Penrose tiling fractality in coordination Cayley’s tree graphs representation

Physica A 389 (2010) 4127–4139

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Penrose tiling fractality in coordination Cayley’s tree graphs representation A.N. Mihalyuk, P.L. Titov, V.V. Yudin ∗ Faculty of Physics and Engineering, Far Eastern National University, 690950 Vladivostok, Russia

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Article history: Received 16 July 2009 Received in revised form 26 November 2009 Available online 15 June 2010 Keywords: Penrose tiling Coordination Cayley’s tree graphs Percolation Fractality

abstract We offer the mathematical apparatus for mapping lattice and cellular systems into the generalized coordination Cayley’s tree graphs. These Cayley’s trees have a random branchiness property and an intralayer interbush local intersection. Classical BetheCayley tree graphs don’t have these properties. Bush type simplicial decomposition on Cayley’s tree graphs is introduced, on which the enumerating polynomials or enumerating distributions are built. Within the entropy methodology three types of fractal characteristics are introduced, which characterize quasi-crystalline pentagonal Penrose tiling. The quantitative estimate for the frontal-radial fractal percolation on a Cayley’s tree graph of a Penrose tiling leading to the overdimensioned effect is calculated. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Since the discovery of quasicrystals by Shechtman et al. [1], quasi-crystalline symmetries have become the subject of numerous investigations. When talking about quasi-crystalline symmetries theory, one of the major problems is the orderdisorder degree diagnostics. Moreover, the quantitative identification method, for instance, for studying quasi-crystalline symmetry organization, has to be the universal one. According to the main tendencies of mathematical crystallography originated from the Voronoi-Delaunay decompositions [2,3], paving, coating and partition procedures are considered in terms of absolutely solid tiles. The problem is to define which symmetry transformations are possible here. From our point of view it shouldn’t be restricted only by the ‘‘tile’s’’ alphabet in the problem of quasi-crystalline paving synthesis. It is necessary to take into account some direct coordinations, incidencies between the neighboring tiles. It is the coordination component of lattice structures that usually stands aside. In recent years, a number of studies have appeared, which avoid the problem of tiles matching [4–6]. These papers may be related to the physical direction of the quasicrystal growth problem, avoiding the ‘‘tile’s formalism’’. Particularly in paper [6], the authors use a coupled short-range potential, which in the modeling process leads to the typical quasicrystal aperiodicity. The physical interpretation of this mechanism is given in paper [7]. It should be noticed that in paper [6], at the same time, consideration via the cellular structure of Penrose tiling boundaries is carried out. The icosahedral property, which is often used in the theory of amorphous solids we perceive as an extra excessive load in the model. Under the physical approach the integral criteria such as an average macrodensity of the filling out procedure or internal energy of a growing cluster prevail [6,7]. Here it is necessary to remember the ideas of Penrose [8], that modeling of Monte-Carlo cluster forming will demand the introduction of relaxation procedures and integral rearrangement procedures for the whole cluster. The strategy of successive atoms in conjunction with corresponding minimizing of growing cluster cannot be the solution in the problem of quasicrystal structure synthesis. In our opinion, one should not eliminate the task of tiles matching and should

∗

Corresponding author. Tel.: +7 914 658 00 32. E-mail address: [email protected] (V.V. Yudin).

0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.06.008

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a

c

b

Fig. 1. The Penrose tiling (a) with it’s alphabet (b), and tree Cayley graph (c).

