“Soft” System of Coordinates in Regular Simplexes

Abstract

In different areas of knowledge, there are common problems of coding and ordering of multicomponent
compositions of objects as well as representation of their processes of change. These problems are solved by the language method RHAT. It can be considered as a coordinate system of regions in space limited by regular simplex. Here R is a sequence of composition components by decrease – names of sectors in the simplex distinguished by hypermedians, a semiquantitative substantial characteristic of compositions, "word" of a new type; H – Shannon information entropy – entropy of mixing; A – anentropy – entropy of separation; T – tolerance. The arrangement of coordinates allows obtaining an alphabetical hierarchical periodic system of compositions (HPSC) of objects of different nature, in particular, the Hierarchical Periodic System of Chemical Compositions (HPSCC) that uses the Periodic System of Elements (PSE) as an alphabet, as well as the Hierarchical Periodic System of Molecular Compositions (HPSMC) that uses HPSCC as an alphabet. Diagrams HA or HT are designed to display random and ordered sets of compositions of any nature as the most appropriate means of studying the processes of separation and mixing. The applicability of the method has been tested on analytical materials in natural and
social science fields.

Full text

International Journal of Intelligent Information Systems
2015; 4(1): 1-7
Published online January 23, 2015 (http://www.sciencepublishinggroup.com/j/ijiis)
doi: 10.11648/j.ijiis.20150401.11
ISSN: 2328-7675 (Print); ISSN: 2328-7683 (Online)
“Soft” system of coordinates in regular simplexes
Tomas Georgievich Petrov
St. Petersburg State University, Institute of Earth Science, St. Petersburg, Russia
Email address:
tomas_petrov@rambler.ru, tgpnever@gmail.com
To cite this article:
Tomas Georgievich Petrov. “Soft” System of Coordinates in Regular Simplexes. International Journal of Intelligent Information Systems.
Vol. 4, No. 1, 2015, pp. 1-7. doi: 10.11648/j.ijiis.20150401.11
Abstract:
In different areas of knowledge, there are common problems of coding and ordering of multicomponent
compositions of objects as well as representation of their processes of change. These problems are solved by the language-
method RHAT. It can be considered as a coordinate system of regions in space limited by regular simplex. Here R is a
sequence of composition components by decrease – names of sectors in the simplex distinguished by hypermedians, a
semiquantitative substantial characteristic of compositions, "word" of a new type; H – Shannon information entropy – entropy
of mixing; A – anentropy – entropy of separation; T – tolerance. The arrangement of coordinates allows obtaining an
alphabetical hierarchical periodic system of compositions (HPSC) of objects of different nature, in particular, the Hierarchical
Periodic System of Chemical Compositions (HPSCC) that uses the Periodic System of Elements (PSE) as an alphabet, as well
as the Hierarchical Periodic System of Molecular Compositions (HPSMC) that uses HPSCC as an alphabet. Diagrams HA or
HT are designed to display random and ordered sets of compositions of any nature as the most appropriate means of studying
the processes of separation and mixing. The applicability of the method has been tested on analytical materials in natural and
social science fields.
Keywords:
Regular Simplex, Coordinate System of Region, Information Entropy, Entropy of Separation, Anentropy,
Ultrapurification, Tolerance, Information Language, Hypermedian, Hierarchical Periodic System
1. Introduction
Background of the method, common for the natural
sciences, is presented in [3, 9, 10]. Mostly it relates to the
multicomponent nature of objects (cenoses, communities,
mixtures), namely: uncertainty of boundaries between
compositions of such objects, which is significant in nature
and in all areas of human activity, including dispensing
medicines and food, choice of winner, study of isomorphism
in crystals, financial capacity determination, military
solutions, etc.) and uncertainty of naming what is inside the
boundaries, difficulties in visibility and ordering of object
compositions, diversity of content measurement units and
disproportionality of large and small in the same composition,
difficulty in identifying the general direction of the evolution
of compositions under multi-directional changes in
individual components. To reduce the complexity of solving
these problems, information language method, or rank-
entropy method (hereinafter - way or method) for displaying,
particularly, coding of compositions RHAT has been
proposed. Here R is a sequence of symbols of composition
components by reduction of their contents; H is C. Shannon
information entropy as entropy of mixing; A – anentropy [6-
13, 15, 20, 21,], as entropy of separation [13, 15]; T –
tolerance [10, 15] - as "entropy of purification". The method
has been repeatedly described in [6 9-13; 17], and will be
outlined briefly here. Details are presented in [21].
