SS symmetry Article
Structure Transformations in Broken Symmetry Groups—Abstraction and Visualization Valery G. Rau 1, *, Igor A. Togunov 1, *, Tamara F. Rau 2, * and Sergey V. Polyakov 1, * 1 2
*
Russian Presidential Academy of National Economy and Public Administration, Vernadsky Prospekt, 119571 Moscow, Russia General and Theoretical Physics Department, Vladimir State University, Gorkogo Street 87, 600000 Vladimir, Russia Correspondence:
[email protected] (V.G.R.);
[email protected] (I.A.T.);
[email protected] (T.F.R.);
[email protected] (S.V.P.)
Received: 16 July 2018; Accepted: 18 September 2018; Published: 27 September 2018
Abstract: The work reports the finding and the study of transformation groups with two conditional elements (binary transformations of abstract structures of the finite numerical sets with broken symmetry). The term Broken Symmetry Group (BSG) is introduced. Transformation examples of relevant structures are studied with computer visualization and application in real structure study. A special type of BSG was discovered, which describes the subsets of “evolutionary trees” with convergent and divergent properties of the oriented graph (orgraph) with structure-development direction edges and “growth spirals”. Keywords: groups of substitution symmetry; Cayley tables; broken symmetry; permutation multiplying tables; orgraph; broken symmetry groups; convergent and divergent transformation graphs; evolutionary trees; growth spiral
1. Introduction All the known physical effects in crystals are described based on the Curie-Neumann principle: the symmetry of the structure and the symmetry of the “impact” on the structure are summated in a way where only shared transformations remain. The growth of monocrystals in particular starts in one spot (nucleant-defect), where the symmetry of medium uniformity and isotropy is broken. In a general case, systems interaction also results the transformations which break the symmetry and lead to the restructuring of each system [1]. This circumstance, as with many other factors, allows highlighting of the unity of two major principles of development, growth, accumulation, evolution, and stability of systems. These are, firstly, when expanded to all the systems, Curie’s Broken Symmetry Principle, which determines the cause of new occurrences, and secondly Le Chatelier’s symmetry conservation principle, which determines the system’s response, counteracting any infraction. The latter claims a conservation of the previous state or a transition to a new one, but also sustainable (symmetrical) state. According to the Noether’s theory, in a physical system of material particles, every differentiable symmetry of the action of a physical system has a corresponding energy-impulse conservation. In this case, the four-dimensional spatial occurrences continuum performs a “function” of possibility space. Lately, the following terms became common in scientific literature: phase space, economical space, personal psychological space, viral genome space, configurational space, informational space, and many others, which often are referred to as many-dimensional. Existing software tools such as Multidimensional Scaling (MDS) allow the projection of all the processed information on a familiar three-dimensional space. In all those
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dimensions, formally abstract variables create an “image” of the system and its reactions, which becomes a method of abstraction “visualization”. A substantial contribution to the studies of the symmetry and natural substances’ properties within a crystal lattice state was made by Russian crystallographers [1,2]. These are the classic works of Fedorov (230 spatial groups of symmetry), Shubnikov (black-and-white symmetry), and Belov (color symmetry) etc. Starting midway through the last century, a numerous amount of works devoted to systems with broken symmetry had been written. In particular, super-symmetry groups and multidimensional groups [3] were introduced for the description of in commensurately modulated phases and quasicrystals. For more complex cases, a description based on “interlacement groups” [4] was introduced, which, in the final analysis, may depict the classic groups of substitutions. A research on pseudo-symmetry during a monocrystal growth may be referred to as an example [5]. Historically, symmetry groups theory takes the mathematical course starting with the work of Galois, describing the theory of symmetry of equation roots permutation. All the classic algebra [6] studies and combinatorics [7], geometry, and topology relate to the group theory basics. General classification of binary relation sets by groups, semi groups, field of scalars, vectors, tensor quantities and rings relate directly to the maze, diagrams, nets, graphs, orgraphs and other mathematical structures as well as to real objects. In particular, in modern quantum field theory a diagram technique of the description of the broken symmetry processes was offered by Feinman. The works of other Nobel Prize laureates, including Higgs, led to the understanding of the gauge symmetry of the physical vacuum state and of the moment of the “beginning” of the Universe [8], starting with the Higgs boson. Therefore, the accordingly reinterpreted quantum theory of nonabelian gauge fields with spontaneously broken symmetry lays the basis of scientific conceptions of this research area. Feinman’s diagrams will be studied in further works within the scope of Broken Symmetry Group (BSG) analysis. The article does not aim to present the fundamental essence of symmetry and the mechanisms of broken symmetry in different types of matter (physical, biological, social etc.), but attempts to search for a broken system elementary mathematical model and possible “visualization”, for which purpose some studies previously developed and presented in the field of computer techniques had been used [9,10]. Group theory shows that any finite symmetry group is isomorphic to a permutation group accordingly (Cayley theorem). Therefore, to describe symmetry, a “matrix representation” may be implemented as well as a two-rowed matrices substitution representation. In this article, the substitution is the choice due to, firstly, symmetry operation record simplicity and, secondly, substitution coding regardless of space characteristics. This has a particular importance when dealing with the state space or phase space, where topology and sizing may be indeterminate enough to enter a certain virtual possibilities space to visualize a calculation result. As shown in the study [9], quaternion abstract symmetry, Pauli matrices, Dirac matrices and the local symmetry can be analyzed with substitution groups, and its visualization can be performed on periodic packing space discretion in the lattice [9]. This article will report the results describing translational symmetrical structures as well as non-periodic packing spaces structures and their discretion graphs. We will implement broken symmetry mathematical groups for restructuring processes description, in accordance with Curie’s principle. This was briefly reported by the article’s authors at the First Russian Crystallography Congress [11]. 2. Transformation Models in Structures with Broken Symmetry In our research, the first case of the broken symmetry group (BSG) finding and new properties showing up was the structure of the complex octahedral cation [Me(urea)6 ]2+,3+ (reference code “WOKNIT” in the database [12] with chelate hydrogen bonds [10,13]. Previously, the overview article [14] reported a crystalo-chemical research of possible combinations of directed H-bonds in different structures of complex compounds with carbamide. The description of the symmetry of complex cation [13] with double hydrogen bond required the “defect” substitutes entry to be in the form of two-rowed matrices as follows: ( 012345 223550 ), or, briefly, with the use of the “coding“ (lower row):
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g[1] = (223550). Therefore themultiplication classic multiplication substitution tabletable) (Cayley table) of octahedral a complex (223550). Therefore the classic substitution table (Cayley of a complex octahedral symmetry group in the orgraph model of directed bonds was not possible to apply. The symmetry group in the orgraph model of directed bonds was not possible to apply. The octahedral octahedral was symmetry brokendirected regarding directed symmetry brokenwas regarding bonds [12].bonds [12]. A series of computer experiments were performed to to research such modifications in symmetry [11]. A series of computer experiments were performed research such modifications in symmetry Now we will intentionally implement a defect (“accidental coding mistake”) in the entry of a certain [11]. Now we will intentionally implement a defect (“accidental coding mistake”) in the entry of a substitution cyclic subgroup. certain substitution cyclic subgroup. A regular pentagon (with aa fixed fixed central central point) point) will will be be perceived perceived as as aa classic simple A regular pentagon (with classic symmetry symmetry simple geometric structure structure (Figure (Figure 1). A singular singular transformation transformation g[0] g[0] will will be be left left without without the the defect; defect; however however geometric 1). A ◦ turn), which is coded by matrix g[1] instead of a classic permutation (72 = (023451), formally we class instead of a classic permutation (72° turn), which is coded by matrix g[1]class = (023451), formally we will enter a different operation: g[1] = (023431). Evidently a defect shows up within the fifth place in will enter a different operation: g[1] = (023431). Evidently a defect shows up within the fifth place in the permutation permutation coding coding row. row. the
Figure Figure 1.Groups 1. Groupscalculation calculationresults resultswith withmultiplication multiplication table table and and aa program program window window screen screen shot. shot.
