


Valentin Tzvetanov Penev
Ludmil Lubomirov Konstantinov
Central Laboratory of Mineralogy and Crystallography Bulgarian Academy of Sciences Acad. Georgi Bonchev Str., bl. 107, 1113 Sofia, BULGARIA Email: vpenev@interbgc.com; vpenev@bas.bg; v_penev@mail.bg Web site: http://www.clmc.bas.bg
We define and analyze the notions chemical point, free and bonded chemical point and chemical graph, which are geometric images in V_{S}(6) of free simple chemical objects, bonded simple chemical objects and complex chemical objects. It is shown that: (à) these notions are sufficient for geometrical representation of arbitrary set of chemical objects; (b) the role of chemical figures in the space of chemical structures V_{S}(6) is played by chemical points and graphs, while that of transformations in this space is played by translations and rotations in the ordinary physical space V_{E}(3) and in Mendeleev's space V_{M}(3); (c) from a geometrical point of view, the study of various chemical processes is a study of the transformation of the corresponding sets of chemical figures in V_{S}(6); (d) chemistry is a science for chemical figures in V_{S}(6), for their mutual disposition and for the size of their parts as well as for the transformations of chemical figures. Using the developed mathematical formalism and the introduced basic notions an entirely geometric criterion for equivalency of chemical objects (i.e. of chemical figures in V_{S}(6)) is formulated in a general form.
INTRODUCTION
In previous publications [1, 2, 3, 4, 5] we represented the whole conceptual logical and mathematical formalism necessary for entirely mathematical representation of both simple and complex chemical objects as well as of arbitrary chemical processes (both simple and complex) through the physical conceptual scheme for representation changes. At the same time, we represented a lot of the results concerning the investigation of the relationship between the geometrized chemistry and geometry. To realize this fact it is sufficiently to keep in mind that all basic elements of the developed conceptual, logical and mathematical formalism (spaces, coordinate systems, mathematical images of chemical and physical objects, operations with these images, etc.) are in fact parts (elements) of the conceptual and notional formalism of geometry.
In the present publication we shall:
GEOMETRICAL IMAGES OF CHEMICAL OBJECTS
The geometrical images of simple chemical objects in V_{S}(6) are introduced as follows:
DEFINITION 1: By the notion chemical point we shall denote such a 6dimentional point in the space of chemical structures V_{S}(6)^{1} that is a unique mathematical image of some simple chemical object.
It becomes clear from this definition and from the definition of V_{S}(6), that:
1. Each chemical point in V_{S}(6) is of a nonzero chemical charge^{2}, i.e. with a nonzero triad of coordinates in the Mendeleev space V_{M}(3). Here, a nonzero triad of coordinates means such a triad in which at least one of the three coordinates is not equal to zero.
2. The first triad of coordinates of each chemical point is defined with respect to K_{E}(3) in V_{E}(3) and represents the position of the corresponding simple chemical object in the ordinary physical space V_{E}(3), while the second one is defined with respect to K_{M}(3) in V_{M}(3) and represents the species of the corresponding simple chemical object (i.e. the chemical charge assigned to the corresponding point in V_{E}(3)).
3. Each chemical point in V_{S}(6) is at the same time a mathematical point, but not any mathematical point in V_{S}(6) is a chemical point (because not to any point in V_{E}(3) a nonzero chemical charge is attributed, i.e. not any point in V_{S}(6) is mathematical image of simple chemical object).
4. At any moment of time the set of chemical points in V_{S}(6) is discrete and, thus, countable (in contrast to the set of mathematical points in V_{S}(6) which is continuous and coincides with V_{S}(6)).
The geometric images of the complex chemical objects in V_{S}(6) are introduced as follows:
DEFINITION 2: By the notion chemical graph we denote such a mathematical graph in V_{S}(6), which is a unique mathematical image of some complex chemical object.