Method wasinitially proposed particularly for convolution
of chemical compositions of rocks and minerals, but later its
usefulness has been shown in solving conceptual tasks in
various fields of knowledge [4, 13, 15,16, 19, 25; 26].
Method is usually set out in terms of algebra, information
theory. Here, we present it at the elementary level as a way of
describing – coordination – coding of an area in space
limited by a regular simplex with equal distances from the
center to all vertices, and distances between all vertices. Of
all simplexes, triangle has long been a “sacred cow” of
mineralogists, petrologists, and many other geologists, at the
same time becoming a dead end in the representation of
multicomponent systems on paper. Background demand of
visibility at preservation of the traditional approach has not
enabled to pass into the volume and work in it, since even
imaging of tetrahedron content is problematic.
Representation of 5-10-50-component analysis without a

2 Tomas Georgievich Petrov: “Soft” System of Coordinates in Regular Simplexes
radical convolution of analysis is impossible, and, at the
same time, chemical analyses with 50 or more elements have
become quite common. On the other hand, actively working
"naturalists" and "humanitarians" accumulate thousands of
chemical, mineral, biospecies, age, linguistic "analyses".
There are problems in putting them in order, discovering
duplicates, defects, seeing the material as a whole (rather
than its individual components, or the relations between two
or three of them), seeing the relationship between
compositions, concentrations and rarefactions, banality and
originality of individual analyses, identifying the direction of
change and path length of the evolution of composition, and,
amid all this, in detecting a law or prohibition of a certain
already formed hypothesis: all that was worth of the efforts
spent on the method development and is worth of the work to
become acquainted with it.
The method can be considered both as a way of coding,
and as a "soft" system of coordinates, meaning by its
"softness" lack of an exact solution of the inverse problem
with the number of variables exceeding the number of
equations. Thus, 40-element composition analysis of a
complex mineral, tourmaline, contains 50 unknowns
convolved in 4 equations: H, A, T and Sump
i
=1.
Under "soft" coordinate system we mean description of
area position (and not point, as in a "hard" one of some
space, in this case, space of a regular simplex).
Purpose of the article is to pay attention to the universal
encoding method of multicomponent compositions RHAT, as
well as to the problems in mathematics arising from the use
of the method in different fields of knowledge.
2. R, H, A, T as Coordinates in a Simplex
Space
Figure 1. Simplex with names of sectors distinguished by medians
First coordinate: R, rank formula of composition, rating, is
a sequence of symbols (or names) of n components (selected
from the total possible in this branch of knowledge number N)
by reduction of their contents p, when Sump
i
=1. This series
(e.g. "quartz-feldspar-mica", a sequence by reduction of
contents of principal rock minerals – granite) is a sector
name in a regular simplex having three vertices when split by
medians in a triangle (Fig. 1). Rating R, a semiquantitative
while substantial characteristic of composition, is regarded as
the major semantic part of the composition convolution in
the form of RHAT code. Four-component composition
requires a tetrahedron when split by median planes, and
further – an unthinkable regular polyhedron with median
hyperplanes. In the simplex vertices all p = 1, that is, here are
absolutely pure components (not present in nature, but more
on that below). In the simplex center p
1
= p
2
= p
3
==…. p
n
=1/n. In volume of a simplex with n vertices, the number of
sectors К= n! An elementary example of sector naming is
shown in Fig. 1.
Contents of the first component are in the range 1/n-1. In
other ranks, 0-1/n.
