BASIC MODULES IN SOME SYSTEMS AND PREDICTION

N.L. Smirnova

Dept. of geology, Univ. of Moscow, Russia, crystal@geol.msu.ru.

Keywords: prediction, ordered isomorphy, parsimony, modules, symmetry, systems, nets

Ordered isomorphy and parsimony are universal phenomena. These phenomena are a basis of all organized systems. In accordance with isomorphism all systems (objects) are composed of 0 - 3 dimensional modules. According to parsimony the number of essentially different types of at least one of constituent modules in a system tends to be small (1 - 4). The concept of modules is used in various areas of science. As it is accepted in mathematics a module is the numerical characteristic of any object. In informatics, a module is a complete unit either in software or hardware which can be used with other compatible units to build up a larger structure. In astronautics, a module is an autonomous compartment in the spacecraft. The notion module (modulus) is seldom used in phisics. In construction, a module is a rather large fragment (building unit). In chemistry modules are chemical elements and their combinations. In crystalogy modules are fragments of the crystalline space. In phylosophy conceptions part and hole express relation between total combinations of objects or elements of objects and connection that unite things and leads to appearence in a hole new integrative properties and laws. Between parts exist consolidating structural genetic subordinating managing connections. One can draw a conclusion based on the above reasons that modules are parts constituting the object.

The module can be in many different objects. The module can be simple and complex. The last one consists of the same or different simple modules. Every module is constructed of a sceleton constant part and a varying topotactic part. Sequences of modules can be represented by letters, projections, coordinates. Chemical elements, quarks, basic physical units, regular polygons and polyhedrons, rocks, codons and so on are basic modules.

Let us consider a number of system. To this end let us isolate in the system several constituting modules, say four modules in accordance with the parsimony law. Let us denote these constituents by unity elements. The unity elements in the 1, 2, 3, 4 columns of the table are referred to 1, 2, 3, 4 basic constituents. The 1-, 2-, 3-, 4- ary combinations of constituents correspond to combinations of the unity elements of the systems. Four of these combinations (modules), are unary, six are binary, four are ternary and one quaternary. These combinations are a models for other systems (table). In the atomic environment types (AET) four constituents are a triangle (t), a quadrangle (q), pentagon (p), hexagon (h). The polygons of each of 15 combination of the matrix share the same vertex (have one common vertex). It is easy to see, that four of them with ph combinations are absent. In the s, f, d, p system of the cations of chemical compounds and minerals all 15 modules are realized. The last sfdp module recently was found in the mineral christovite. In the system of the cation charges 15 modules were found for 1, 2, 3, 4; 1, 2, 3, 5; 1, 2, 3, 6 charges with one exception - 1236. Even for additional combinations [45], [46], [56] all modules are quaternary. Derived units in the system of fundamental properties consist of 7 basic units: space (s), time (t), mass (m), electricity (e), light (l), temperature (), mole (n). All 15 modules of s, t, m, e are realized but several ones with l, t, n are missing. In the system of quarks the only the 4-ary module is absent. Let us consider space group systems. It is well known that all crystalline structures are build of 11 Kepler - Schubnikov nets. Tetragonal and cubic structures are composed of 3 nets: e - 4⁴, n - 3²434, v - 48² perpendicular to axis 4. The nets are in planar sections of space groups.

The symmetry of these planar sections are described by three plane symmetry groups (out of 17 ones): p4 (1), p4m (2), p4g (3). These sections are at arbitrary hight z. Most of tetragonal groups also have five sections at a fixed hight: 1, 2, 3, c4 (4), c4m (5). Combinations of arbitrary and fixed section (the numbers of space groups taken from International tables are in parentheses) are: 1 (75, 79, 83, 103, 104), 11 (83), 2 (99, 107), 22 (123), 3 (100, 108), 33 (127), 12 (89), 112 (124), 13 (90), 113 (128), 123 (97), 14 (85), 114 (87), 124 (126), 134 (130), 25 (129), 225 (139), 35 (125), 335 (140). Among 15 theoretical combinations of 1 - 4 sections 4, 23, 24, 34, 234, 1234 are absent, whereas for section 5 only 25, 225, 35, 335 combinations exist (table).