It is clear from this definition that:
1. The nodes (points, vertices) of each chemical graph are always chemical points and represent the species and position with respect to K_{E}(3) in V_{E}(3) of the simple chemical objects building the complex chemical object represented by the given chemical graph.
2. Each chemical graph in V_{S}(6) is at the same time a mathematical graph, but not any mathematical graph in V_{S}(6) is at the same time a chemical graph (because not any mathematical graph in V_{S}(6) is mathematical image of a some complex chemical object). In other words, the nodes (points, vertices) of each chemical graph are always chemical points, they always have an exactly specified chemical meaning, whereas the vertices (nodes, points) of nonchemical graphs are not chemical points and, consequently, they have not any chemical meaning at all.
The next definition classifies the geometric images of simple chemical objects depending on whether the corresponding simple chemical objects are bonded to form complex chemical objects or are free.
DEFINITION 3: By the term bonded chemical point we shall denote such a chemical point in V_{S}(6) which is a node (vertex) of some chemical graph, whereas by the term free chemical point we shall denote such a chemical point in V_{S}(6) which is not a node (vertex) of any chemical graph at all.
It becomes clear from this definition that the bonded chemical points are geometric images in V_{S}(6) of such simple chemical objects that take part in the chemical composition of complex chemical objects (and, thus, they are bonded). Respectively, the free chemical points are geometric images in V_{S}(6) of such simple chemical objects which do not take part in the chemical composition of any complex chemical object (and, thus, they are free).
From these definitions one can formulate the following conclusions:
Conclusion 1: At any moment of time:
Conclusion 3: The structure^{3} of each complex chemical object is represented entirely geometrically in V_{S}(6) through defining the corresponding chemical graph, representing the considered complex chemical object.
Conclusion 4: The composition of each complex chemical object is represented entirely geometrically in V_{S}(6) through defining the set of nodes (vertices) of the corresponding chemical graph, representing the considered complex chemical object, i.e. through defining the corresponding set of bonded chemical points.
Conclusion 5: The construction of each complex chemical object is represented entirely geometrically in V_{S}(6) through defining the set of edges (lines) of the corresponding chemical graph, representing the considered complex chemical object.
SOME ASPECTS OF THE RELATION BETWEEN GEOMETRY AND CHEMISTRY. GEOMETRYC CRITERION FOR EQUIVALENCY OF CHEMICAL OBJECTS.
In the introduction we mentioned that all basic elements of the developed formalism (spaces, coordinate systems, mathematical images of chemical and physical objects, operations over these mathematical images, etc.) are in fact parts (elements) of the main conceptual and notional formalism of geometry. Now we shall formulate some conclusions and comments enlightening further important aspects of the relation between geometry and chemistry. Firstly however, we shall quote a definition of the notion geometry:
"Geometry, in the initial meaning of this notion, is a science dealing with the figures, mutual disposition and size of their parts as well as the transformations of these figures. This definition is in a total accordance with the definition of geometry as a science dealing with spatial forms and relationships. Indeed, the figures, as considered in geometry, are spatial forms; the mutual disposition and size are determined through the spatial relationships; and finally, the transformation, as considered in geometry, is also some kind of relationship between two figures, namely the considered one and that obtained after its transformation ([6], ð. 143).”
According to this definition, it becomes clear that from a geometric point of view:
CONCLUSION I: The role of chemical figures in the space of chemical structures V_{S}(6) is played by the chemical points and chemical graphs, while the role of transformations in this space is played by the wellknown translations and rotations in the ordinary physical space V_{E}(3) and by the translations and rotations , _{ } and in Mendeleev's space V_{M}(3), defined in [4].
CONCLUSION II: Chemistry is the science dealing with the chemical figures in V_{S}(6), the mutual disposition and size of their parts as well as the transformations of these figures.
These two conclusions are quite important, as they define the subject of chemistry entirely in terms of geometry and, thus, reveal clearly the relation between the languages of chemistry and geometry.