Besides the said role, R is a means of component selection
if the sample contains analyses of different lengths, which is
critical for subsequent calculations of composition
characteristics. Namely, for comparability of the processing
results of analyses with varying n detail, after ranking of the
initial data, further computing includes the amount of
components not greater than n of the shortest analysis. This
does not eliminate the possibility to find in the analysis a
component, the content of which is smaller than would be in
the case of a more complete analysis. Therefore, there is a
choice: increasing of details and reduction of confidence in
the correctness of the result, or increasing of confidence at
reduction of analysis details.
2.1. Information Entropy H or En
The second coordinate: H, Shannon information entropy
up to a constant is an analog of the thermodynamic entropy
of mixing [2] and the measure of complexity, diversity of
system composition. (The diversity problem is discussed in
[26]). Evaluation by formula H = – Sump
i
lnp
i
. To reduce to
the interval 0-1 we use En = H/lnn, where lnn=Hmax.
Maximum value of contribution of one component in entropy
–p
i
lnp
i
for logarithms with any base is at p =0.368…. In a
simplex-triangleisolines of entropy normalized to 1 are
shown in Fig. 4. In accordance with isoline appearance,
entropy can be regarded as a generalized distance from
simplex vertices to the composition point.
Figure 2. Isolines of entropy normalized to the interval 0ч1.
2.2. Anentropy A or An
The third coordinate, anentropy [6, p.34; 9, p.17] is
regarded as a measure of smallness of small components and
calculated by formula A = –1/n*Sumlnp
i
– lnn. According to
the formula, A exists in the range 0-+∞. To reduce to the

International Journal of Intelligent Information Systems 2015; 4(1): 1-7 3
interval 0-1, convenient for many applications
diagrammaticform (in combination with entropy), A of a real
composition is divided by anentropy of a "simple analytically
perfectly clean" system[9, p.18]. This is necessary, since zero
in composition responds to the logarithm of minus infinity.
For maximum pure composition is taken composition, in
which p
1
= 1–(n–1)*ẟ, p
2
=p
3
=…=p
n
= ẟ, where ẟ are
contents equal to half sensitivity of the analysis method. This
trick is dictated by V.I. Vernadsky "ubiquity principle" (as it
is called, slightly reinforcing the position of its author) [1.
p.518-527], according to which any macroobject contains all
elements of the Periodic System. Let us add: tricks are useful
as well (with their ẟ) when considering compositions of any
real systems, whether mineral, national, age etc., in which the
smallest contents are almost always underestimated - to the
detriment of the meaning of quantitative estimates.
Anentropy isolines in a regular triangle are shown in Fig. 5.
In accordance with their nature, anentropy can be considered
a generalized distance from the simplex center to the
composition point.
Anentropy is regarded as entropy of separation, since
many observations of trajectories of processes of separation,
for example, by grain size in sedimentary rocks, population
age, during mineral processing, show its increase [13, p.24;
22].
Figure 3. Isolines of anentropy normalized to the interval 0-1.
2.3. Tolerance – Т
Tolerance as a coordinate [10, p.30; 13, p.24], the
contribution to which is 1/p, is calculated by formula T=
ln[1/n*Sum(1/p
i
)] –lnn. We do not show isolines of
normalized tolerance because they are pressed to the simplex
vertices even in a greater extent than anentropy isolines. This
characteristic is most applicable for estimating the degree of
composition closeness to the simplex vertex, to a state of
perfect purity of composition.
To distinguish it from the entropy of separation (as a
means to display widely used methods of relatively coarse
separation), tolerance, the means to show processes of
extremely pure substances, can be called entropy of
purification.
From the viewpoint of information theory, the value of
tolerance is the logarithm of average signal waiting time at
their uniform arrival.
Contributions of three quantitativecomposition coordinates
are shown in Fig. 6.
Figure 4. Contribution into entropy , anentropy - , tolerance - .
Contributions into A and T are functionally dependent on
the contribution into entropy. If we take the contribution into
entropy (– plnp) for a primitive function, its first derivative
with respect to p minus one will be contribution into
anentropy (– lnp), contribution into tolerance – the second
derivative with respect to contribution into entropy and the
first one with respect to contribution into anentropy (1/p).
Thus, both anentropy and tolerance are natural outcomes,
subsequent development stages of the ideology embedded in
C. Shannon information entropy. The transition from the
widely applicable one-dimensional data representation of the
complexity of systems to the two-dimensional, diagrammatic
and three-dimensional, spatial, that is to the means more
filled (or richer) in content is implemented this way.