If thereafter we use a simple rule of two-row matrices multiplication, multiplication, then then we we will will get an 2 3 2 3 = g[2] = (034342), (g[1]) == g[3] = = (043433) and (g[1])44 == g[4] = operations operations set g[1] = (023431), (g[1]) = (034344) as other elements of a substitution cyclic group. We will “expand” this final subgroup up to the “9th order group” by the operation with a defect defect g[5] ==(012343).As wewe willwill getget the the finalfinal multiplication tabletable of substitution with awith defect (012343).Asa result a result multiplication of substitution a (Table defect 1). (Table 1). For the structure, new operations with “a defect” cannot be considered permutations. However, Table 1. Substitution with a defect subset andbecause a multiplication table. at the same time they can still beoperations called number substitutions, the term “substitution” is wider than the term “permutation”. Permutation retains the numbers set, and substitution may g[0] = (0 1 2 3 4 5); g[0] g[1] g[2] g[3] g[3] g[5] g[6] g[7] g[8] change them. g[1] = (0 2 3 4 3 1); g[1] g[2] g[3] g[4] g[3] g[1] g[2] g[3] g[4] g[2] = (0 3 4 3 4 2); g[2] g[3] g[4] g[3] g[4] g[2] g[3] g[4] g[3] = (0 4 3 4operations 3 3); g[4] g[3] g[4] g[3] g[3] g[4] g[3] g[4] Table 1. g[3] Substitution withg[3] a defect subset and a multiplication table. g[4] = (0 3 4 3 4 4); g[4] g[3] g[4] g[3] g[4] g[4] g[3] g[4] g[3] g[0]==(0(0112233443); 5); g[0] g[1] g[2] g[3] g[3] g[5] g[6] g[7] g[8] g[5] g[5] g[6] g[7] g[8] g[7] g[5] g[6] g[7] g[8] g[1]==(0(0223344334); 1); g[1] g[2] g[3] g[4] g[3] g[1] g[2] g[3] g[4] g[6] g[6] g[7] g[8] g[7] g[8] g[6] g[7] g[8] g[7] g[7] g[7] g[8] g[7] g[8] g[7] g[7] g[8] g[7] g[8] g[2]==(0(0334433443); 2); g[2] g[3] g[4] g[3] g[4] g[2] g[3] g[4] g[3] g[8] g[8] g[7] g[8] g[7] g[8] g[8] g[7] g[8] g[7] g[3]==(0(0443344334); 3); g[3] g[4] g[3] g[4] g[3] g[3] g[4] g[3] g[4] g[4] = (0 3 4 3 4 4); g[4] g[3] g[4] g[3] g[4] g[4] g[3] g[4] g[3] g[5] = (0 1 2 3 4 3); g[5] g[6] g[7] g[8] g[7] g[5] g[6] g[7] g[8] For the structure, new operations with “a defect” cannot be considered permutations. However, g[6] = (0 2 3 4 3 4); g[6] g[7] g[8] g[7] g[8] g[6] g[7] g[8] g[7] at the same time they can still be called number substitutions, because the term “substitution” is wider g[7] = (0 3 4 3 4 3); g[7] g[8] g[7] g[8] g[7] g[7] g[8] g[7] g[8] than the term “permutation”. retains set,g[7] andg[8] substitution may change them. g[8] = (0Permutation 4 3 4 3 4); g[8] g[7] the g[8] numbers g[7] g[8] g[8] g[7]
visualization of three operations, randomly chosen from Table 1, is presented in Figure 2 [15].
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Evidently, resulting table table isisnot notaaclassic classicmultiplication multiplication Cayley table, inverse elements Evidently, the the resulting Cayley table, as as inverse elements are (b) are absent and properties change “by rows” and “by columns”. Therefore, we may state that “a absent and properties change “by rows” and “by columns”. Therefore, we may state that “a mistake” mistake” in coding elementincoding in symmetry groups to a broken symmetry. Weconsider will consider [8] each that in element symmetry groups led to aled broken symmetry. We will [8] that each transformation from the elements of the transformation set may be visualized by analyzing transformation from the elements of the transformation set may be visualized by analyzing the the substitution substitution itself. itself. Each Each modification modification [9] [9] can can be be visualized visualized by byanalyzing analyzingthe thesubstitution substitutionitself. itself. Substitution Substitution visualization of three operations, randomly chosen from Table 1, is presented in Figure 2 visualization of three operations, randomly chosen from Table 1, is presented in Figure 2[15]. [15]. Figure 2. Visualization of substitutions g[1], g[8] Table 1 in one dimensional space (а) and g[0], g[1], g[8] in two-dimensional space (b).
Definition 1. The product set of two-rowed matrices of substitution numbers, which are not permutations, will be called broken symmetry groups (BSG). The order of the finite group broken symmetry is determined by the amount of transformations. The above example with nine operations can be marked as BSG 9. In one of the computer experiments [11] using a designed program of binary matrices (b) multiplication with classic and non-classic substitutions (Figure 3) numerical set of 34 number values made a substitution set consisting of 91 BSG operations. Out of the set, two operations subgroups can be highlighted (including g[0]), which create small finite subsets (Table 2) BSG 6 and BSG 4. Таble 2. 6 and 4 transformations subsets of BSG-91 and their multiplication tables. Figure 2. Visualization of substitutions g[1], g[8] Table 1 in one dimensional space (a) and g[0], g[1], Figure of substitutions in30one g[0], g[4] g[1], g[0] =g[8] (0 1 2in3 two-dimensional 42. 5 6Visualization 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21g[1], 22 23g[8] 24 25Table 26 27 28129 31 32dimensional 33); g[0] space g[1] (а) g[2]andg[3] space (b). g[1] =g[8] (3 5 7in 9 11 13 15 17 19 20 22 24 26 27 28 29 16 12 23 25 3 7 30 32 4 8 31 5 33 6 10 12 18 14); g[1] g[2] g[3] g[4] g[5] two-dimensional space (b).
