality of its parts, even with the unlimited increase of
the quantity of parts in the totality.
Continuing the investigation of this model (i.e., analyzing the variability of
the values of the relations of the neighboring numbers of the Fibonacci
sequence), one can discern also the presence here of a certain likeness of
forms developing at different levels with different volumes of contentfulness
(with the variation of the number of parts or of their sizes in the composition
of these different-level forms), but the repetition (i.e., the equality or
identity) of forms belonging to different levels with different volumes of
contentfulness is unrealizable and impossible in principle! About this again
testifies the infinite incompleteness and unattainability (in computation) of
the exhaustively exact correlation designated by the irrational number "φ",
observed (the speech is about the incompleteness and unattainability) in the
investigation of the character of the change of the values of the relations of
the neighboring members of the increasing Fibonacci sequence.
The golden proportion, or the special correlation of the whole and the part
designated by the number "φ", is an indicator of the heightened
contentfulness of forms-relations of a definite type, manifesting itself through
the optimality of properties-characteristics and through the aesthetic
perfection of these forms. The irrationality of this number testifies both to the
unattainability of the exhaustive expression of the highest contentfulness by
the forms of such proportions, and to the possibility of the unlimited
approximation to its exhaustive expression. The coming of the event of the
exhaustive expressedness of the highest contentfulness by the form of the
golden relation would mean that, beginning with some step of the realization
of the algorithm of the computation of the constant "φ", the correlation of
the whole and the part (of the golden proportion) would remain unchanged,
i.e., limited by some quantity of signs (by any their quantity, but necessarily
limited), and therefore unchanged. However, the infinity and non-periodicity
of the fractional part (i.e., the irrationality) of the number displaying this
correlation (as in the case with the number "π") speaks of the opposite.

As in the case with the number "π", the infinity and non-periodicity of the
fractional part (the irrationality) of the number "φ" means that very barrier
which separates two sides-constituents of one and the same relation
{content/form} that are incompatible in exactness and non-interchangeable.
As in the case with the same "π", again we note the infinite "run-up" or the
unending transition from one component of reality to the other (from one
region of reality — into the other) in their relation. If the case with the
number "π" speaks of the impossibility of the whole expression of the
boundless continuity and curvature by the limited discreteness and
rectilinearity, then the case with the number "φ" testifies to the principled
inexpressibility in an exhaustive manner of the contentful whole (again
uncountable and continuous) by means of the totality of formally limited
parts (i.e., by means of the countable and discrete). Despite the
inseparability and interconnectedness of the contentful and formal sidesconstituents in the known relations "π" and "φ", the barrier between the
contentful and formal components (in both relations) is essential to such a
degree that it cannot be overcome by any successive or gradually-measured
steps-approximations, however long these steps-approximations might be
carried out. Even human thinking with its enormous potential is not capable
of overcoming this barrier by measured gradualness; it is overcome
exclusively by a leap. However, the latter always conceals in itself something
not fully understood and not fully said. Partly for this very reason the nature
and meaning of the fundamental world constants up to now nobody can
properly explain.
Thus, the number "φ" is such a special quantitative correlation between the
components in the relation-form {whole/part} which answers to the
maximally possible contentful filling (or the highest contentfulness) of this
form-relation.
At that, the irrationality of the number "φ" testifies to the principled
impossibility of the establishment in exhaustive form of the final numerical
correlation between the components of the relation-form {whole/part},
corresponding to its maximally contentful filling, and thereby to the
impossibility of the acquisition of the highest contentfulness by this formrelation.
The impossibility of the exhaustive expression of the highest contentfulness
by the indicated form-relation is ascertained in the course of the modeling of
the process of the acquisition (achievement) of this highest contentfulness by
the procedure of the computation of the correlations of the neighboring
members of the infinitely-increasing Fibonacci sequence.
Undoubtedly, the number "φ", fixing a definite correlation of the components
(sides-constituents) in certain concrete forms-relations, is connected with a
certain optimality of these concrete forms-relations, conditioned by such a
remarkable-kind correlation of components realized in them. At that, one

cannot but see that the internal contentful fullness of these forms is primary,
while the optimality and the remaining external qualities-characteristics of
such forms are derivative and secondary; in practice, however, everything is
just the opposite — the formal side is taken for the initial one, and only then
through it do they attempt to express the rationality and irrationality of all
internal connections and relations.
The heightened internal contentfulness of forms-relations, marked by the
golden correlation of their components, sets the most optimal intensity of the
development of such forms, ensuring their prolonged and productive
contentful existence at certain stages. However, a similar scenario can be
true and realizable only at definite segments of the development of these or
those forms; with the change of conditions and circumstances (i.e., in the
course of the inevitable further universal development) these or those
bearers of golden proportions may lose their miraculous potential. For this
reason it is very imprudent to be carried away by the excessive
absolutization of the special correlations of components and of the formsrelations corresponding to them, predicting for such forms the main roles for
all times. Real forms are different and all sorts, and therefore the entire
spectrum of developing forms-relations is by no means exhausted by one
(this or that) concrete correlation of their components. At the given stage of
development precisely such a correlation of the constituent sides within
certain forms-relations is and looks most optimal, stable, and expedient; at
another stage under other conditions and circumstances this correlation may
be forcedly transformed (just as over the centuries change the sympathies
toward the proportions and beauty, say, of the female figure). Therefore to
attempt to reduce all forms-relations to the golden relation, i.e., to
absolutize, fetishize, mystify this relation and to spread the known myth
(about the through presence of this correlation) wholesale over the entire
host of unique developing forms — is absurd. Moreover, in the indicated
connection the main task of man and of his thinking is to learn to foresee and
anticipate the appearance of new correlations (including optimal ones, i.e.,
distinguished by heightened contentfulness) of the contentful and formal
sides-constituents in their relations-forms at new stages of development,
which, in fact, will determine the further progress of mankind.
In general, all manner of assertions about the "final" discovery of the most
final and perfect of forms (be it the golden section, or some "universal"
mathematical formula, or some "super" technologies) contradict the very
spirit of development, which, despite any "absolutes," breaks open and
overcomes by its contentful side any form attempting once and for all to fix
in itself and by itself any contentful movement-development; from
definiteness => through indefiniteness => to new definiteness, — such is the
most general form of the chain of the accomplished development. At that,
the elements of indefiniteness and unpredictability of the course of
development were, are, and will always be — in accordance with the
formulation of the principle of self-development (the return to oneself, but
every time to oneself changed, enriched, and developed). This context