The next important conclusion is:
CONCLUSION III: The conceptual and mathematical formalism developed in the framework of the project "Geometrization of the fundamentals of chemistry" is totally accorded with both the conceptions of stereochemistry and the use of the graph theory for representing chemical objects and processes in contemporary theoretical chemistry.
This conclusion shows clearly that:
Criterion for chemical equivalency: Two chemical objects A and B are equivalent then and only then when there exists such a combination of translations and rotations in the ordinary physical space V_{E}(3) that transforms the geometrical image in V_{S}(6) of A into the geometrical image in V_{S}(6) of B.
This entirely geometric criterion for equivalency is valid for both simple and complex chemical objects and can be expressed mathematically in the following way:
A º B Û $ , such that  (1) 
where: and are chemical points if A and B are simple chemical objects; and are chemical graphs if A and B are complex chemical objects; is the operator of translation at a distance a along the udimension of V_{E}(3); is the operator of translation at a distance b along the vdimension of V_{E}(3); is the operator of translation at a distance c along the wdimension of V_{E}(3); , , are operators of rotation by angles a, b and g in V_{E}(3).
From the formulated criterion for equivalency it becomes clear that:
1. The equivalent (i.e. the equal within to translations and rotations in V_{E}(3)) chemical figures differ from one another only in their first triad of coordinates defining the position in V_{E}(3) of the corresponding chemical objects, represented geometrically through the corresponding chemical figures.
2. If in transforming the geometrical image of a complex chemical object into that of another one, beside translations and rotations in V_{E}(3), operations of reflection in V_{E}(3) are also used, one considers these complex chemical objects as nonequivalent from a chemical point of view. (Examples for complex chemical objects, whose geometrical images are transformed from one into another through operations of reflection in V_{E}(3), are the left and rightorientated quartz, sugar, etc.)
After we defined the basic geometrical notion equivalency for arbitrary chemical figures, we should explicitly emphasize that the problem of geometric definition of the notion similarity for chemical figures (i.e. the problem of geometric formulation of a criterion for similarity of chemical objects) is much more complex. We consider this problem as well as the closely related to it problem of studying these transformations in the space of chemical structures V_{S}(6), which are possible from the chemical point of view (i.e. which describe possible chemical processes), are to be solved quite further in the process of geometrization of the language of chemistry. However, we should explicitly note that before formulating mathematically any criterion for similarity of chemical objects, one must clarify in the field of chemistry itself the problem "when two chemical objects can be considered as similar chemically?". In other words, in chemistry itself one must firstly clarify in a logically correct manner the problem of the chemical similarity. Only then, one may think about formulating a mathematical (and in particular geometrical) criterion for similarity of chemical objects. In our opinion, as starting point in a future clarification of the problem of the chemical similarity one could accept the following statement:
HYPOTHESIS: Two complex chemical objects are assumed as chemically similar, if:
CONCLUSION IV: The discrete mathematics will possibly play also the role of theoretical basis of a future entirely mathematical reformulation of the language of chemistry.
CONCLUSION V: The reformulation of the language of chemistry in the terms of discrete mathematics will substantially simplify and accelerate the widespread usage in chemistry of powerful cybernetic methods and technologies, both theoretical and applied.
The latter conclusion is confirmed also by the fact that there is a quite clear relation between the method for construction of unique mathematical images of simple chemical objects, proposed in the framework of this project, and the cybernetic problem for recognition of images (i.e. the taxonomic problem in cybernetics).
We already said that the set of all chemical points (both free and bonded in chemical graphs) in an arbitrary volume V in the ordinary physical space V_{Å}(3) is always discrete. In other words, chemical meaning in V_{Å}(3) has always a discrete set of points only. Consequently, if we look after at the V_{Å}(3) as chemists only a discrete (i.e. nonArchimedean) set of points would be seen, not a continuum at all. This fact suggests that:
CONCLUSION V²: Chemistry is maybe a particular type of nonArchimedean geometry^{6}. In other words, as a result from a eventual future reformulation of chemistry in geometrical terms one may possibly obtain some particular type of nonArchimedean geometry.