3. Systematics – Ordering of
Composition Codes
RHAT composition coordinates (with descriptions of
specific objects) are recorded in a row and arranged in a
column, as shown in Table 1. For this a set, series of RHAT
coordinates is taken as a word that uses three alphabets. One
is letter, for rank formulas playing the role of "roots",
semantic meaning of a word, and two numerical: for entropy
("suffix") –by reduction of H values and for A and T – by
their increase ("inflexions"). Ordering is done according to
the known linguistic principle, consistently. First roots are
ordered, then suffixes, and further inflexions.
In general, it is desirable to use any "intensional" alphabets
as symbols in rank formula R[25, 28]. They are understood
as alphabets which have semantic links between adjacent
symbols (entities), as opposed to "natural" alphabets, in
which such links either do not exist or are very weak [28]. It
is known that dictionaries using "natural" alphabets usually
put close words and form groups having similarity in
inscription (alt, altar, alter, altimeter, altruism…) and rare

4 Tomas Georgievich Petrov: “Soft” System of Coordinates in Regular Simplexes
with a similar sense. Unlike the latter, when using the most
famous intensionalalphabet in encoding chemical
compositions of geological objects, the Periodic Table of
Elements, groups of rank formulas OSiAlH..OSiAlNa…
OSiAlK contain different, but related in chemical
composition objects. They have a common name -
"alumosilicates"; some of them are presented in Tab. 1. When
rank formula length increases, certainty of allocating a
portion of space, to which the composition belongs, and, on
the other hand, the degree of generalization at its reduction
are growing.
In case of equal R, ordering of composition coordinates is
made by reduction of H, the preferential direction of entropy
change during separation [29]. (Separation is understood as
formation of two or more systems from one that differ in
composition. Mixing is the opposite process of forming a
relatively homogeneous system of two or more differing in
composition). In case of equal H, ordering of coordinate lines
is made in the usually opposite direction of A and T changes,
i.e. by their increase.
Such ordering results in a Hierarchical Linear Periodic
System of Compositions, in particular, chemical, when
coordinates of chemical compositions of objects are ordered
[12]. Table repeated from [12] shows a sample of code
coordinates from the existing Database of Chemical
Compositions that contains more than 80 000 entries. This is
a negligible part of a full table (probably impossible to
construct), given the number of stable elements (88?) and
taking into account n= 10 elements in R. Horizontal lines in
the table are dividers revealing the hierarchical structure of
the system. Equal signs mean that p
n
/p
n+1
≤1.15, i.e. indicate
the proximity of contents of the adjacent components in a
rank formula.
Table 1. Sample from the "Chemical Compositions of the Universe" Database
Rank formula En An Object
H
O
C
N
Ca=
P
K=
S
Na
Cl
0.428
0.434
human body
H
O
N
Cl
Si
Li
B =
S
C
Ca
0.278
0.980
water, geyser, Kamchatka
O
C
Ca
Mg
Fe
Si
P
Al
Mn
K
0.561
0.210
carbonatite, Sallanlatva
O
Mg
Si
Fe
Al
Ca
Na
K =
Cr
Ti
0.542
0.301
Mars
O
Mg
Si
Fe
Al
Ca
Na
Cr
K
Ti
0.511
0.305
The Earth, mantle+crust
O
Si
H
Al
C=
Ca=
Mg=
Fe=
K
Na
0.578
0.166
Quaternary clay
O
Si
H=
Al
Fe
K
Mg
C=
Ca
Ti
0.361
0.401
sandstones, Kazakhstan
O
Si
Na
Mg
Al=
Ca
Fe
Mn
W
Ti
0.286
0.804
quartz, Transbaikalia
O
Si
Mg
Al
Ca=
Fe
Cr
Ti
Mn
Na
0.554
0.274
pyrope, Urals
O
Si
Mg
Fe
Al
Ca
Na
Mn
S
K
0.567
0.193
meteorite, Zhmerinka
O
Si
Al
Na
K
H
Fe
Ca
Mg
Ti
0.488
0.247
granite, average of 2,485 an.