presupposes as natural the representation of the fact that the moment of
indefiniteness is also an objective and inalienable element of every
development.
Examples of the description of the optimal kind of all sorts of forms and
processes, the construction, formation, or development of which submits to
the regularities of the golden proportion, can be found in such printed works:
[2, 14, 15, 19, 25, 40, 50].
§ 3. With the examination of the essence of the number "e" the
matter stands still more complicated
In the scientific and educational literature the constant e is identified as the
base of natural logarithms, which by no means introduces any additional
clarity in our case. The constant e is not connected in an obvious way with
simple geometric relations like the constants π (pi) and φ (phi) examined
above. If the number π serves for the most part for the description of quite
static forms-relations, the number φ — for the description of transitional
forms ready for dynamic development, then the number e is present in the
description of the most dynamic, in the fullest sense of the word, developing
forms-relations-processes-systems, the components of which prove to be,
first of all, such sides-constituents of reality as: {motion/rest},
{time/space},
{energy/mass},
{kinetic
energy/potential
energy},
{temperature/pressure}, etc.
As the authors of modern publications consider: "If π is responsible for
space, then e — for time and likewise manifests itself almost everywhere"
[20]. In other words, e is responsible not for statics, but for dynamics, for
the process, for changing and progressing forms (and not for forms for the
most part object-like and static).
Proceeding from this, one should expect that the interconnection and
correlation of the components in their relation, designated by the number e,
are more complicated and more contentful than in the relations of π and φ,
since with respect to the constant e the speech goes about components of
another "floor" of reality, of another level of contentfulness, than those levels
which are occupied by the components of the relations of π and φ.
Correspondingly to this, the numerical value of e fixes another, than π and
φ, "boundary" and transitional correlation between the mentioned
components in their relations.
Besides, from the position of our representations about the development of
reality (i.e., from the position of the concept of development) the aboveenumerated set of relations ({motion/rest}, {time/space}, {energy/mass},
{kinetic energy/potential energy}) enters into the "clip" or "sphere of
influence" of such a generalized relation as {change/basis}. On this
foundation, further investigations of the essence of the constant e we shall

conduct in the context of precisely this, most informative for us, initial
relation.
The relation {change/basis} — this is a completely different "class," a
different level of forms-relations, than the relations {curvature/rectilinearity}
and {whole/part} examined by us earlier (in the consideration of the
constants π and φ). The form-relation {change/basis} differs from the two
latter forms-relations by a more expressed dynamism, mobility, liveliness,
changeability.
But all this, as before, clarifies little; for a magic carpet by description may
be both square and Persian, and take part in air parades, but the main thing
(in the channel of our mood for the revelation of the inconspicuous essence)
would still be the elucidation of that — what makes it a carpet-plane.
Therefore, again, as in the cases with the numbers π and φ, let us turn to
the most capacious definitions and interpretations of the constant e, as well
as to the history of its appearance and origin.
The initial and most effective, in our opinion, push or key to the
comprehension of the sense of the number e is provided by its classical
definition [8, p. 231; 26, p. 354]. Scrupulously examining this definition of
the number e from the position of the concept of development mentioned
already more than once, one can discover that under the very definition of
the given number there hides a peculiar algorithm of the realization of the
most effective way of the accumulation and assimilation of the accreting
content by some form developing in the direction of the acquisition of the
highest (maximally attainable) contentfulness! And again the speech goes
about a certain contentfulness, which likewise served for us as a key moment
in the course of the revelation of the essence of the constants π and φ. The
form realizing itself by such an algorithm represents an optimal formation
thanks to the way, following from the given definition, of the appropriation of
the maximally possible volume of content accreting on the basis of a very
limited initial contentful volume of the form itself and over a quite definite
finite interval of time.
On what is this assertion based? On the comparison of the textbook definition
of the constant e with the way of the calculation of the compound bank
interest accrued on a certain sum of the initial money deposit in the course of
a definite time interval [31]. Strictly speaking, the history of the origin of the
number e, as is noted in many literary sources, is connected precisely with
the seeking-out of the maximal limit of the growth of the money deposit
accreting with respect to its initial sum under the conditions of infinitely
shortening intervals of time between the moments of the accrual of interest
on the successively accreting at those same moments sum of this deposit.
Only, in contrast to the history with the origin of the number e, in the case of
the investigation of the definition of this number from the position of the
mentioned concept of development under the sum of the deposit and the