DESCRIPTIVE, INDUCTIVE, DEDUCTIVE AND AXIOMATIC STAGES (PHASES) IN THE DEVELOPEMENT OF DIFFERENT SCIENCES AND SCIENTIFIC DISCIPLINES
After discussing some basic aspects of the relation between chemistry and geometry we shall consider the possibilities for axiomatic reformulation of the language of chemistry. Before that however, we shall make some comments. So, according to the R. Blanshe [8] the development of each scientific discipline or science goes through four consequent phases; descriptive, inductive, deductive and axiomatic. During the initial two phases, descriptive and inductive, the development accentuates on the accumulation of empirical knowledge. During the third, higher, deductive phase the accent is shifted on organizing the already accumulated (and still continuing to be accumulate) empirical knowledge in the form of a deductive theory (or a set of deductive theories), i.e. on developing such deductive formulations of a given scientific discipline or science, which are based on a few appropriately selected basic statements^{7}. During the highest axiomatic phase of development the accent is on the strict logical analysis of the deductive formulations of the considered scientific discipline or science, developed during the preceding period, in order to reformulate them axiomatically. However, it should especially be noted that the boundaries between the different phases in the development of a given scientific discipline or science are quite conditional. This holds especially for the boundary between the deductive and axiomatic phases.
A perfect illustration of the above scheme are the stages in the development of elementary (Euclidean) geometry; descriptive and inductive phases  from the geometry coming of into being till Euclid's times (3^{th} century B.C.); deductive phase (with increasingly stronger axiomatic elements)  from Euclid's till Lobachevski's times (more than 22 centuries!!!); and axiomatic phase  from Lobachevski's times till to nowadays. Indeed, from the history of mathematics, one has known [9], that the Euclid's "Elements" has, in fact, systematized the accumulated till then empirical geometric knowledge in the form of a deductive theory. At the same time, in David Hilbert's "The Foundations of Geometry" (appeared 23 centuries after Euclid's times) has already presented a really strict axiomatic formulation of the existing till then deductive formulations of the elementary geometry.
The same stages have also been observed in the development of classical mechanics; descriptive and inductive stages  from ancient till Newton's times; deductive stage (with increasingly strong axiomatic elements)  from Newton's times till to now (lasting already more than 3 centuries!!!); so far there is no strict axiomatic formulations of the classical mechanics. Similar is the situation in almost physical disciplines  they are still either in the descriptive and inductive phases of their development, or relatively soon (as compared with Euclidean geometry) have entered in the deductive phase.
Starting from the described scheme, we can accept that the descriptive and inductive phases in the development of chemistry is lasting from the ancient till Mendeleev's times. We think that with the appearance of the Periodic System chemistry has entered in fact in deductive stage of its development, where the accent is shifted on organizing the accumulated empirical knowledge in the form of deductive theories based on a few appropriately selected basic principles. It should especially be noted that the deductive phase in the development of chemistry is still continuing till to nowadays.
We shall complete the preliminary comments with an important remark: It is impossible to develop deductive formulations (and, even less, axiomatic ones) of a scientific disciplines without specifying, at least generally, their logical structure and defining logically in a correct manner their systems of basic notions and relationships, as well as without developing entirely mathematical representations of these basic notions and relationships.
From the comments given above and from the results presented in [15], it becomes clear that:
CONCLUSION VI²: By fulfilling the project "Geometrization of the Fundamentals of Chemistry" we have completed, in principle, the preliminary work, necessary both for an entire reformulation of chemistry in deductive form and for developing really axiomatic formulations of the chemical disciplines.
Indeed, in the projects's framework, we have clarified the general logical structure of the language of chemistry, defined logically correctly the set of basic chemical notions and relationships, and developed their unique spatial mathematical representations. In addition, we have formulated mathematically, analized and proved a lot of chemical statements of various logical ranks (definitions, axioms, theorems, corollaries, etc.), which form a substantial part of the future deductive and axiomatic formulations of chemistry.