O
Si
Al
Ca=
Fe
Mg
Ti
Na
K
Mn
0.552
0.236
basalt, Moon
O
Si=
Ca
C
H=
Fe
P
F=
K
Al
0.617
0.138
carbonatite, Malawi
O
Ca=
C
Fe
Mg
P
Si
Al
Sr
Na
0.519
0.278
carbonatite, Kovdor
O
Ca
Fe
P
Mg
Si
Al
Na
Mn
Ti
0.569
0.268
phoscorite, Kovdor
F
Ca
Ba=
Ti
Zr
O
Be=
Al
Bi
Mn
0.281
0.962
fluorite, Transbaikalia
S
Fe
As
Sb
Zn
Pb
Co=
Ni
Bi
Se
0.282
0.967
pyrite, Sibay
Cu
Sn
As
Fe
Sb
Pb
Ni=
Ag
Bi=
Co
0.069
0.526
bronze, knife, Alekseevka
The database of object descriptions contains a reference to
the source, the number of table and analysis in it, as well as
other specific information about the object.
4. Properties of Databases Systematized
Using RHAT Method
Database, a sample of which is shown in Table, has the
following properties:
Versatility – with respect to the possible diversity of
chemical compositions (hence the database name), since
there is no chemical (let us add: and any other) composition,
which would be impossible to imagine in RHA(T) form;
Linearity - has no branching and therefore is extremely
simple;
Algorithmicity – when constructing;
Openness – to expand the list of components;
Stability, i.e., the system allows for the removal and
inclusion of new records without changing the order of the
others;
Countability – the maximum number of rank formulas
U(from Universum) can be defined as the number of
permutations without repetitions of N possible components
of this alphabet by n, i.e. U=N!/(N–n)!. For N=88 and n=10
we have U= 1.64E+19. The chemical Universe can be placed
in this catalog.
Completeness – for a certain alphabet length and a certain
length of rank formula there are no and cannot be other rank
formulas over the number of permutations, as defined above.
Therefore, for these N and n the sequence is a Universum of
existing, possible and impossible compositions in this field
of knowledge;
Hierarchicity– ordering, as in alphabetical dictionaries, is
primarily done by the first, the highest rank (first letter).
Further, inside the rank formulas with the same first rank
ordering takes place by the second, lower rank (second letter),
etc. As a result, we have a hierarchical ordering of
codes.Rank formula with length n is an ordered list of all
higher taxa;
Periodicity – rank formulas of objects with similar
composition are arranged in table in groups, between which
there are other, much different ones; thus, in Table 1 rank

International Journal of Intelligent Information Systems 2015; 4(1): 1-7 5
formulas OCCa…OCaC… located at opposite ends of the
table belong to rocks of the same facies – these are carbonate
varieties.
Arrangement of objects in a system manifests connection
with the procedure of changing some properties of object
ordering in the original alphabet (here, in the Periodic Table).
Thus, for chemical compositions, from the classification
beginning to its end, average atomic masses of objects, their
densities grow statistically, occurrence in nature decreases
statistically.
5. Entropy diagrams HA and HT
Values of all three numeric coordinates H,A,T depend on
the number of components. Therefore, if desired to deal with
full analyses, very often having different n, their comparison
will be either difficult, or impossible, or this fact can be used
(but at a sufficiently large amount of data) to identify the
structure of the information space. Specifically, to image
variety of materials by their detail (analyses length), that is
clearly manifested inHA diagram of age distributions of
population in countries of the world [14].
Bearing in mind great importance of the issue let us
partially repeat. For comparability of data, R lengths are
standardized on some general level and, after renormalization
of sums of truncated analyses to the unity, HAT is calculated.
To account for the fullest possible information, standard
should be brought into proximity with the length of the
shortest analysis in the sample. It should be borne in mind
that the shorter the analysis, the greater the chance to lose
underestimated components and include in the calculation
lower values than would have been at a more complete
analysis, respectively, the higher anentropy can be [21, p.68].