bank interest there are implied a certain form of the initial content (basis)
and a certain abstract content "winding" onto the basis (change), accreting
to the basis by the algorithm corresponding to the above-mentioned way of
the accrual of the bank interest.
According to the algorithm used for the calculation of e, providing for ever
more frequent (accelerating) interest accruals on the correspondingly
accreting sum of the deposit, with such an increasing frequency accretes also
the very sum of the deposit; a similar algorithm is realized, as we have
already established, also in the computation of the ever more exact value of
the constant φ by means of the infinite sequence of Fibonacci numbers. Only
in contrast to the forms, the static proportions of which are determined by
the measures of the number φ, the forms formed after the patterns of the
number e are built by the dynamic path, with the participation of the stepby-step changing component of time, by virtue of which also in this latter
case the accumulating content does not succeed in being expressed by a
form wholly, — the state of contentful "saturation" is not attained (does not
set in) in principle, since there is always, there is always present a "beyondform" leakage of content, despite the process of the successive growth,
perfection, and raising of the degree of the "all-embracingness" of the form
of the representation of this content. Hence, even in the regime of the
rapidly developing dynamic layering of content (change), predetermined (the
speech is about the regime) by the indicated algorithm of the computation of
the number e, and even with the limited content of the isolated initial form
(basis), it is impossible in the limit (fully and without remainder) to cover,
absorb, and appropriate exhaustively all the accreting content by any final
form whatsoever! Here we again (as in the case with the number φ)
encounter the principally unattainable limit of the accumulation and
expression by any perfect form whatsoever of exhaustively all the
unswervingly accreting (in the course of the development of this form)
content.
It must be said that mathematicians, describing e as a remarkable limit of a
certain functional relation, do not connect the algorithm or procedure of the
computation of the value of this relation with any of the physical processes
[2]; in other words, for the mathematical cast of mind the examination and
solution of the task in the most general abstract formulation is a natural and
habitual matter.
In the case examined by us of the use of the algorithm of the computation of
the number e for the calculation of contentful accumulations (changes)
proceeding from the initial limited contentful volume (basis), the
"denominator" (i.e., the basis) of the relation {change/basis} adduced
earlier, like the "numerator" (change), directly participates in the contentful
accretions and accumulations: changes do not occur on an empty place, but
always "cling" to the basis (to the initial deposit, to some starting capital or
contribution), and this concerns not only the development of specific forms-

relations in the domain of economics and finance, but also the development
of any forms whatsoever and in any spheres whatsoever.
A similar point of view on the correlation (interconnection) of the indicated
components in the mentioned relation is considered self-evident (can one
really separate the change itself in its "pure" form from that which
changes?), — see, for example, about the connection of change and the
stability of the properties, structure, and laws of the existence of bodies [44,
p. 161]. Since the numerator of the relation ("change") is not essential by
itself (outside the "basis"), then under the "change" of the numerator one
should understand the "contentfully changing basis" (or the "basis accreting
with content"). Under the "basis" of the denominator there is understood the
"basis of the initial content." So, the analysis of the given algorithm shows
that the maximal sum of the increment of the content of the changing basis,
i.e., of the "numerator" of the relation, in comparison with the content of the
initial basis, i.e., with the "denominator" of the relation, under the conditions
of the realization of such a special algorithm (specific way of the increment of
content) cannot go out beyond the limits of the excess of the contentful
primary basis in e (as a certain definite number) times.
Let us now examine, in what the specific algorithm of the calculation of the
compound bank interest (i.e., of contentful accretions and accumulations in
the context of our concept of development), coinciding in form with the
algorithm of the computation of the constant e, differs from the other
algorithms of bank accruals.
According to the mechanism of the realization of the given algorithm, the
maximal increment by content of a certain form of limited initial
contentful volume in the course of a finite interval of time is attained
under the condition that every infinitely small part of the contentful
volume of the given form ensures an increment, externally accreting
to this part of the contentful volume, proportional to the sum of the
initial contentful volume of this part and of the contentful volume
accreting to it in the course of periodically repeated contentful
additions throughout the mentioned interval of time. By this somewhat
ornate phrase here there is merely verbally transmitted the essence of the
well-known mathematical expression for the computation of the constant e
[8, p. 231; 26, p. 354], — and nothing more! In other words, thanks to such
a way of accumulation (contentful enrichment), every particle, every smallest
(infinitely small) element (at first of the initial, and then of the "layer-bylayer" accreting form) participates in full measure in the general single
process of the accumulation and assimilation (assimilation) of the externally
arriving content. And even under such a condition, the total volume of the
content of the initial form (i.e., of the basis) and of the content externally
arriving or accreting to the content of the initial form (i.e., of the change) is
not capable of exceeding the contentful volume of the initial form (basis)
more than in e times.