g[5] g[3] g[2] = (9 13 17 20 24 27 29 12 25 3 30 4 31 5 33 6 16 26 32 8 9 17 10 18 11 19 12 13 14 15 22 26 23 28); g[2] g[3] g[4] g[5] g[3] g[4] g[3] = (20 27 12 3 4 5 6 26 8 9 10 11 12 13 14 15 16 31 18 19 20 12 22 23 24 25 26 27 28 29 30 31 32 33); g[3] g[4] g[5] g[3] g[4] g[5] Definition The product setset two-rowed numbers, which are not will Definition 1.13 The product of which are notg[4]permutations, g[4] = (3 5 26 9 1. 11 15 31 19 20 22 24 26of 27of 28two-rowed 29 16 12 23 matrices 25 3matrices 26 30 32of 4 8substitution 31 5substitution 33 6 10 12 18 14);numbers, g[4] g[5] g[3]permutations, g[5] g[3] g[5] =be (9 called 13broken 31 20 24 27 29 12symmetry 25 3 30 4groups 31 5 33 6 (BSG). 16 26 32 8 9The 31 10order 18 11 order 19of 12 the 13 of 14finite 15 26group 23 28); g[5] g[3] g[4] g[3] byg[4] be called symmetry broken symmetry is determined the will broken groups (BSG). The the22 finite group broken symmetry isg[5] determined by g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33); g[0] g[1] g[2] g[3] amount nine operations be as BSG 9. g[3] the nine canmarked be marked asg[3] BSG 9. g[1]amount = (10of 14 transformations. 18of 21transformations. 25 27 29 21 25 4 31The 4 31 The 6above 16 6above 16example 8 16 8example 10 18with 10 18with 12 16 12 14 14operations 16 23 16can 23 28); g[1] g[2] g[2] = (31 16 16 18 16 14 16 18 16 25 16 25 16 29 16 29 16 25 16 25 31 16 31 16 31 16 31 16 16 16 18 g[2] g[3] g[3] g[3] 16 18 14); In In one one of of the the computer computer experiments experiments [11] [11] using using aa designed designed program program of of binary binary matrices matrices g[3] = (16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 g[3] g[3] g[3] g[3] multiplication with classic and non-classic substitutions (Figure 3) numerical set of 34 number values multiplication with classic and non-classic substitutions (Figure 3) numerical set of 34 number values 16 16 16);
made made aa substitution substitution set set consisting consistingof of 91 91 BSG BSG operations. operations. Out Out of of the the set, two operations operations subgroups subgroups can can Some transformations visualization from the table presented on2) Figure be (including which small finite (Table be highlighted highlighted (including g[0]), g[0]), whichcreate create small finiteissubsets subsets (Table 2)BSG BSG63.6and andBSG BSG4.4. Таble 2. 6 and 4 transformations subsets of BSG-91 and their multiplication tables. (b)
g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33); g[1] = (3 5 7 9 11 13 15 17 19 20 22 24 26 27 28 29 16 12 23 25 3 7 30 32 4 8 31 5 33 6 10 12 18 14); g[2] = (9 13 17 20 24 27 29 12 25 3 30 4 31 5 33 6 16 26 32 8 9 17 10 18 11 19 12 13 14 15 22 26 23 28); g[3] = (20 27 12 3 4 5 6 26 8 9 10 11 12 13 14 15 16 31 18 19 20 12 22 23 24 25 26 27 28 29 30 31 32 33); g[4] = (3 5 26 9 11 13 15 31 19 20 22 24 26 27 28 29 16 12 23 25 3 26 30 32 4 8 31 5 33 6 10 12 18 14); g[5] = (9 13 31 20 24 27 29 12 25 3 30 4 31 5 33 6 16 26 32 8 9 31 10 18 11 19 12 13 14 15 22 26 23 28); g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33); g[1] = (10 14 18 21 25 27 29 21 25 4 31 4 31 6 16 6 16 8 16 8 10 18 10 18 12 16 12 14 14 16 23 16 23 28); g[2] = (31 16 16 18 16 14 16 18 16 25 16 25 16 29 16 29 16 25 16 25 31 16 31 16 31 16 31 16 16 16 18 16 18 14); g[3] = (16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16);
g[0] g[1] g[2] g[3] g[4] g[5] g[0] g[1]
g[1] g[2] g[3] g[4] g[5] g[3] g[1] g[2]
g[2] g[3] g[4] g[5] g[3] g[4] g[2] g[3]
g[3] g[4] g[5] g[3] g[4] g[5] g[3] g[3]
g[2]
g[3]
g[3]
g[3]
g[3]
g[3]
g[3]
g[3]
g[4] g[5] g[3] g[4] g[5] g[3]
Figure 3. BSG 91 operations visualization as a model structure with 34 dots. Transformation structure g[1] Table 2 of the 6th order BSG subgroup (a) and transformation structures g[1]–g[3] Table 2 of the Some transformations visualization from the table is presented on Figure 3. 4th order BSG 4 subgroup (b). (b)
g[5] g[3] g[4] g[5] g[3] g[4]
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Table 2. 6 and 4 transformations subsets of BSG-91 and their multiplication tables. g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33); g[0] g[1] g[2] g[3] g[4] g[5] g[1] = (3 5 7 9 11 13 15 17 19 20 22 24 26 27 28 29 16 12 23 25 3 7 30 32 4 8 31 5 33 6 10 12 18 14); g[1] g[2] g[3] g[4] g[5] g[3] g[2] = (9 13 17 20 24 27 29 12 25 3 30 4 31 5 33 6 16 26 32 8 9 17 10 18 11 19 12 13 14 15 22 26 23 28); g[2] g[3] g[4] g[5] g[3] g[4] g[3] = (20 272018, 12 3 4 10, 5 6 26 9 10 11 12 13 REVIEW 14 15 16 31 18 19 20 12 22 23 24 25 26 27 28 29 30 31 32 33); g[3] g[4] g[5] g[3] g[4] Symmetry x 8FOR PEER 5g[5] of 12 g[4] = (3 5 26 9 11 13 15 31 19 20 22 24 26 27 28 29 16 12 23 25 3 26 30 32 4 8 31 5 33 6 10 12 18 14); g[4] g[5] g[3] g[4] g[5] g[3] g[5] = (9 13 31 20 24 27 29 12 25 3 30 4 31 5 33 6 16 26 32 8 9 31 10 18 11 19 12 13 14 15 22 26 23 28); g[5] g[3] g[4] g[5] g[3] g[4] g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33); g[0] g[1] g[2] g[3] 3.25BSG operations as12a16model with 34 dots. g[1] = Figure (10 14 18 21 27 29 91 21 25 4 31 4 31 6 16 6visualization 16 8 16 8 10 18 10 18 12 14 14structure 16 23 16 23 28); g[1] Transformation g[2] g[3] g[3]structure g[2] = g[1] (31 16 16 18 16 14 16 18 16 25 16 25 16 29 16 29 16subgroup 25 16 25 31 16 (а) 31 16and 31 16 transformation 31 16 16 16 18 16 18 14); g[2] g[3] g[3] Table g[3] 2 of the Table 2 of the 6th order BSG structures g[1]–g[3] g[3] = (16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16); g[3] g[3] g[3] g[3]
4th order BSG 4 subgroup (b).
Some transformations from thetable tableof is presented on Figure 3. from Table 2, i.e., Of particular interest visualization is the multiplication the 4th order subgroup Of particular interest is the multiplication table of the 4th order subgroup from Table 2, i.e., product table g[i] × g[k], which can serve as the grounding for the following statement: subgroups product table g[i]properties × g[k], which can serve as the grounding for the following statement: with convergent of the transformation structure orgraph can exist within BSGs.subgroups with convergent properties of the transformation structure orgraph can exist within BSGs. Evidently, for each table (with any finite number of elements), structurally created analogically for each table (with2,any number elements), created analogically withEvidently, the multiplying table (Table 4thfinite order), there isofalways one structurally transformation which inevitably with the multiplying table (Table 2, 4th order), there is always one transformation which inevitably exists, and which can be registered as a single value of integer R. A particular case of a group structure exists, and an which can be operation registered is aspresented a single value of integer R. A particular case=of with such inevitable in Table 3. Operation g[R] = g[7] (7a7group 7 7 7 7structure 7 7). with such an inevitable presented Table 3. Operation = g[7] = (7 73 7for 7 7g[3], 7 7 7). The point where alloperation the pathsiscross withininthe orgraph structureg[R] (as per Figure R = 16)
in a word will be called “Rome” (as per old saying: “All roads lead to Rome”), and the elements set Table 3. Broken symmetry group with the “Rome” point on the 8 points set and the group of such a table, where “Rome” point exists, will be called “the Rome Transformations Set”. multiplication table.