Hexagonal and cubic space groups are composed of 7 nets d - 3⁶, p - 3636, f - 6³, l - 3⁴6, i - 3464, s - 3.12², - 46.12 perpendicular to axis 3 and 6. The nets are in planar sections of space groups. The symmetry of these planar sections are described by five plane symmetry groups: p3 (1), p3m1 (2), p31m (3), p6 (4), p6m (5). These sections are at arbitrary hight z. Most of hexagonal groups also have these five sections at a fixed hight z 0, 1/12, 1/6, 1/4, 1/3, 1/2. Combinations of arbitrary and fixed section (the numbers of space groups are in parentheses) are: 1 (143, 146, 158, 159, 161, 173), 11 (174), 2 (156, 160, 186), 22 (187), 3 (157, 185), 33 (189), 4 (168, 175), 44 (184), 5 (183), 55 (191), 12 (149), 112 (188), 13 (150, 155), 113 (190), 14 (147, 148), 114 (176), 25 (164, 166), 225 (194), 35 (162), 335 (193), 45 (177), 445 (192), 123 (182), 124 (163), 134 (165, 167). Among 15 theoretical combinations of 1, 2, 3, 4 sections 23, 24, 34, 234, 1234 are absent, whereas for section 5 only 25, 225, 35, 335, 45, 445 combinations exist.

Let as consider the system of bonds. The plane network of intersecting bonds was represented as a graph where edges are lines and vertexes are knots (k). The knots can be on one line, in intersections of two, three, four lines. They are denoted as 1, 2, 3, 4. The frequency of 143 knots is: 1 - 116, 2 - 19, 3 - 7, 4 - 1 (100, 17, 6, 1%). The loops (l) of net are designated by the combinations of knot numbers. The loops are devided into four sets: 1, 2, 3, 4. Indices are the sums identical knots.

There are 29 loops with identical and different indices, 21 among them with different indices, and 10 different loops without indices (the sums of the loops with the same symbols are in parentheses): 1: 1 - (0), 2: 2 - 2₃ (1), 12 - 1₇2₄, (1); 3: 3 - 3₄ (1), 13 - (0), 23 - 2₁3₁ (1), 2₂3₁ (2), 2₃3₁ (1), 2₁3₂ (4), 2₂3₂ (2), 2₁3₃ (2), 123 - 1₁2₂3₁ (1), 1₂2₂3₁ (1), 1₁2₁3₂ (1), 1₁2₃3₁ (2), 1₁₆2₄3₁ (1), 4: 4 - (0), 14 - (0), 24 - 2₁4₁ (1), 124 - 1₂2₃4₁ (1), 34 - 3₁4₁ (1), 3₂4₁ (1), 134 - (0), 234 - 2₂3₂4₁ (1), 1234 - 1₁2₂3₁4₁ (1), 1₁2₂3₂4₁ (2). If the knots 1 are not taken into account then a loop will contain the knots 2, 3, 4. The sums of these knots are two (b), three (t), four (q), five (p). These loops are designated b - binary, t - ternary, q - quaternary, p - pentary respectively. Among 29 loops designated as b, t, q, p there are 16 different loops: b: 2₁3₁(1), 2₁4₁ (1), 3₁4₁ (1), t: 2₃ (1), 2₂3₁ (4), 2₁3₂ (5), 3₂4₁ (1), q: 2₄ (1), 3₄(1), 2₁3₃(2), 2₂3₂ (2), 2₃3₁ (3), 2₃4₁ (1), 2₂3₁4₁ (1); p: 2₂3₂4₁ (3), 2₄3₁. (1). Every knot can be designated as a combination of attached b, t, q, p loops. Among 27 knots there are 21 with indices and 9 (table) without them. The knots correspond to four sets - B, T, Q, P. B: b - (0), T: t - t₂ (2), bt - (0), Q: q - q₁ (1), bq (0), tq - t₁q₁ (4), t₁q₃ (1), t₂q₂ (1), t₂q₃ (2), btq - (0),P: p - p₁ (1), bp (0), tp - t₁p₁(2), btp - (0), qp - q₁p₁ (1), q₂p₁ (2), q₃p₁ (1), bqp - b₁q₁p₁ (1), b₂q₁p₁ (1), tqp - t₂q₃p₂ (1), t₃q₂p₁ (1), t₄q₂p₁(1), dtqp - b₁t₂q₂p₂ (1), b₂t₁q₂p₃ (1). The system consists of three constituents: circle (II), central part (I), the periphery outside the circle (III). Knots frequency in I, II, III is different: knot1: 5, 0, 111; 2: 4, 10, 5; 3: 4, 3, 0; 4: 1, 0, 0. Polymeric chains 21_n and 31_n, n =1 - 12 are start from 14 knots

Combinations of 1 - 6 modules. Feasible combinations are designated by italics.