Thus, the number e is located on the boundary of the demarcation of the two
components in the relation {change/basis} and binds them together by a
definite dependence, conditioning the change (accumulation) of the content
accreting to the basis (i.e., "sticking" or "winding" onto the initial,
deliberately limited volume of the basis) in the total calculation, not
exceeding the volume of the primary basis more than in e times.
As it can now be seen, the algorithm of the calculation of e resembles the
formulation of the principle of self-development. And in the one, and in the
other cases there is present the dynamics of the development of the form in
time and space. The difference consists only in the fact that the first variant
describes the particular case of the development of a form of initially limited
contentful volume (initial capital, a definite sum of a money deposit in a
bank, some process in an isolated spatial volume, etc.) throughout a
concrete finite time interval; the second, generalized variant is applicable to
the description of the development of any forms (including open developing
systems, as well as the totalities of systems as a whole) without any spacetime limitations.
The conditions of the realization of such a special algorithm (answering to the
above-noted particular case) presuppose the presence of explicit limitations
both of the contentful and of the formal plan; there is evident the limitedness
both in time of the accrual of the interest, and in the volume of the initial
deposit (i.e., limitations of the space-time character). But along with this, all
the advantages of such an isolated process are likewise demonstrated in full
force.
Being a particular case of the general principle of self-development, the
algorithm of the calculation of the number e (similar in form to the process of
the "winding" of content described by the formulation of the principle of selfdevelopment) allows separate forms under the specific conditions of the
activity of their limited contentful primary basis in the course of finite time to
accumulate the maximally possible content by virtue of the circumstance that
every small particle, every smallest element of these separate forms
participates in a direct and full-valued way in the process of
contentful accumulation.
Thus, the number e, or more precisely the algorithm of its obtaining
(computation, calculation), represents a description of the most optimal way
(procedure) of the assimilation and accumulation by some concrete separate
(isolated) form of the maximal volume of the content arriving from outside.
If in the cross-section of what has been said one imagines that the
developing reality (i.e., the entire world surrounding us) as the universal
form of a single complex process (not having limitations neither of the
formal, nor of the contentful character, i.e., not limited neither in space, nor
in time) were composed ("woven") of such maximally effective (optimal)
small "e-forms," then this single universal big complex form would be

characterized by the highest contentfulness. However, the "harsh" reality
introduces its corrections in the form of the interference and interpenetration
of forms into the "fates" of one another. The "atypicalness" of such "e-forms"
is conditioned by the circumstance that the given form is built on the basis of
an initial form, contentfully limited and quite isolated from the interaction
with other forms; a similar algorithm of development is not viable in
application to the incomparably more widespread open forms-processes,
which, by virtue of the forced interaction with other forms, develop by more
general and less optimal patterns. Meanwhile, the "e-form" — this is a vivid
example of the most optimal and effective development of a form within the
framework of content-form limitations (i.e., limitations, in this case, of the
space-time character).
Because of the described limitations, a similar specific algorithm (after the
type of the calculation of the number e) is not realizable in full measure with
respect to the open forms-systems, having universal distribution and
developing by the general patterns of the generalized formulation of the
universal principle of self-development. The capitalization of a limited bank
deposit or starting capital — this is almost the only specimens of really
existing more or less closed isolated systems, to the description of the
development of which the algorithm of the computation of the number e is
ideally applicable: probably, for this very reason all literary sources, as if on
command, refer exclusively to the example of the calculation (sanctified by
the name of Bernoulli) of the compound usurious interest, — as to the
example of a really functioning isolated system, guaranteeing the maximally
attainable contentful accumulations under the conditions of known
limitations.
However, this by no means signifies that the advantages of such special
forms are in any noticeable degree not realized in practice; the fact that the
specific "e-forms" by urgent necessity are widely used by man in the form of
dimensionless values in the description of a huge number of all sorts of real
processes signifies that the real forms are oriented toward development after
the image and likeness of the most contentful "e-forms," but by virtue of the
merciless external influence, interference, and interpenetration on the part of
the same forms the trajectory of their development can be substantially
distorted and significantly differ from the most optimal one. In other words,
the designation of the number e (preserving in the "folded" form the
algorithm of the obtaining of this number), being present in various
combinations with other parameters, variables, and arguments in formulaic
expressions describing all sorts of real processes, indicates thereby although
far from the highest, but still some optimality (heightened contentfulness) of
the forms of such processes. Besides, the need for "e-forms" in the
description of complex forms-processes is conditioned also by the
circumstance that by these small forms there are "patched" the ruptures or
steps on the boundaries of the division between the sides-constituents of
reality, objectively taking place under the conditions of the development of
big forms and processes; compromise small forms in similar cases are simply