Table 3. Broken the “Rome” point g[4] on theg[5] 8 points and the group g[0] = (0symmetry 1 2 3 4 5 6 7); group g[0]withg[1] g[2] g[3] g[6] set g[7] g[1] = (1 2 3 4 5 6 7 7); g[1] g[2] g[3] g[4] g[5] g[6] g[7] g[7] multiplication table. g[2] = (2 3 4 5 6 7 7 7);
g[2]
g[3]
g[4]
g[5]
g[6]
g[7]
g[7]
g[7]
7); g[3] g[6] g[7] g[7] g[0] = (0 1 2 g[3] 3 4 =5 (3 6 47);5 6 7 7 7g[0] g[1] g[4] g[2] g[5] g[3] g[4] g[5] g[7] g[6] g[7] g[7] g[4] = (4 5 6 7 7 7 7 7); g[4] g[5] g[6] g[7] g[7] g[7] g[7] g[7] g[1] = (1 2 3 g[5] 4 5 =6 (5 7 67);7 7 7 7 7g[1] g[2] g[3] g[4] g[5] g[6] g[7] g[7] 7); g[5] g[6] g[7] g[7] g[7] g[7] g[7] g[7] g[6] = (6 7 7 7 7 7 7 7); g[6] g[7] g[7] g[7] g[7] g[7] g[7] g[7] g[2] = (2 3 4 5 6 7 7 7); g[2] g[3] g[4] g[5] g[6] g[7] g[7] g[7] g[7] = (7 7 7 7 7 7 7 7); g[7] g[7] g[7] g[7] g[7] g[7] g[7] g[7] g[3] = (3 4 5 6 7 7 7 7); g[3] g[4] g[5] g[6] g[7] g[7] g[7]ROME g[7] g[4] = (4 5 6 7 7 7 7 7); g[4] g[5] g[6] g[7] g[7] g[7] g[7] g[7] g[5] = (5 6 7 7 7 7 7 7); g[5] g[6] g[7] g[7] g[7] g[7] g[7] g[7] The the orgraph (as g[6] point = (6 7 7where 7 7 7 7all 7); the paths g[6] cross g[7]within g[7] g[7] structure g[7] g[7]per Figure g[7] 3 for g[3], g[7] R = 16) in a word will be called “Rome” (as per old saying: “All roads lead to Rome”), and the elements g[7] = (7 7 7 7 7 7 7 7); g[7] g[7] g[7] g[7] g[7] g[7] g[7] g[7] ROMEset of
such a table, where “Rome” point exists, will be called “the Rome Transformations Set”. A the orgraph orgraph with withdivergent divergentvectors vectors(“All (“Allroads roadsstart startinin Rome”), formally A transition to the Rome”), formally cancan be −1 , where upper and lower rows switch places in substitutional −1 be seen as “reverse” operation, g[1] seen as “reverse” operation, g[1] , where upper and lower rows switch substitutional matrices from Table 3 operations is matrices[15]. [15]. A A graphic graphicrepresentation representationofofthe thereverse reversetotothe thetransformations transformations from Table 3 operations shown onon Figure 4 as a divergent process (diverging “development spiral”). is shown Figure 4 as a divergent process (diverging “development spiral“).
−1 –g[7]−1 visualization in a virtual state space G, g[7]−1 —reverse Figure 4. Operations Operations −1–g[7] −1 visualization in a virtual state space G, g[7]−1—reverse modification Figure 4. g [0]g[0] modification with “Rome point” R = 7. with “Rome point” R = 7.
A virtual “possibility space”, or “latitude rate” (Figure 4) is necessary for such an “expansion”. Not a single system may develop without this condition. This circumstance requires additional systemic philosophical analysis (for instance within the “fractal-facet model” [16]). The g[7] operation in the Roman set from Table 4 acts as “zero”, since its application to other
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multiplied”. Therefore, the Roman set of binary operations has “unity” (g[0]) and “zero” (g[7]) but does not have inverse elements. Symmetry 2018, 10, 440 6 of 12 The orgraph divergent property leads to a situation where all the points of the structure become distinguishable (not identical), i.e., they only pass into themselves. This is the finite operation of “unity” g[0]−1 (Figure 4). On the contrary, the convergence leads to absolute symmetry when all points A virtual “possibility space”, or “latitude rate” (Figure 4) is necessary for such an “expansion”. of the structure become indistinguishable (identical). This is the finite operation of “zero” g[R] Not a single system may develop without this condition. This circumstance requires additional (“Rome point”). systemic philosophical analysis (for instance within the “fractal-facet model” [16]). Table 4 and Figure 5a represent transformations types and a multiplying table for a 14 dots The g[7] operation in the Roman set from Table 4 acts as “zero”, since its application to other system (7 × 2) with 6 stages of the evolutionary spiral. The system expanding is done in accordance operations of the set of elements results in g[7]: “zero remains zero, regardless of how it is multiplied”. with the work’s formulations [9]. Similarly, Table 5 and Figure 5b represents a development spiral of Therefore, the Roman set of binary operations has “unity” (g[0]) and “zero” (g[7]) but does not have the expanded system of 24 dots with 7 stages (8 × 3). inverse elements. Table Table4.4.Transformations Transformationstypes typesand anda amultiplying multiplyingtable tablefor fora a1414dots dotssystem. system. g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13); g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13); g[1] 7 811 9 10 g[1] = (2 3=4(2 5 63 74 85 96 10 12 11 13 12 12 13 13);12 13); g[2] = (4 5 6 7 8 9 10 11 12 13 12 g[2] = (4 5 6 7 8 9 10 11 12 13 12 13 12 13 13);12 13); 7 811 9 10 g[3] =g[3] (6 7=8(6 9 10 12 11 13 12 12 13 13 12 12 13 13 12 12 13 13);12 13); g[4] =g[4] (8 9=10(811 12 11 13 12 12 13 13 12 12 13 13 12 12 13 13 12 13 13);12 13); 9 10 g[5] =g[5] (10 11 12 11 13 12 12 13 13 12 12 13 13 12 12 13 13 12 12 13 12 13 13);12 13); = (10 g[6] =g[6] (12 13 12 13 13 12 12 13 13 12 12 13 13 12 12 13 13 12 12 13 13 12 12 13 13).12 13). = (12
g[0] g[1] g[2] g[3] g[4] g[5] g[6]; g[0] g[1] g[2] g[3] g[4] g[5] g[6]; g[1] g[2] g[3] g[4] g[5] g[6] g[6]; g[1] g[2] g[3] g[4] g[5] g[6] g[6]; g[2] g[3] g[4] g[5] g[6] g[6] g[6]; g[2] g[3] g[4] g[5] g[6] g[6] g[6]; g[3] g[4] g[5] g[6] g[6] g[6] g[6]; g[3] g[4] g[5] g[6] g[6] g[6] g[6]; g[4] g[5] g[6] g[6] g[6] g[6] g[6]; g[4] g[5] g[6] g[6] g[6] g[6] g[6]; g[5] g[6] g[6] g[6] g[6] g[6] g[6]; g[5] g[6] g[6] g[6] g[6] g[6] g[6]; g[6] g[6] g[6] g[6] g[6] g[6] g[6]. g[6] g[6] g[6] g[6] g[6] g[6] g[6].
Table 5. Transformations types to anda asituation multiplying tableall forthe a 24 dots system. The orgraph divergent property leads where points of the structure become distinguishable they pass into themselves. This theg[4] finite operation of g[0] = (0 1 2 3 4 5 (not 6 7 8 9identical), 10 11 12 13 14i.e., 15 16 17 18only 19 20 21 22 23); g[0] g[1] g[2]isg[3] g[5] g[6] g[7]; g[1] = (3 4 5− 6 17 8(Figure 9 10 11 124). 13 14 16 17contrary, 18 19 20 21the 21 21convergence 21 21 21); g[1]tog[2] g[3] g[4] g[5] g[6] g[7] g[7]; “unity” g[0] On15the leads absolute symmetry when all g[2] =of (6the 7 8 9structure 10 11 12 13become 14 15 16 17indistinguishable 18 19 20 21 21 21 21 21 21 21 21 21);This isg[2] g[4] operation g[5] g[6] g[7]of g[7] g[7]; g[R] points (identical). theg[3] finite “zero” g[3] = (9 10 11 12 13 14 15 16 17 18 19 20 21 21 21 21 21 21 21 21 21 21 21 21); g[3] g[4] g[5] g[6] g[7] g[7] g[7] g[7]; (“Rome point”). g[4] = (12 13 14 15 16 17 18 19 20 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21); g[4] g[5] g[6] g[7] g[7] g[7] g[7] g[7]; Table transformations types a multiplying forg[7] a 14g[7] dots system g[5] = (15 4 16and 17 18Figure 19 20 215a 21represent 21 21 21 21 21 21 21 21 21 21 21 21 21 21and 21 21); g[5] g[6] g[7]table g[7] g[7] g[7]; g[6] = (18 19 20 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21); g[6] g[7] g[7] g[7] g[7] g[7] g[7] g[7]; (7 × 2) with 6 stages of the evolutionary spiral. The system expanding is done in accordance with = (21 21 21 21 21 21 21 21 21 21 21 21 21 21Table 21 21 21 21 21 Figure 21 21 21 21 g[7] g[7]ag[7] g[7] g[7] g[7] g[7] g[7].of the theg[7] work’s formulations [9]. Similarly, 5 and 5b21). represents development spiral expanded system of 24 dots with 7 stages (8 × 3).