CONCLUSION VI²I. The basic theoretical problems to be solved in each particular chemical discipline are the clarification of its logical structure and its entirely mathematical reformulation using the developed mathematical formalism. By solving these theoretical problems one will perform the preparation for developing real axiomatic formulations of the particular chemical disciplines.
CONCLUSION IÕ. Only after developing entirely mathematized deductive formulations of the chemical disciplines, based on the physical conceptual scheme, one could consider for their strictly axiomatic reformulation on the basis of the geometrical conceptual scheme [4].
We shall finish the discussion on the possible axiomatic reformulation of the language of chemistry with a remark. As seen from the considered examples, the deductive phase in the development of the diverse scientific disciplines and sciences has been of a great duration, more than 22 centuries for the elementary geometry, and more than 3 centuries for the classical mechanics. Thus, it is not realistic to hope that the deductive phase in the development of chemistry will be completed soon!!!
REFERENCES
[1]  Penev V., Geometrization of the Fundamentals of Chemistry. RIVA Publ. House, Sofia, 1997, 224 pp. (a monograph in Bulgarian with summary in English) (Full text and summary are available through the official site of the project "Geometrization of the language of chemistry" – http://www.clmc.bas.bg/staff/V_Penev/gcl/) 
[2]  Penev, V., L. Konstantinov, M. Marinov, Logical structure of the fundamentals of chemistry, conceptual schemes of its geometrization and spatial mathematical models of the sets of different species of simple chemical objects. – Academic Open Internet Journal, vol. 2 (2000), Part 2: Chemistry – http://www.acadjournal.com/2000/v2/part2/p3/ 
[3] 
Penev, V., L. Konstantinov, Geometrization of the language of chemistry: Mathematical representation of the complex chemical objects. – Academic Open Internet Journal, vol. 11 (2004), Part 2: Chemistry – http://www.acadjournal.com/2004/v11/part2/p3/ 
[4] 
Penev, V., L. Konstantinov, Mathematical representations of chemical processes: Part I. Conceptual schemes for representing changes; space and coordinate system; method. – Academic Open Internet Journal, vol. 11 (2004), Part 2: Chemistry – http://www.acadjournal.com/2004/v11/part2/p4/ 
[5] 
Penev, V., L. Konstantinov, Mathematical representation of chemical processes: Part II. Representation of the free physical particles and of the laws for preservation of the electric charge and the mass in chemical reactions. – Academic Open Internet Journal, vol. 11 (2004), Part 2: Chemistry – http://www.acadjournal.com/2004/v11/part2/p5/ 
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[7]

Manev K. Introduction in the discrete mathematics, Sofia, “New Bulgarian University”, 1996. 
[8] 
Blanshe R., L’ axiomatique, Paris, Press universitaires de France, 1965. 
[9]

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NOTES
^{1} The space and the coordinate system of chemical structures, V_{S}(6) and K_{S}(6), are defined as follows: [3]: V_{S}(6) = V_{M}(3) x V_{E}(3) and K_{S}(6) = K_{M}(3) x K_{E}(3), where V_{M}(3) and K_{M}(3) stand respectively Mendeleev's space and Mendeleev's coordinate system, defined in [1, 2], while V_{Å}(3) and K_{Å}(3) are respectively the ordinary 3 dimensional physical space and the properly chosen coordinate system in it. On its turn, the space and the coordinate system of chemical processes, V_{P}(7) and K_{P}(7), are defined as follows [4]: V_{P}(7) = V_{S}(6) x V_{T}(1) = V_{T}(1) x [V_{Ì}(3) x V_{Å}(3)] and K_{P}(7) = K_{S}(6) x K_{T}(1) = [K_{M}(3) x K_{E}(3)] x K_{T}(1), where V_{T}(1) and K_{T}(1) stand respectively for 1 dimensional space of time and a properly chosen coordinate system therein.