necessary for the adequate and correct description of such really developing
big processes and forms.
Every form-relation-process has as the direction of its development the
optimization of itself (which, strictly speaking, constitutes also the essence of
the formulation of the principle of self-development). By this same
circumstance there is conditioned also the presence-presence of optimizing
values ("e-forms") in the formulaic expressions-descriptions of diverse real
complex forms-relations-processes. As it follows from our investigation, the
most effectively the task of the above-mentioned optimization is solved
nevertheless by the "e-form," which, not having by virtue of the limitations
accompanying it any noticeable distribution in the "pure" form, enters,
however, as a component part and is extremely in demand in the description
of optimally developing real general forms and processes.
Along with this, the limitedness by the number-limit e — not the only
remarkable feature of the scenario of the development of forms in the
channel of the algorithm investigated by us. The second remarkable
peculiarity of the given scenario — the principled unattainability of the limit
of the highest (maximal) contentfulness and the inexpressibility of this limit
in an exhaustively exact digital form. The process proceeding by such an
algorithm has neither a platform, nor a point of "saturation." And again we
encounter the relation of two components, whose transition from one to the
other within the framework of the relation is impossible by a smooth
measured course. In the role of the interacting and opposing components, as
has already been noted, in this case there act the sides-constituents
"change" and "basis." The relation composed of these components of reality
(analogously to other, more concrete relations of the given type:
{motion/rest}, {time/space}, {energy/mass}, {temperature/pressure},
etc.) is doomed to the presence (between the sides-components of this
relation) of a barrier in the image of an irrational number, answering to a
certain objective, although not expressible exhaustively, limit correlation of
these components. And, finally, the third remarkable feature of the same
scenario consists in the possibility of its infinite development and
approximation to the expression of the highest contentfulness of the
mentioned relation arbitrarily closely.
Neither practically, nor theoretically is it possible to establish exhaustively
exactly the correlation, and consequently, also the boundary of the division
between the accreting contentful volume ("change") and the initial contentful
volume ("basis"); in an analogous way it is impossible to determine also the
exact boundary of the division between the components in the "kindred"
relations
from
the
same
"clip":
{motion/rest},
{time/space},
{energy/mass}... With whatever frequency and however long one may
attempt to realize similar steps of approximations in the course of the
realization of the algorithm analyzed by us, there will always remain some
contentful part, not expressed, not calculated, and not subject to calculation
by any rational form. And this barrier between the expressed content and the

content incapable of being expressed in all its fullness by any form
whatsoever — is insurmountable in principle!
In this very thing consists the sense of the irrational number e. The infinite
non-periodic fraction of this number personifies that same infinite "run-up"
before the "jump" from one domain of reality (from the rational reality of the
form) — into its other domain (to the irrational reality of the content), i.e.,
from one (rational) component of the relation — to the other (irrational)
component of this same relation.
Thus, the number e is such a special quantitative correlation between the
components in the relation-form {change/basis}, which answers to the limitpossible contentful filling (or the highest contentfulness) of this form-relation.
At that, the irrationality of the number e testifies to the principled
impossibility of the establishment in an exhaustive form of the final
numerical correlation between the components of the relation-form
{change/basis}, corresponding to its maximally contentful filling, and
thereby to the impossibility of the acquisition of the highest contentfulness
by this form-relation.
The impossibility of the exhaustive expression of the highest contentfulness
by the indicated form-relation is ascertained in the course of the modeling of
the process of the acquisition (attainment) of this highest contentfulness by
the specific procedure of the accrual of the compound bank interest with the
unlimited shortening of the intervals of time between the moments of its
accrual.
As the experience of observations shows, in the descriptions of optimally
proceeding real complex processes (with the maximal for definite conditions
contentful increment) there are always present "e-forms," indicating thereby
the belonging of the forms of these general complex processes to the number
of optimal forms. On this score in the scientific literature there are presented
numerous examples of the descriptions of the forms of the most contentful
development of all sorts of processes in the most diverse domains of reality
[9, 10, 11, 12, 17, 20, 29, 30, 31, 33, 36, 38, 41, 43, 46, 49, 55].
§ 4. Having determined ourselves with the nature, meaning, and
necessity of the introduction by man of the world constants into
practical circulation, let us make yet several more clarifications and
explanations
As it has been possible to establish in the course of the investigation, all
three constants examined by us — π, φ, and e — are limits, toward which
functions-relations of a certain kind strive with the unlimited increase of their
arguments. The historical excursus allows revealing the circle of particular
practical tasks, in the course of the solution of which the first mentions of the
above-named values appeared. The solutions of the indicated tasks are















connected with the finding of the algorithms of the computation of the
following limits:
in the case of the constant π — of the limit of the relation of the
increasing perimeters of the polygons inscribed in the circumference (or
circumscribed around it) to the diameter of this circumference [42, p. 43];
in the case of the constant φ — of the limit of the relation of the
increasing total whole to its parts on the basis of the modeling of the
computational procedure by means of the ascending sequence of Fibonacci
numbers [39];
in the case of the constant e — of the limit of the relation of the content
accreting in a definite way (up to "saturation") onto the content of the initial
form (i.e., of the "numerator") to the content of the initial form (i.e., to the
"denominator") on the basis of the modeling of the procedure of the
computation of the compound bank (or usurious) interest [31].
The further analysis gives the possibility to discern the relations of the
general sides-components of reality hiding behind the relations of the
particular sides of the practical tasks:
in the case of the constant π such sides of reality are the components of
the relation {curvature/rectilinearity}, as well as the components of a
series
of
other
"kindred"
relations:
{continuity/discreteness},
{boundlessness/limitedness}, etc.;
in the case of the constant φ — the components of the relation
{whole/part}, as well as the components of a series of other similar
relations: {general/singular}, {function/structure}, and the like;
in the case of the constant e — the components of the relation
{change/basis}, as well as the components of a series of other similar
relations: {motion/rest}, {energy/mass}, {time/space}, etc.
A scrupulous study helps to understand that behind the above-indicated
constants there hide ways or devices of the expression and representation of
the objective, but not quite comprehensible for man (irrational) sides of
reality through the no less objective and considerably more comprehensible
for man (rational) sides of it. In this sense the constants examined by us
prove to be for human consciousness peculiar "transitions" or "bridges" from
the domains of the rational — into the domains of the irrational:
in the case of the constant π — this is the transition from "rectilinearity"
to "curvature";
in the case of the constant φ — this is the transition from the "part" to
the "whole";
in the case of the constant e — this is the transition from the "basis" to
the "change."
The above-mentioned limits are precisely the personification of the barriers,
which must be overcome in order to "get over" from one domain of the
relation into its other domain, from one component of reality to its other
component. The infinite non-periodic string of digits of the fractional parts of
the irrational numbers π, e, φ (and they are known today already with