not substitute for purely physical approach. It is necessary to give such representation of lattice, cellular structures, where both ‘‘tiles’’, and coordination components would be peer. Generalized tree Cayley graphs (TCG) [9–11] are the correct such objects. This paper is organized as follows. In Section 2 the algorithm of coordination TCGs construction and their mathematical properties are discussed. In Section 3 the tangential fractal dimension of TCG in approaching of it outgoing bushes is introduced, which corresponds to the frontal percolation type. This characteristic is the supremum or the capacitive one from the information-theoretical point of view. In Section 4 the Mandelbrot’s shell (MSh) for the frontal representation of percolation on the TCG is introduced. Mandelbrot’s shell is a geodesic zigzag line on each streamer of TCG. It is possible since our coordination TCG are the Markovian ‘‘jungles’’ [9,12]. There is no such conception at all in typical Bethe–Cayley tree graphs, which do not have an interbush local intersection, even if they possess a random branchiness property. Mandelbrot’s shells may be considered as a generalization of the plane Huygens fronts. Thereby the Mandelbrot’s shells provide wave percolation on the TCG in radial direction, which fractality is computed by two methods. The presence of the abovementioned intersection brings in certain structural redundancy into the tree of coordinations. The estimate of redundancy degree or the Penrose tiling semantic level, as a linguistic system at the abovementioned alphabet with tree graph grammatics is given here [7,13]. In Section 5 the conception of streamer as a ‘‘bushy’’ fractal is introduced in opposite to the ‘‘fat’’ fractal or the Mandelbrot’s shell. Final results and their interpretation according to three methods of streamer fractal dimension estimation are given. Here the strongest result is an estimate of streamer’s ensemble on the TCG with accounting alphabet [2q × 2p] and probabilistic structure. 2. Informodynamical method for studying of cellular, lattice systems In our papers [9–12] a set of studies dedicated to searching the general representation method of quasi-crystalline lattices and cellular systems of mesodefects of quartz glasses and amorphous films was carried out. The formalism for mapping these objects into coordination tree Cayley graphs was introduced [10,11]. It is necessary to indicate three important points, which distinguish general TCG from Bethe–Cayley tree graphs [9]. The variable degree of branching of the bushes is accepted for such TCG. This fact in turn will be the indicator, that such a coordination TCG may be endowed by probabilistic measures and distributions. Finally, the third and a quite important moment, is that our coordination TCG allows a local interbush intralayer intersection. These aspects and other mathematical characteristics of topological, metrical character are also subject of our consideration. SuperTCG cannot be studied without applying decomposition methods. We chose the simplicial decomposition. Requirement of simpliciality is extremely important in the theory of complex systems [14]. This requirement in turn strongly correlates with the wellknown Mandelbrot’s fractality definition. The coordination tree Cayley graphs (TCG) equally reflect both tiling part in their own vertices and direct coordinations (incidences) in their own edges. If to consider our superTCG, which represents lattice, cellular systems then we eo ipso achieve a complete description of both tiling (element) components, and coordination components (ways of neighboring). That’s why in our papers the conception of hierarchical alphabet consisting of the elementary typical modules is used. For example, in papers [11,15] it was shown that for pentagonal Penrose symmetry the alphabet [2q × 2p] is sufficient. Fig. 1 shows Penrose tiling (a), three-level hierarchy tile’s alphabet (b), and it tree Cayley graph (c). In paper [8] this nonrecursive procedure was carried out at the second alphabet level. This procedure may be simplified by employing block coding conception from the information theory. Then the synthesis of such lattices is realized by the third alphabet level (by the pair of decagons Fig. 1(b)), though minimal crossings by modules of second alphabet level have to be assumed here. Still, heuristics of quasi-crystalline mosaics synthesis at the third level of alphabet looks much simpler than nonrecursive procedure at the second alphabet level. The mapping process of generalized lattices, and cellular systems into coordination

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tree Cayley graphs works as follows: (i) Any module from the dual alphabet of second level is accepted as the TCG vertex. Then the rosette of initial coordinations, adjacencies, vicinities is being built. (ii) From the reached vertices analogous straight vicinities directed only forward are built. This cascade of coordinations in principle is infinite, thus we obtain growing TCG (Fig. 1(c)). All vertices and edges for Penrose mosaic may be described by alphabet [2q × 2p], where the q-coordinate indicates the vertex type (2-nd alphabet level), and the p-coordinate characterizes the straight contact with the other lower standing module. For the planar case two contact and vertex types are possible. Fig. 1(c) shows the coordination TCG for the Penrose mosaic. For the planar case two contact and vertex types are possible. Fig. 1(c) shows the coordination TCG for the Penrose mosaic. By own topology it substantially differs from the widely known Bethe–Cayley tree graphs [16,17]. In our method the variable (even random) branchiness degree of outgoing bushes is assumed. So even in this sense such TCGs are endowed by the appropriate probabilistic measures, distributions. Besides, our coordination TCGs have property of intralayer local interbush intersection, which is usually absent in the classical Bethe and Cayley’s tree graphs [16,17]. TCGs are endowed by internal metrics (ultrametrics). Meanwhile the external metrics of the paving surface will be the Euclidean one. Then we have to deal with non-Euclidean mapping of TCG onto plane with Euclidean topology. In our paper [12] in goal to empathize abovementioned effect of intersection on the TCG, we introduced the term Markovian ‘‘jungles’’. One may equivalently present evolving TCG in the percolation terms. Any functionals on the TCG are the subject to percolation [9–11]. In general, as we already noticed, tree graph has the quasistochasticity property, and this factor is exclusively internal and not the externally imported. In our case, in Fig. 1(c) one may readily see, that we have no deal with any stochasticity. This system seems not to be the stochastic, but the chaotic one. We want to especially emphasize the significant difference between stochasticity and chaos categories. Chaos may possess some ordering degree, which is observed in the corresponding fractal dimension values. Shown in Fig. 1(c), TCG may be described by conventional forming logic:

( G=

XX

N =

i

ji

nj i ;

nj ni [ i [ i

) x(lji+1 /lji ); K (lji )

(1)

ji

where i—is the TCG level, ji —is the level index of i-level, K (lji )—is the branchiness index of ji -vertex at the i-level, x(lji+1 /lji )— is the outgoing S S edge from ji to ji+1 (conventional logic), nji —are the vertices at the i-level, N—is the total quantity of TCG vertices, i ji —are the consolidation operators by ji -vertices and i-levels. Let us empathize that in Eq. (1) edges are presented in the conventional logic, where ji —is a starting index, and edge is directed to the address ji+1 . Like for any graph, in Eq. (1) vertices are described, and the most important—the branchiness degree of outgoing bush is defined. This term (bush) prompts that in the process of tree graph analysis appropriate decomposition methods must be applied, among which we especially empathize the simplicial decomposition [14,18]. One may conveniently realize it in the bush approach. Any bush may be presented as the sum of outgoing branches: K (lj )

gx (K (lji )) ≡

Xi

x(lji+1 /lji ).

(2)

ji+1

In Eq. (2) the partial bush with branchiness (lji ) is considered, which starts from the corresponding vertex. The summation procedure is accomplished for partial bush by ji+1 index. x(lji+1 /lji )—are branches in the conventional logic. Consequently the TCG (Eq. (1)) has to be decomposed into simplicial bush decomposition:

( G=

N;

nj ni X i X i

) gx (K (lji )) .

(3)

ji

Eq. (3) in comparison with Eq. (1) is a considerable advancement, because here hierarchical bush decomposition is used. The Markovian expansion is allowed by i-index, i.e. we deal with the radial M-chain. It should be noted that Markovian feature in the tangential direction is not satisfied since local interbush intralayer intersection property presence. On the lattice, cellular structures the flow of coordinations is the subject to percolation, this flow is topologically reflecting in the growing TCG. The conception of simplicial decomposition in fact means the decomposition of superTCG into bushes (Eq. (3)). Taking into account the character of TCG internal metric, we will introduce the ultrasimplex concept, which interprets the partial bush. Therefore simplicial decomposition of TCG may be treated as the method of hierarchical decomposition of supersimplexes into additive composition of ultrasimplexes. This principle is quite constructive, because automatically assumes the topological similarity of supersimlex and its constituent modules—partial ultrasimplexes. Such an approach is known in the system analysis of complex structures [14]. From the other side, one may remember the fractal methodology of Mandelbrot [19,20], where the fact of decomposition of big complex system into similar in certain criteria modules is the necessary requirement for the fractality.

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a

b

Fig. 2. The branchiness statistic (a) and profiles of PEP distributions (b) for TCG of Penrose tiling.

Further inquiry of superTCG is based on the applying of enumerative theory [18] to the tree-graph structures. In the classical approach enumerating polynomials (EP) are calculated. It may be specified for each tree hierarchy level:

Πi (xr ) =

r X

Ti (xk ) · xk .

(4)

k

Any EP is specified by two freedom degrees: k—is the bush branchiness degree, Ti (xk )—are coefficients of EP in the absolute form. Coefficients point to the number of bushes with fixed branchiness at the corresponding TCG level. It should be noted that EP in that form assumes the algebraic direction of graph’s studying. Further, when appropriate normalization procedures are accomplished one may proceed from EP to probabilistic EP (PEP), and consider these PEP as probabilistic distributions (already no polynomials):

Πi (xr ) =

r X

ti (xk ) · xk ; ti (xk ) =

T i ( xk ) r

P

k

.