(а)
(b)
Figure (а) 77 steps steps in inthe thestructure structureof Figure5.5.Two Twoversions versionsofof“evolution “evolutionspiral” spiral”in in broken broken symmetry symmetry groups: (a) 24dots. dots. of77×× 2 = 14 dots; and (b) 8 steps in the structure of 8 × × 33==24
Thepoint point of sets in in mathematics, obtained by the symmetry method method generally The setsclassification classification mathematics, obtained bybroken the broken symmetry stand as symmetry and semi groups There grouping. was no contravention the generally stand as subgroups, symmetry rings subgroups, rings andgrouping. semi groups There wasofno associativity requirement detected in all the examined cases. Below are a few cases of BSG to describe contravention of the associativity requirement detected in all the examined cases. Below are a few symmetry broken symmetry. cases of BSGand to describe symmetry and broken symmetry.
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3. Applied Method of Symmetry Groups Visualization
Table 5. Transformations types and a multiplying table for a 24 dots system.
Example 1. As shown in the study [5], within the crystal growth the internal intermolecular g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23); g[0] g[1] g[2] g[3] g[4] g[5] g[6] g[7]; interaction system two local symmetry into system with one g[1] = (3 4can 5 6 7convert 8 9 10 11 12the 13 14 15 16 17with 18 19 20 21 21 21 21centers 21 21); (LC) of g[1] g[2] g[3] g[4] g[5]ag[6] g[7] g[7]; g[2] = (6 7 8 9 10 11 12 13 15 16 17 18 19 without 20 21 21 21 21 21 21 21 21 21); modifications g[2] g[3] g[4](Figure g[5] g[6] g[7] g[7]; (classic global center (GC) of 14symmetry the property 6, g[7] g[1]) g[3] = (9 10 11 12 13 14 15 16 17 18 19 20 21 21 21 21 21 21 21 21 21 21 21 21); g[3] g[4] g[5] g[6] g[7] g[7] g[7] g[7]; substitution). Suppose have А21(with a 21); global symmetry center from g[4] = (12 13 14 15 16 17 18we 19 20 21 21two 21 21subgroups: 21 21 21 21 21 21 21 21 21 g[4] g[5] g[6] g[7] g[7] g[7] g[7]Table g[7]; 6 and g[5] = (15 16 17 18 19 20 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21); g[5] g[6] g[7] g[7] g[7] g[7] g[7] g[7]; В (with two LCs and broken symmetry from Table 7 and Figure 7. g[6] = (18 19 20 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21); g[7] = (21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21).
g[6] g[7] g[7] g[7] g[7] g[7] g[7] g[7]; g[7] g[7] g[7] g[7] g[7] g[7] g[7] g[7].
Table 6. Subgroup А with broken symmetry with a global symmetry center.
3. Applied Method of Symmetry Groups g[0] = (0 1 2Visualization 3 4 5 6 7); g[0] g[1] Е g[1] = (1 0 7 6 5 4 3 2); g[1] g[0] GC Example 1. As shown in the study [5], within the crystal growth the internal intermolecular interaction can convert the system with two local centers (LC) of symmetry into a system with one Table 7. Subgroup В with broken symmetry and two local centers. global center (GC) of symmetry without the property modifications (Figure 6, g[1]) (classic substitution). Suppose we have two subgroups: symmetry g[0] = (0 1 2 3 4 5 6A7);(with a global g[0] g[1] g[2] center g[3] from Table 6 Еand B (with two LCs and broken symmetry and Figure 7. g[1] = (4from 5 6 7Table 0 1 273); g[1] g[0] g[2] g[3] LC We will calculate A5×0 B5 to a joint existence of the structure with a global g[2] = (0 0 5assess 0 5); a possibility g[2] of g[3] g[2] g[3] BS and local symmetry one structure. The calculation g[3] =centers (4 1 4 1within 4 1 4 1); g[3] g[2] g[2] g[3] result is presented BS in Table 8.
Figure Figure6. 6. Global Global center center (GC) (GC) regain regaing[1]’ g[1]’== g[1]. g[1]. Table 6. Subgroup A with broken symmetry with a global symmetry center.
We will calculate А × В to assess a possibility of a joint existence of the structure with a global g[0] = (0 1 2within 3 4 5 6 7); and local symmetry centers one structure.g[0] Theg[1] calculation result is Epresented in Table 8. g[1] = (1 0 7 6 5 4 3 2);
g[1] g[0]
GC
Table 8. BSG table of the group А × В binary transformations and a multiplying table. Table 7. Subgroup B with broken symmetry and two local centers.
g[0] = (0 1 2 3 4 5 6 7);
g[0] g[1] g[2] g[3] g[4] g[5] g[6] g[7]; E g[0] g[1] g[2] g[3] E g[1] g[0] g[3] g[2] g[5] g[4] g[7] g[6]; GC g[1] g[0] g[2]g[1] g[3]g[4] g[5] g[6] g[7]; LC g[2] g[3] g[0] LC g[2] g[3] g[2] g[3] BS g[3] g[2] g[1] g[0] g[5] g[4] g[7] g[6]; TC g[3] g[2] g[2] g[3] BS g[4] = (4 1 4 1 4 1 4 1); g[4] g[6] g[7] g[5] g[4] g[5] g[6] g[7]; NC g[5] = (1 4 1 4 1 4 1 4); g[5] g[7] g[6] g[4] g[5] g[4] g[7] g[6]; NC Table 8. BSG table of the group A × B binary transformations and a multiplying table. g[6] = (5 0 5 0 5 0 5 0); g[6] g[4] g[5] g[7] g[5] g[4] g[7] g[6]; NC g[0]= =(0 (0 5 1 20354 50 657);0 5); g[0] g[1]g[5] g[2] g[3] g[5]g[4] g[6] g[7]; E g[7] g[7] g[4]g[4] g[6] g[5] g[6] g[7]. NC g[0] g[1]= =(0(11 20 37 4655647); 3 2); g[1] g[2]= =(4(45 65 76 0710213); 2 3); g[2] = (0 5 0 5 0 5 0 5); g[3] = (5 4 3 2 1 0 7 6); g[3] = (4 1 4 1 4 1 4 1);
g[1] = (1 0 7 6 5 4 3 2); g[2] = (4 5 6 7 0 1 2 3); g[3] = (5 4 3 2 1 0 7 6); g[4] = (4 1 4 1 4 1 4 1); g[5] = (1 4 1 4 1 4 1 4); g[6] = (5 0 5 0 5 0 5 0); g[7] = (0 5 0 5 0 5 0 5);
g[1] g[0] g[3] g[2] g[5] g[4] g[7] g[6]; g[2] g[3] g[0] g[1] g[4] g[5] g[6] g[7]; g[3] g[2] g[1] g[0] g[5] g[4] g[7] g[6]; g[4] g[6] g[7] g[5] g[4] g[5] g[6] g[7]; g[5] g[7] g[6] g[4] g[5] g[4] g[7] g[6]; g[6] g[4] g[5] g[7] g[5] g[4] g[7] g[6]; g[7] g[5] g[4] g[6] g[4] g[5] g[6] g[7].
GC LC TC NC NC NC NC
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Figure 7. Structural images of transformation in a full group.