billions of significant digits after the decimal point [2]) looks like an unending
"run-up" before the "jump" from one domain of reality into another and
simultaneously indicates the impossibility and unrealizability of the smooth
overcoming of this barrier — the latter is overcome only by a leap in
thinking, i.e., at the level of the notions and representations about the
interacting contradictory sides-components of reality in their relations.
Being expressed by irrational numbers, the given constants symbolize not
simply transitions from one component of reality to another in their actual
relations, but transitions infinite, uncompleted, unfinished, unclosed...
Precisely this circumstance allows making one of the most important
conclusions: if π, φ, e and can be called numbers in reliance on the
established terminology with respect to the definition of an irrational number,
then "constants" one cannot call them in any way!
The symbols π, φ, e designate "living," uncompleted transitional processes,
by which, strictly speaking, is caused the use in formulaic expressions and
descriptions of any and all real forms and processes precisely of symbols,
and not of concrete numbers.
These are special forms of the expression of specific transitional contentful
processes, which it would be proper to name precisely so: "π-transition," "φtransition," "e-transition" or "π-form," "φ-form," "e-form," — proceeding
from the corresponding understanding and handling of them as of the
designations of actual transitional forms and processes.
The mathematical form of the expression of such limit transitions is the
function. In the great multitude of descriptions of other all-possible formsprocesses (i.e., in the diverse formulaic expressions) these functions are
present not as constants, but as forms of processes, in principle not having
the possibility of being expressed by constant finite values; in the final
analysis π, φ, and e — these are infinite and unrealizable to the end
transitions from the rational to the irrational, clothed in mathematical form.
As soon as and when these transitions are reduced to numbers with a limited
quantity of significant digits after the decimal point, only then do they cease
to be both processes, and transitions, and functions inside the descriptions of
more complex and composite processes; only then do these forms turn into
constants, the presence of which instead of the corresponding symbols limits
the contentful potential (fullness and depth) of the mentioned descriptions.
Thus, under the designations π, φ, and e there are implied "folded"
algorithms (modeling the actual transitional processes) of the computation of
the limit numerical values of the correlations of the components in the
relations
{curvature/rectilinearity},
{whole/part},
{change/basis}
respectively.
Besides, in the course of the examination it was ascertained that all the
irrational numbers π, φ, e examined above designate limits or limit
correlations of the components in their relations, answering to the highest
contentfulness of the relations of these components. If one imagines that the

mentioned relations {curvature/rectilinearity}, {whole/part}, {change/basis}
can be characterized by all-possible values of the correlations of their
components in the ranges (determining the measure of the contentfulness of
these correlations) from the conditional "zero" (i.e., of small contentfulness)
up to some highest contentfulness, then the limit numerical values of the
"round π," the "golden φ," the "saturated e" will correspond precisely to the
maximal contentfulness of the above-enumerated relations (which, as has
been noted more than once, in itself is likewise in principle unattainable and
representable only in the form of an unattainable limit).
Observations and practical experience show that by no means all formsrelations (objects, processes, phenomena) belong to the number of the
"round," "golden," or "saturated." The overwhelming majority of the real
"ordinary" forms-processes-relations occupies the most diverse steps of
contentfulness within the limits of the ranges indicated above; true, all these
"ordinary" forms-relations-processes likewise have as the directionality of
their development the attainment of all the same heightened (and in the limit
expression — the highest) contentfulness. But the main thing in this
connection is nevertheless that among the boundless diversity of formsrelations man has managed to single out such which are characterized by
heightened contentfulness, which is precisely what distinguishes these formsrelations from all the rest; at that, to the singled-out forms-relations it has
been possible to put into correspondence the limit correlations of the
components of these relations established by man in the form of more or less
concrete numbers, which subsequently became the indicators of the
heightened contentfulness of the given forms-relations. The singling-out and
advancement of such forms from all the rest was and is accompanied by a
burst of human emotions (admiration, tenderness, and delight), which in
reality prove to be a reaction by no means to the internal heightened
contentfulness of these forms in comparison with others, but to their external
signs, characteristics, and properties. And the matter here, perhaps, is not
even in the values and magnitudes of the numbers π, φ, and e themselves,
but in the fact that such special numbers exist and they correspond to special
(standing out from the rest) relations-forms of the development of reality.
As has already been noted, the symbols π, φ, e designate the "boundary"
correlations of the components in the most contentful specific formsrelations-processes. But these are a kind of particular cases among the
multitude of other more general forms, the development of which is
described by the generalized algorithm (formulation) of the principle of selfdevelopment. Therefore the presence of the symbols π, φ, e in the
description of other more general complex processes and forms indicates the
optimality and heightened effectiveness of these other complex forms and
processes in comparison with the configurations of all the other real
processes and forms. Besides, in the presence of objectively taking place
ruptures or steps between the sides of reality at the boundaries of their
division, in a series of cases the adequate description of really developing