There is reason to believe that the delay in the use of entropy
in some branches of knowledge, particularly in geology, is
associated with ignoring the need to standardize the analysis
details before computations.
Field of allowable EnAn values for a 10-component
system is shown in Fig. 5. The upper limit was obtained by
mixing of compositions described in detail in [9, p.17].
Figures near the points along the upper curve correspond to
the number of components with equal contents, assuming
that the remaining ones (10 minus "figure"), supplementing
the analysis to ten components, had contents of half
sensitivity of the analysis method taken for 0.00005. (This
value corresponds approximately to the sensitivity of "wet"
analyses that have been ubiquitous until recently and did not
lose value with the advent of new technologies). The lower
curve meets the condition: p
1
= 1– (p
2
+p
3
+…+p
10
), thereby:
p
2
=p
3
=p
4
=…=p
10
.
Abbreviations S denote point of original compositions
(Start), F (Final) - final product compositions. The diagram
shows the typical directions of composition change during
separation (two S-F patterns on the left) and mixing
processes (two patterns on the right), found in tracing the
evolution trajectories of compositions in dozens of real
processes of change in composition of rocks and minerals.
Featured direction ratios during mixing – combination of two
different compositions into one, and during separation –
formation of two or more different compositions from one
illustrate Yu.V. Shurubor’s theorems of preferred directions
of entropy change during mixing and separation [29].
Figure 5. Limits of EnAn field and typical directions of composition
evolution during separation and mixing (explanation in the text).
Separation and mixing processes shownin Figure 5, "pull"
points of compositions to vertex No.1, if content of one
component is growing in composition (native elements, or
close to mono-ethnic community); to No.2, if, except for two
components with equal contents, the remaining components
are removed (purification of rock salt NaCl); to No.3, if a
three-element compound of KOH, NaOH type is purified,
and so on. Reverse movement, towards the simplex center
(No. 10 in the diagram) corresponds to mixing processes.
Neither simplex center nor its vertices contain real (or at least
chemical) compositions, since to achieve these points, non-
existent in nature and in laboratory ideal pure substances are
required. Everything happens between these two extremes in
a cyclic implementation of separations and mixings. For
example, in geology, formation of granite is separation, then
hydration of certain minerals, mixing, further disintegration
of rock with removal of having become mobile solutions and
clays – separation, entering of granite degradation products
in streams and soil–mixing and so on, for each preserved
chemical element of the primarysystem, an eternal change of
participation in mixing and separation processes. In
biocenoses, birth and death of organisms is imposed on
separation-mixing processes, but the reciprocating motion of
a point noting the seasonal cycles in HA diagram will occur
monotonically as long as the ecosystem is not "sick" or starts
to disintegrate.
According to the data published in [5] and calculations
conducted by the author, it is known that the scatter in EnAn
data in the middle ofthe diagram when using standard
chemical analysis in geology for n=10 is approximately
±0.005 in both axes.
Description of the diagram ispresented in [9, 13] andin
most detailin [21]. Model of the separation-mixing process in
qualitative presentation is given in[15].

6 Tomas Georgievich Petrov: “Soft” System of Coordinates in Regular Simplexes
6. Unresolved Problems
Development of the method has mainly taken place
"broadwise", towards the testing of opportunities and
meaningfulness of results when using the method in various
areas of knowledge, which are already about 20. At the same
time, the mathematical aspect has been apparently developed
insufficiently.
Exceeding the number of unknowns on the number of
equations implies uncertainty when solving the inverse
problem, inability to determine p
i
for HA or HT data.
However, in each triple combination: R
i
, H, A, (or R
i
, H, T),
p
i
range is not unlimited. There is a task to find a way for
calculating p
min
and p
max
for each R
i
of the system with
cardinal n for the set HA, HT, EnAn at p
min
= ẟ. In other
words, what are shapes of "bodies of R
i
contents" in
projections on the fields of EnAn, HA, and HT diagrams?