Evidently, in a joint BSG, g[1] and g[2] visualize a global symmetry center (inversion) (1) and (2) two LCs of inversion. g[3] describes a structure with 4 centers and t-transfer (ТC). Example 2. Let the two subsets of elements—dots numbered from 0 to 14 and from 15 to 25– interact in various ways. The possible three scenarios of its “behavior” are: (1) with the symmetry center (equality) between the dots of internal convergence (”Rome points” 0 and 15); (2) with numbers 0 and 15; (2) without the local symmetry center of convergence dots and (3) with a directed bond between them (Figure 8). First, lets refer to the case dynamics of the third scenario (Table 9). Figure 7. Structural images of transformation in a full group. Figure 7. Structural images of transformation in a full group.
Scenario 3. Directed bond.15 → 0.Ultimately Rome g[4]. Evidently, in a joint BSG, g[1] and g[2] visualize a global symmetry center (inversion) (1) and (2) Evidently, in a joint g[1] and g[2] visualize global symmetry center (inversion) (1) and (2) two LCs of inversion. g[3]BSG, describes structure with 4a centers and t-transfer (TC). Table 9. Stepaby step system dynamics in Scenario 3. two LCs of inversion. g[3] describes a structure with 4 centers and t-transfer (ТC). Example 2. Let the two subsets of elements—dots numbered from 0 to 14 and from 15 to g[0] = (02. 1 2Let 3 4 5the 6 7 two 8 9 10subsets 11 12 13 of 14 15 16 17 18 19 20 21numbered 22 23 24 25); from g[0] 0g[1] g[3] g[4]; Example elements—dots to g[2] 14 from 15 to 25– 25–interact in various ways. The possible three scenarios of its “behavior” are: (1) and with the symmetry g[1] = (0 0 0 1 1ways. 2 2 3 3The 4 4 5 possible 5 6 6 0 15 15 16 17scenarios 16 17 21 21of 20its 20);“behavior” g[1]are: g[2] g[3] g[4] g[4]; interact in various three (1) with the symmetry center (equality) between the dots of internal convergence (”Rome points” 0 and 15); (2) with numbers g[2] = (0 0 0 0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 15 15 15 15 17 17 16 16); g[2] g[3] g[4] g[4] g[4]; center (equality) between the dots of internal convergence (”Rome points” 0 and 15); (2) bond with 0 and 15; (2) without dotsg[3] andg[4] (3)g[4] with directed g[3] = (0 0 0 0 0 0 the 0 0 0local 0 0 0 0symmetry 0 0 0 0 0 0 0 center 0 0 15 15of 15convergence 15); g[4]ag[4]; numbers 0 and 15; (2) without the local symmetry center of convergence dots and (3) with a directed betweeng[4] them (Table = (0 (Figure 0 0 0 0 0 08). 0 0First, 0 0 0 0lets 0 0 refer 0 0 0 0to 0 0the 0 0 case 0 0 0);dynamics of the third g[4] scenario g[4] g[4] g[4] g[4].9). bond between them (Figure 8). First, lets refer to the case dynamics of the third scenario (Table 9). Scenario 3. Directed bond.15 → 0.Ultimately Rome g[4]. Figure 8 represents a visualization of the transformation scenario. Scenario 3. Directed bond.15 → 0.Ultimately Rome g[4]. Table 9. Step by step system dynamics in Scenario 3. g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25); g[1] = (0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 0 15 15 16 17 16 17 21 21 20 20); g[2] = (0 0 0 0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 15 15 15 15 17 17 16 16); g[3] = (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 15 15 15); g[4] = (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0);
g[0] g[1] g[2] g[3] g[4]; g[1] g[2] g[3] g[4] g[4]; g[2] g[3] g[4] g[4] g[4]; g[3] g[4] g[4] g[4] g[4]; g[4] g[4] g[4] g[4] g[4].
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Figure 8 represents a visualization of the transformation scenario.
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Figure 8. Step Stepby bystep stepsystem systemdynamics dynamics Scenario Figure 8. inin Scenario 3. 3.
In microbiology in particular, the development model can characterize two branches of interaction (“archaea” and “viruses” in g[1]) with a formation of a single “viruses’ life tree” (in g[4]) with possible transformation stages in the genome space. Evolution within the genome space is studied separately in reference [17].
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Table 9. Step by step system dynamics in Scenario 3. g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25); g[1] = (0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 0 15 15 16 17 16 17 21 21 20 20); g[2] = (0 0 0 0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 15 15 15 15 17 17 16 16); g[3] = (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 15 15 15); g[4] = (0 0 0 0 0 0 0 0Figure 0 0 0 0 8. 0 0Step 0 0 0by0 step 0 0 0 system 0 0 0 0 0); dynamics in Scenario 3.
g[0] g[1] g[2] g[3] g[4]; g[1] g[2] g[3] g[4] g[4]; g[2] g[3] g[4] g[4] g[4]; g[3] g[4] g[4] g[4] g[4]; g[4] g[4] g[4] g[4] g[4].
In Figure microbiology in particular, the of development model scenario. can characterize two branches of 8 represents a visualization the transformation interaction (“archaea” and “viruses” in g[1]) with a formation of a single “viruses’ life tree” g[4]) In microbiology in particular, the development model can characterize two branches of (in interaction with possible and transformation in the genome space. (“archaea” “viruses” instages g[1]) with a formation of a single “viruses’ life tree” (in g[4]) with possible Evolution within the genome space is studied separately in reference [17]. transformation stages in the genome space. The calculation process of the evolution within separately BSG convergent model[17]. is registered in BSG-7 Evolution within the genome space is studied in reference table (Table 10). Figure 9b represents a reverse divergent process of the evolutionary development The calculation process of the evolution within BSG convergent model is registered in BSG-7 bytable time (Table axis (Figure 9а). Each of the few highlighted columns in the multiplying table characterizes by 10). Figure 9b represents a reverse divergent process of the evolutionary development a separate of the process. Thehighlighted direction incolumns each group is multiplying “interpreted” consequentially time axis“branch” (Figure 9a). Each of the few in the table characterizes a “from bottom-up”. separate “branch” of the process. The direction in each group is “interpreted” consequentially “from bottom-up”.
Table 10. Transformation group BSG-7 multiplying table (Figure 9b). Table multiplying table22 (Figure g[0] = 10. (0 1Transformation 2 3 4 5 6 7 8 9 10group 11 12 BSG-7 13 14 15 1617 18 192021 23); 9b). g[1] = (0 0 0 1 1 2 3 3 4 2 6 6 11 5 10 9 77 8 81515 14 14); g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 18 192021 22 23); 0 35310 6 29 6772 83381515 4 49914 1014); 10); g[1] =g[2] (0 0=0(0 1 10 20 30 30 40 21 61 61 11 g[3] = (0 0 0 0 0 0 0 0 0 0 1 1 3 0 3 0 11 1 122 6 6); g[2] = (0 0 0 0 0 0 1 1 1 0 3 3 6 2 6 2 33 4 499 10 10); 0 110122 00 06 000 g[3] =g[4] (0 0=0(0 0 00 00 00 00 00 00 10 10 30 00 30 01 11 6); 3 3); g[5] = (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 000 g[4] = (0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 00 0 000 3 3); 1 1); g[5] =g[6] (0 0=0(0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 1); 0 0). 0 000000 00 01 000 g[6] = (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 000 0 0).
(a)
(b)
Figure 9. The process of prokaryotes’ “life tree” formation within genome space a digital Figure 9. The process of prokaryotes’ “life tree” formation within thethe genome space (а) (a) andand a digital model of the process based BSG. model of the process (b)(b) based on on BSG.