forms and processes without the attraction of the "π-," "φ-," and "e-forms"
is simply impossible.
The following assertion will be the answer to the question posed at the
beginning of the article: all three algorithms for the calculation of the limit
correlations of π, φ, e — these are particular cases of the generalized
algorithm (formulation) of the principle of self-development, which lies
precisely in the basis of the fundamental regularity of development,
conditioning the necessity of the seeking-out and practical application of such
"constants." In all three cases the speech goes about conditionally "ideal,"
conditionally "exceptional," limit-contentful forms-relations, in some way
isolated (especially in the case of the "e-form") from the interaction with all
the other real forms. In the "harsh" reality of such exceptional (ideal) forms
there are very few, they are — special; the forms of the open real world,
however, are the product and consequence of the confrontation and
interaction of the whole host of the forms of the open world. And with
respect to such a kind of open real forms and processes the speech is
conducted already not about their absolute perfection, exceptionality, and
ideality (as in the case of the particular specific forms-relations), but about
their movement-development in the direction of a certain optimality (i.e.,
of the maximally attainable contentfulness), but under the conditions of the
real interference and intervention of other forms. And in the same way the
sign of the orientation toward optimality and the highest contentfulness, but
already in the case of the real complex open processes-forms, is the
presence of the small "π-," "φ-," and "e-forms" in the descriptions of these
big real open processes-forms.
Thus, π, φ, and e — these are the algorithms of the particular specific formsrelations, characterized by the expressed directionality toward the attainment
of the absolute perfection and exceptionality of their special forms, whereas
the algorithm (formulation) of the principle of self-development determines
the universal directionality of the movement-development of any all-possible
forms-relations of reality, including also the specific particular forms
investigated by us.
Any and every SOMETHING is a relation, accessible to expression with
respect to everything else and in mathematical or digital form. The number
in the given case acts as the measure of any and every relation; the number
is the quantitative characteristic of the relation, indicating the correlation of
the components in their integral relation, the degree of the participation of
the sides-components of the relation in one another, the degree of the
expressiveness of one constituent by means of its other, the position of the
boundary of division between the sides-components in their relation. The
known examples of the found remarkable limits of π, φ, and e speak of the
real and quite realizable possibility of the numerical expression (cognition) of
one through the other: of the whole through the part (or through the totality
of parts), of the boundless continuity through the limited discreteness, of the
developing content (change) through the existing form (basis), of the

uncertainty through the certainty, of the unknown through the known (but
every time with some differences from the already known), etc., etc.
Precisely in such a way (by way of the expression of the unknown or of the
contentful not "graspable" by consciousness in full measure through the
known or cognized form-like) is realized the process of cognition (as,
incidentally, also development itself), and another way and other path simply
does not exist. The numbers π, φ, and e — these are examples of the
resultative attempts of man to express one (insufficiently cognized) through
the other (sufficiently comprehended).
However, besides the load — to be "bridges" between the sides-components
of reality — the so-called "constants" examined by us fulfill yet one more
important function: all three "constants" — these are boundary pillars
(milestones), standing on different frontiers and outlining the boundaries of
the objective physical level of reality; the first two (π and φ) are located on
the boundaries of the division of the domains connected with the external,
for the most part static geometric (spatial) forms, the last one (e) — on the
boundary of the division of the components of the more abstract, more deeplying and dynamic (temporal) forms-relations.
Further, taking into account that π, φ, and e are located on the boundaries
of the components of completely differently planned and differently leveled
(in any case, obviously not intersecting) forms-relations, which are the
relations-forms
{curvature/rectilinearity},
{whole/part},
and
{change/basis}, one must seek the quantitative connection between these
forms-relations (and, correspondingly, between the numbers answering to
them) with the account of this differently-leveledness, i.e., more
meaningfully. It is not excluded that with time such an internal connection
will be established or there will be discovered a system of numeration (and
maybe, also a system of reference), in which the world "constants," like the
chemical elements of the Mendeleev table, will in an ordered way occupy
definite little cells of the table, thanks to which it will prove possible to
forecast the appearance and introduction (dictated by practical necessity)
into circulation of new world "constants."
As for the other world "constants," and in particular the so-called physical
"constants," then, probably, in a similar way one can investigate and
establish the genuine essence both of the speed of light [4, pp. 98–107; 5, p.
37], and of the gravitational constant, and of the Planck constant, etc. At
that, it is necessary to remember that every world "constant," still
insufficiently comprehended and for the time being passing for an
inexplicable and unsolved number, — this is always a barrier or at least a
"boundary pillar" on the frontier between the incompatible and differing from
one another components-constituents of a certain relation, i.e., between the
objective sides of reality. Moreover, it is no less important also to understand
that the mentioned relations of the noncoinciding and opposing sides
of reality are not located somewhere there, in the "far distance" from
one another, and are not separated between themselves by some