Another question is very important in the practice of
studying evolution of compositions. All mathematically
made mixings (interpolations) of compositions, that is when
signs of change are preserved, and the vast majority of the
examined processes of evolution of compositions of natural
objects give arcs convex downward or downward – to the
right (such as given in [13] in Fig. 7). [3] describes the
process, which has a curve of the opposite form: bulge
upward – to the left. As has been well established, it took
place in two stages: the first one, with the normal trend of
separation – decrease of H, growth of A, and the second one
with the opposite trend, mixing. There was a task to
determine whether this type of curve may correspond to a
"one-way" ("monotonic") process, that is, without changing
the signs of change in content? If we prove that this is
impossible, there will be confidence that the processes of this
type are not monotonic.
7. Some Results and Prospects
RHAT is a universal method for describing compositions,
one of the methods along with the principal component
analysis, cluster analysis, and others that may be included in
generalizing discipline "Composistics" enabling to work with
compositions, intensities, weights, values of components of
any complex systems [8 p.271]. The method provides a wide
range of possibilities; alternatives for applying it are
described in [11]. The same publication lists three types of
work, namely: with primarily disordered analytical materials,
with a database, and with materials ordered in space and time.
Proposed in 1971, the method can be said to lie long, it
was not properly claimed in geology due to several reasons,
among them probably the most important: a) unconventional
measurement units – atomic fractions accepted by the author
for a universal approach – for chemical composition (instead
of remaining since the 19th century oxides such as SiO2,
Al2O3, FeO…); b) integrated approach to complex
compositions at prevailing in diagrams consideration of only
2-3 components of composition, c) more than once
mentioned logarithmic curve. It is also very likely: d) distrust
of a geologist-crystallographer who encroached on the
solution of problems that appeared to be insoluble in the 70-
ies: uniform ordering of all chemical and mineral
compositions, and placing in a single diagram, having a
genetic interpretation, compositions of objects of any nature,
and thus being able to display and study the evolution of the
whole world of multicomponent compositions in diagrams,
and not using one-dimensional sequences of information
entropy. The proposed method for construction of a two-
parameter alphabet was the first step to address the issue of
content encoding of crystal-chemical structural formulas of
minerals [16] (and not only them), in combination with the
R-dictionary-catalog of chemical compositions of minerals
[18]; it offered an opportunity to create a unified structural-
chemical classification of real minerals and to develop on its
basis the looming "RHAT dictionary-catalog of crystal-
chemical compositions of minerals".
Work on the method and several others is supported by
original Petros3 software enabling to deal with 100 alphabets,
to form databases, including bibliographic one, to perform
calculations of correlation coefficients and distances of
several kinds, convolutions of large amounts of information
in the form of "generalized rank formulas" etc. [10].
8. Conclusion
The author will consider his mission accomplished, if he
draws attention to the fields of mathematics, that enable a
consistent encoding of compositions of objects of any nature
and development of an algorithm for constructing rank-
entropy alphabetically ordered Hierarchical Periodic Systems
of Compositions, in contrast to the original Periodic System
of Elements, in addition, showing the processes of
composition change, revealing common in disparate at first
glance processes in stagnating, living and social, in particular,
regarding them as separation and mixing – implementation of
differentiation and integration, recognized fundamental by
Herbert Spencer.
Acknowledgements
The author thanks his long-time collaborator N.I.
Krasnova for dedicated efforts to disseminate the method,
S.V. Moshkin for programming and implementation of the
endless requests to add another method in the software, to
explain how-to, A.G. Bulakh, Yu.L. Voytekhovsky, and
recently deceased V.V. Gordienko for deep understanding
and high public estimation of the method, S.V. Chebanov for
continued support and strengthening methodological
fundamentals of the method, A.A. Andriyanets-Buyko, V.A.
Glebovitsky, V.Ya. Vasiliev, V.N. Dech, I.S. Sedova, P.B.
Sokolov, O.I. Farafonova, V.V. Khaustov, A.V. Shuisky who
contributed to the author in his work towards formation and
development of the method, and all others who resisted the
method development in different ways, prompting the author
to a more thorough understanding and description of his
work. Special thanks to O.Yu. Bogdanova, whose translation

International Journal of Intelligent Information Systems 2015; 4(1): 1-7 7
of the papers promoted their publication in the
multidisciplinary scientific environment.
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