The presentedexample example scientifically scientifically and confirms the the validity of theofstudied methods The presented andobjectively objectively confirms validity the studied of broken symmetry visualization in biological systems methods of broken symmetry visualization in biological systems Example 3. Currently, a new field of supramolecular bonds chemistry is being developed which relates to the synthesis of three-dimensional branched polymers and oligomers called dendrimers (1. Hoffman Allan S. The origins and evolution of “controlled” drug delivery systems // J. Contr. Release. 2008. V. 132. pp. 153–163. 2. Nanoparticulate Drug Delivery Systems/Ed. by D. Thassu;
Example 3. Currently, a new field of supramolecular bonds chemistry is being developed which Symmetry 2018, 10, x FOR PEER REVIEW 10 of 12 relates to the synthesis of three-dimensional branched polymers and oligomers called dendrimers (1. Hoffman Allan3. S.Currently, The originsaand of “controlled”bonds drug delivery systems // J.developed Contr. Release. Example newevolution field of supramolecular chemistry is being which 2008. V. 132. pp. 153–163. 2. Nanoparticulate Drug Delivery Systems/Ed. by D. Thassu; M. Deleers; relates to the synthesis of three-dimensional branched polymers and oligomers called dendrimers (1. Symmetry 2018, 10, 440 10 of 12 Y. Pathak.—Informa Healthcare, p. 352). This type of drug bonddelivery is of particular since with Hoffman Allan S. The origins and2007, evolution of “controlled” systemsinterest // J. Contr. Release. each event of a dendrimerDrug the amount ofSystems/Ed. branching inbya D. simple caseM. increases 2008.elementary V. 132. pp.growth 153–163. 2. Nanoparticulate Delivery Thassu; Deleers; exponentially. With the increase of molecular mass of such bonds the shape and density of with the Y. Pathak.—Informa Healthcare, 2007, p. 352). This bondThis is oftype particular interest M. Deleers; Y. Pathak.—Informa Healthcare, 2007,type p. of 352). of bond is of since particular molecules alsowith changes it occurs with the modifications dendrimers’ each elementary growth eventgenerally of a dendrimer the along amount of branching in a simple case increases interest since each and elementary growth event of a dendrimer the amount ofofbranching in a physicochemical properties such as viscosity, solubility, density etc. Dendrimers can create exponentially. With the increase of molecular mass of such bonds the shape and density of the simple case increases exponentially. With the increase of molecular mass of such bonds the shape complexes with other molecules, moreover the stability of such complexes is controlled by the molecules changes and also generally it and occurs along itwith thealong modifications of dendrimers’ and densityalso of the molecules changes generally occurs with the modifications of external environmental state. This opens a such potential for dendrimer use in medicine a vehicle forcan a physicochemical properties such as viscosity, solubility, density etc. Dendrimers can create dendrimers’ physicochemical properties as viscosity, solubility, density etc. asDendrimers directed delivery of genes or pharmaceutical substances (vectors). The initial stage of this process is complexes with other molecules, moreover the the stability of of such complexes create complexes with other molecules, moreover stability such complexesisiscontrolled controlled by by the presented in Figure 10а. state. The formation ofaa“branches” ofdendrimer dendrimers gives a task its mathematical This potential use ininmedicine asas a avehicle external environmental state. Thisopens opens potentialfor for dendrimer use medicine vehiclefor fora description with the consideration of the directions of the process, which is, from a mathematical directed delivery of genes or pharmaceutical substances (vectors). The initial stage of this process is a directed delivery of genes or pharmaceutical substances (vectors). The initial stage of this process point of view, a directed graph; it is sort of a “labyrinth” (Figure 10b) or even a net if bonds between presented ininFigure mathematical is presented Figure10а. 10a.The Theformation formationofof“branches” “branches”of ofdendrimers dendrimers gives gives aa task its mathematical the brancheswith form. description with theconsideration considerationofof directions of the process, which is, from a mathematical description the thethe directions of the process, which is, from a mathematical point point of view, a directed it isofsort of a “labyrinth” (Figure or even if bonds between of view, a directed graph;graph; it is sort a “labyrinth” (Figure 10b) 10b) or even a netaifnet bonds between the the branches form. branches form.
(b)
Figure 10. The initial stage of dendrimers synthesis (а) and a “labyrinth” with bonds (b).
(b)
Hypothetically, in ainitial liquid crystal state, carbamide molecule chains which diverge Figure The initial stage of dendrimers synthesis and with (b). Figure 10. 10. The stage of dendrimers synthesis (a) (а) and aa “labyrinth” “labyrinth” with bonds bonds (b). from the “center” could have created a dendrimer. Hypothetically, in a liquid crystal state, carbamide molecule chains which diverge from the Hypothetically, in a liquid crystal state, carbamide molecule chains which diverge from the “center” couldfragment have created a dendrimer. A simple of such a chain, highlighted in Figure 3b, is structurally like such an element “center” could have created a dendrimer. simple fragment of(Figure such a chain, in Figure 3b, is an element of theAlabyrinth structure 10b), highlighted if an end atom (marked asstructurally dot 5) will like not such interact with theof the labyrinth structure (Figure 10b), if an end atom (marked as dot 5) will not interact with the external cycle, therefore bondadirection has changed. A simple fragmentthe of such chain, highlighted in Figure 3b, is structurally like such an external element cycle, therefore the bond direction has changed. For the highlighted element we will find a BSG with an initial (non-identical) transformation of the labyrinth structure (Figure 10b), if an end atom (marked as dot 5) will not interact with the For the highlighted will findhas a BSG with an initial (non-identical) transformation g[1]. g[1]. external cycle, therefore element the bondwe direction changed. Transformation visualization is presented in Figures 11an and 12and and correspondstransformation withTable Table11 11 Transformation visualization Figures 11 and 12 corresponds with For the highlighted element is wepresented will find in a BSG with initial (non-identical) group fragmentdata. data. group g[1]. fragment Transformation visualization is presented in Figures 11 and 12 and corresponds with Table 11 group fragment data.
(a)
(b)
Figure 11. Tetragonal carbamide structure (a) and carbamide molecules “chain” (b).
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Figure 11. Tetragonal carbamide structure (а) and carbamide molecules “chain” (b). Figure 12. An element of the structure with a cycle (by Figure 10b). Table 11. BSG transformations within the 5 dots structure 5. g[0] = (0 1 2 3 4 5); g[0] g[1] g[2] g[3] g[4] g[1] = (1 2 4 1 3 4); g[1] g[2] g[3] g[4] g[1] g[2] = (2 4 3 2 1 3); g[2] g[3] g[4] g[1] g[2] g[3] = (4 3 1 4 2 1); g[3] g[4] g[1] g[2] g[3] g[4] = (3 1 of 2 3the 4 2); g[4] g[1] g[3] (by g[4]Figure 10b). Figure 12. An element structure withg[2] a cycle Figure 12. An element of the structure with a cycle (by Figure 10b). Table 11. BSG transformations within the 5 dots structure 5. of two dendrimers with Example 4. We shall investigate the case of a more complicated structure Table 11. BSG transformations within the 5 dots structure 5. 11 and 15 branching dots, which at the initial stage have a bond only between the directed graph tree g[0] = (0 1 2 3 4 5); g[0] g[1] g[2] g[3] g[4] 3 4 5); g[0]g[1] g[1]g[2] g[2]g[3] g[3]g[4] g[4]g[1] “roots”. g[1] g[0] = (1 =2 (0 4 11324); 1 3 4); g[1]g[2] g[2]g[3] g[3]g[4] g[4] g[1] g[2] g[1] = (2 =4a(1 3 phased 22143); g[1] g[2] occurs (Figure 13) and this As the result of the interaction, rearrangement of the bonds g[2] = (2 4 3 2 1 3); g[2] g[3] g[4] g[1] g[2] g[3] = (4 3 1 4 2 1); g[3] g[4] g[1] g[2] g[3] visual data fully matches the BSG structure presented in the table of the general group of 26 elementsg[4] g[3] = (3 =1 (4 2 33412); 4 2 1); g[3]g[4] g[4]g[1] g[1]g[2] g[2]g[3] g[3]g[4] dots (Table 12).