"spaces" and "times," but are interwoven into one another and are
found in a single bundle in every smallest element of reality,
occupying its different levels-floors.
The "constants" have an important significance, since they stand on the
frontiers between the thoroughly cognized and the insufficiently studied or
not yielding to understanding also as easily and freely as the already
cognized; precisely the world "constants" outline the boundaries of the
rational (from the position of the stereotyped human thinking) world, i.e., of
the habitual and very "definiteness-like" objective (geometrically-physical)
level of reality. If π is located on the boundary of the transition from the
linear (flat) world to the spherical (filled with curvature), then the number e
and a whole series of other world "constants" stands on the threshold of the
transformation of again the habitual for us three-dimensional world
(absolutizing its static spatial-form side-constituent) into the dynamically
changing, so-called space-time continuum. In view of the importance of the
relations fixed by the designations of the world "constants," with each of
these relations and the correlation of their components one must deal
unhurriedly and carefully; offhand such questions cannot be correctly
resolved. In any case one must begin with the study of the history of the
origin of this or that problem form-relation — here at once there may be
discovered the main hint for the disclosure of the secret of its special
properties and, correspondingly, popularity. The following of the principle of
the observance of the unity of the historical and the logical in the conduct of
such examinations — is especially topical.
The understanding of the essence of the world "constants" pushes apart the
horizons of our representations about the developing reality, since the
"constants" are the indispensable participants of the correct modeling of the
behavior of the most multi-leveled reality, being found in the composition of
the mathematical (formulaic) expressions describing this behavior. In this
connection it is most expedient, in our opinion, to adhere to that position
that with respect to each concrete case the best solution, interpretation, and
explanation will be found by the "narrow" professional, who has mastered the
concept of development presented here and mentioned yet again more than
once. Say, the specialist in the domain of the history of the formation of
geometry (with the knowledge both of mathematics and of its history, well
acquainted moreover with the indicated concept of development) will find it
easier than anyone else to come out onto the initial conditions and
circumstances which led to the empirical or intuitive discovery of the
remarkable correlation of π; or the physicist (well acquainted with the
historical background and again with the given concept) will find it simpler to
deal with the number e; and the biologist (zoologist, botanist), who has
mastered the concept of development, — with the regularities of the golden
relation in living forms, etc., etc. The world "constants" are waiting for their
researchers.

In conclusion let us note that the present article serves as an example of the
practical application of the concept of development as an instrument for the
investigation of the surrounding world.
Conclusion
Let us sum up what has been set forth.
1. The world "constants" — these are the external forms of the manifestation
of the internal connections and relations of the sides-components of reality;
these are peculiar "bridges" between the mentioned sides-components in
their real relations; these are the limit correlations of the definite sidesconstituents of reality, expressed by numerical values.
2. The concrete examples of the remarkable limits of π, φ, e testify to the
principled realizability and resultatively realized by man aspirations and
attempts of the quantitative expression (let it be even approximate) of the
insufficiently cognized (irrational) sides-constituents of reality through the
comprehensible and habitual (rational) sides-constituents of it.
3. The irrationality (i.e., the infinity and non-periodicity of the fractional
parts) of the numbers corresponding to the mentioned limits of the relations
of π, φ, and e indicates, in the first place, the principled unattainability of
the expression in an exhaustive form of the irrational side-constituent of
reality through the rational side-constituent of it and, in the second place,
the possibility of an arbitrarily close approximation to such an exhaustive
expression.
4. The same irrationality of the numbers π, φ, and e indicates also the fact
that these numbers — by no means "constants," as is customary to consider,
but certain forms of infinite processes of the transition from one sidecomponent of reality to its other side-component. Perceiving this at the level
of intuition, man for that very reason includes them in the formulaic
expressions and descriptions of real forms and processes not in the form of
concrete numerical values, but in the form of symbols.
5. The world "constants" — these are a kind of "boundary markers" along the
perimeter of the boundary of the most studied by man, comprehensible for
him and habitual objective (physical) level of reality, i.e., precisely the world
"constants" outline the boundaries of the so-called rational (from the position
of the everyday human consciousness) world.
6. The limits of the relations of π, φ, and e — these are also "milestones" on
the boundaries of the division between the components of quite differently
leveled relations, such as: {curvature/rectilinearity}, {whole/part},
{change/basis}; for that very reason to seek the quantitative correlations
between their numerical values without the meaningful penetration into
these relations and their components preceding the given search, in our

opinion, is little effective. It is quite admissible that between the aboveenumerated forms-relations there exists a deep-lying contentful (nonformal)
connection, which may be discovered with time; but in order to speak about
this concretely, further in-depth investigations of these very relations and
their components are needed.
7. The numerical values of π, φ, and e designate the correlations of the
components in the above-enumerated forms-relations, answering to the
heightened (and in the limit — the highest) contentfulness of these formsrelations on their levels, which makes these forms-relations the "happy"
possessors of their remarkable properties and characteristics, fixed by the
human consciousness as something special and exceptional.
8. Since, without resorting to the help of the "constants" investigated by us,
many real processes it is simply impossible to describe correctly and
adequately, man attracts these specific forms-relations (π, φ, and e) for
participation in the descriptions of many other real complex forms and
processes. Being present as the designations of the processes of transition in
the formulaic expressions and descriptions of other, more complex forms and
processes, these specific forms-relations of heightened contentfulness
indicate thereby the optimality of those other forms and processes in
comparison with the multitude of other "ordinary" forms and processes, not
distinguished by a similar optimality. Besides, under the designations of the
"constants" in such descriptions there hide compromise connections
(bridges), "smoothing" the objectively existing ruptures or steps between the
components of reality.
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