g[4] = (3 1 2 3 4 2);
g[4] g[1] g[2] g[3] g[4]
Table BSG table (26 the local symmetry Example 4. We shall12. investigate the dots, case 4th of aorder) morewith complicated structure center. of two dendrimers with Example 4. We shall investigate the case of a more complicated structure of two dendrimers with 11 g[0] and=15 branching dots, which at the initial stage have a bond only between the g[2] directedg[4]; graph (0 branching 1 2 3 4 5 6 7 dots, 8 9 10which 11 12 13 15initial 16 17 18 19 20have 21 22a23 24 25); 11 and 15 at14 the stage bond only betweeng[0] theg[1] directedg[3] graph tree treeg[1] “roots”. = (15 0 0 1 1 2 2 3 3 4 4 5 5 6 6 0 15 15 16 17 16 17 21 21 20 20); g[1] g[2] g[3] g[4] g[3]; “roots”. As= (0 the15result phased rearrangement occurs (Figure g[2] 15 0 0 0of0 the 1 1 1interaction, 1 2 2 2 2 15 0 a0 15 15 15 15 17 17 16 16); of the bonds g[2] g[4] g[3]13) g[4];and As the result of the interaction, a phased rearrangement of the bonds occurs g[3] (Figure 13) and this thisg[3] visual presented of 26 = (15 data 0 0 15fully 15 15 matches 15 0 0 0 0 0the 0 0 BSG 0 0 15structure 15 0 0 0 0 15 15 15 15); in the table of the g[3]general g[4] g[3]group g[4] g[3]; visual data fully matches the BSG structure presented in the table of the general group of 26 elementsg[4] = (0 15 15(Table 0 0 0 012). 15 15 15 15 15 15 15 15 15 0 0 15 15 15 15 0 0 0 0). g[4] g[3] g[4] g[3] g[4]. elements-dots dots (Table 12). Table 12. BSG table (26 dots, 4th order) with the local symmetry center. g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25); g[1] = (15 0 0 1 1 2 2 3 3 4 4 5 5 6 6 0 15 15 16 17 16 17 21 21 20 20); g[2] = (0 15 15 0 0 0 0 1 1 1 1 2 2 2 2 15 0 0 15 15 15 15 17 17 16 16); g[3] = (15 0 0 15 15 15 15 0 0 0 0 0 0 0 0 0 15 15 0 0 0 0 15 15 15 15); g[4] = (0 15 15 0 0 0 0 15 15 15 15 15 15 15 15 15 0 0 15 15 15 15 0 0 0 0).
g[0] g[1] g[2] g[3] g[4]; g[1] g[2] g[3] g[4] g[3]; g[2] g[3] g[4] g[3] g[4]; g[3] g[4] g[3] g[4] g[3]; g[4] g[3] g[4] g[3] g[4].
Figure Figure 13. 13. Visualization Visualization of of the the process process stages stages (by (by the the first first scenario, scenario,example example2) 2)in inaaBSG BSG(26 (26dots). dots).
Broken symmetry research becomes relevant not only for structural chemistry and modern physics of elementary particles, which appear to be a quantum theory of nonabelian gauge fields with spontaneously broken symmetry, but also for other real micro- and macro systems, including biological as well as socioeconomical. Each of them has its own symmetry infractor “defect”, which causes the creation of new properties, along with evolution process occurrence. Therefore, further scientific confirmation is required. Figure 13. Visualization of the process stages (by the first scenario, example 2) in a BSG (26 dots).
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Table 12. BSG table (26 dots, 4th order) with the local symmetry center. g[0] = (0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25); g[1] = (15 0 0 1 1 2 2 3 3 4 4 5 5 6 6 0 15 15 16 17 16 17 21 21 20 20); g[2] = (0 15 15 0 0 0 0 1 1 1 1 2 2 2 2 15 0 0 15 15 15 15 17 17 16 16); g[3] = (15 0 0 15 15 15 15 0 0 0 0 0 0 0 0 0 15 15 0 0 0 0 15 15 15 15); g[4] = (0 15 15 0 0 0 0 15 15 15 15 15 15 15 15 15 0 0 15 15 15 15 0 0 0 0).
g[0] g[1] g[2] g[3] g[4]; g[1] g[2] g[3] g[4] g[3]; g[2] g[3] g[4] g[3] g[4]; g[3] g[4] g[3] g[4] g[3]; g[4] g[3] g[4] g[3] g[4].
4. Conclusions A computer experiment with tables of binary transformation sets with a “defect” discovered new transformation sets with zero, with unity, without the reverse elements, with convergent and with divergent properties of the structure orgraph. They were called BSG. A possibility of a transition from the structure group with convergent orgraph properties to a group with divergent properties. A developed approach clearly demonstrates a creation of evolutionary trees and growth spirals of the orgraph structure; therefore, it emphasizes a general aspect of scientific studies. The researched method of visualization allows easy presentation of the points and transitions of broken symmetry in designated transformations. Author Contributions: Conceptualization, V.G.R. and I.A.T.; Data curation, T.F.R. and S.V.P.; Formal analysis, V.G.R. and I.A.T.; Investigation, V.G.R.; Project administration, V.G.R. and I.A.T.; Resources, T.F.R. and S.V.P. Funding: The topic is supported by RFBR grant N 18-07-00170. Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17.
Bornstein, M.H.; Koptsik, V.A. Symmetry in science and art. Leonardo 1977, 10, 334. [CrossRef] Veinstein, B.K. Modern Crystallography; Nauka: Moscow, Russia, 1979. McKay, J. Graphs, singularities and finite groups. Proc. Symp. Pure Math. 1980, 37, 183–186. Zorky, P.M. Symmetry of Molecules and Crystal Structures; Moscow University: Moscow, Russia, 1986. Chuprunov, E.V. Symmetry and Pseudosymmetry of Crystalls; Nizniy Novgorod State University: Nizniy Novgorod, Russia, 2015. Kostrikin, A.I. Introduction to Algebra; Nauka: Moscow, Russia, 1977. Hall, M.J. Combinatorial Theory; Blaisdell Publishing Company: Waltham, UK, 1967. Nelipa, N.F. Physics of Elementary Particles. Calibration fields; Vischaya Shkola Publishers: Moscow, Russia, 1985. Rau, V.G.; Lomtev, L.A.; Rau, T.F. Non-crystallographic symmetry in packing spaces. Symmetry 2013, 5, 54–80. [CrossRef] Rau, V.G.; Lomtev, L.A.; Rau, T.F.; Nikitin, O.R. Broken symmetry groups with non-classic substitution within a computer experiment. In Proceedings of the Congress Digest of the First Russian Crystallography Congress, Moscow, Russia, 21–26 November 2016. Schulte, E. Polyhedral Structures, Symmetry, and Applications; MDPI: Basel, Switzerland, 2018. The Cambridge Structural Database. Available online: webcsd.ccdc.cam.ac.uk (accessed on 15 April 2018). Rau, V.G.; Lomtev, L.A.; Rau, T.F.; Nikitin, O.R. Broken symmetry group in the octahedral [Me(urea)6 ]2+,3+ cation with Chelate hydrogen bonds. J. Struct. Chem. 2018, 59, 120–125. [CrossRef] Rau, V.G.; Bondar, V.I.; Rau, T.F.; Stepanov, S.V. Problems of Crystal Chemistry; Nauka: Moscow, Russia, 1990; pp. 99–110. Rau, V.G.; Lomtev, L.A.; Rau, T.F.; Gorshkov, K.A.; Nikitin, O.R. Computer experiments in groups of permutations with broken symmetry. Mod. Prob. Sci. Educ. 2017, 43–49. Koonin, E.V. The Logic of Chance: The Nature and Origin of Biological Evolution; FT Press: Upper Saddle River, NJ, USA, 2011. Togunov, I.A. Structural Functional Model of Social Systems; RANEPA: Moscow, Russia, 2016. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