(5)

Ti ( ) xk

k

In that case one may consider the percolation of such statistics on the TCG. The opportunity of studying statistical forms of percolation is opening up here. The theory of dynamical chaos supposes that systems of such kind may be analyzed by statistical methods, in spite of their ‘‘dynamics’’. During computation of enumerating polynomials (EP), we observe that all have a range of branchiness definition 0–7, and the mode of distribution falls on the third bush branchiness degree. This mode corresponds to the tetrad Bethe graph, and its fraction yields 30% in the total PEP statistic. The summary portion of branches with branchiness degrees 3, 4 and 5 yields 65%. The portion of frustrated bushes is rather high and yields 18.5%. These estimates are calculated for cumulative PEP of a 12-level tree Cayley graph. Since there are two types of tree-graphs for Penrose tiling, which are generated by two different ‘‘golden’’ rhombuses, then it is necessary to use a weight-average cumulative PEP. The values that we just gave are such a one. Thus, the pentagonal symmetry of bushes with fourth branchiness degree is the basis of the tree-Cayley graph for a Penrose tiling. Eo ipso, we have a quite simple modal branchiness distribution, which allows us to consider the TCG of Penrose tiling as a chaotic system of a purely geometrical type. As an example, we have decided to show the percolation of bush branchiness distribution over the TCG hierarchy levels in the two views (Fig. 2). This figure shows explicitly a quite articulate periodicity of the PEP percolation dependence, which is of a substantially wave-kind of nature. It is quite possible that this phenomenon is related exactly to the feature of quasicrystallinity of pentagonal symmetry. We should note here that this result is an effect of taking into account probability measures on the TCG of Penrose tiling (Fig. 2). The next stage in the informodynamical method [9–12,15,21–23] is the proceeding to the function level. Here one has to compute functionals over EP, PEP and to consider their percolation on the TCG. We hope that percolation dependencies of entropy functional [24] will give the opportunity to identify the order-disorder character of generalized lattices and cellular systems. Hv [Πi (xr )] =

r X k

ti (xk ) · [1 − ti (xk )] =

r X

ti (xk ) · ti (xk ).

(6)

k

It should be noted that any entropy functional has to obey the necessary and sufficient conditions of Fadeev–Feinstein theorem. From EP and PEP were formed entropy functionals, whose percolation over TCG hierarchy levels is shown on the Fig. 3. In this diagram, the well-defined periodicity of entropy functional over the TCG hierarchy levels is also observed.

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Fig. 3. The percolation of entropy Hv and structuredness factor η for Penrose tiling.

Fig. 3 also shows the structuredness coefficient, which may also be used for estimation of the degree of chaoticity. Here, we may conclude that the tree Cayley graph for Penrose tiling, and Penrose tiling itself, are characterized by the chaoticity level, which is around 90%. So, here in the analysis of the entropy functional, the quasicrystalline pentagonal symmetry is again characterized by wave-like percolation dependence. For the classical crystalline systems [21], such an analysis shows the hyperbolic decaying dependence of entropy functional. The critical index for this dependence is considerably less unity in the case of honeycomb and simplex lattices, and yields unity for the quartet lattice. So, the classical crystalline order is characterized by significant long-range action [21]. The above formulated principle of superTCG simplicial decomposition immediately prompts a quite strong correlation with the fractality conception [20]. As known, at the basis of fractality conception the scaling property is the necessary condition. The simplicial decomposition automatically possesses this property, because bushes are the subtrees in the topological plane. The tree-graph approach gives a new opportunity to formulate the method of fractal dimension estimation for quasi-crystalline mosaics, tilings, cellular systems of mesodefects [9,25,26]. In the fractal theory the dimension principle of Hausdorff–Besicovitch methodology is used, and in recent years a great number of different algorithms for estimation of fractal characteristics were offered. In this paper, fractality estimation methods for tree-graph structures occupy the central position. We will be stating them in entropy terms. Here the freedom degrees are EP, PEP coefficients and bush branchiness degrees. Just for them we offer to define the entropy measures. Eo ipso we get the opportunity to introduce functions in the Radon sense [27]—the dependence of entropy of EP, PEP coefficients from the entropy of ultrasimplex average bush branchiness. Continuing this way one may employ the Radon–Nikodym derivative concept, and in the simplest case to demand its consistency, which in turn will be the corresponding fractal characteristic [27]. 3. Tangential fractal dimension for Cayley’s tree graph of the Penrose tiling Now we will give the fractal dimension estimate for Penrose tiling in the tree-graph representation employing the supremum characteristics of percolation of enumerating structures on the TCG. Particularly the enumerating polynomial (EP) may be taken as the basis for estimation method. EP is characterized by ri —polynomial rank, Ti (xk )—EP coefficient, and ki —bush branchiness degree. The construction logic of tangential fractal dimension in approaching of outgoing bushes was obtained and thoroughly discussed in our papers [15,21,23,28]. Here we just give the final expression:




dtg TΣ 10 (x ) = k




r T 10 (xk ) Σ

1 rCP TΣ 10 (xk ) · ln iΣ 10




·

X

ln TΣ 10 (xk )

k=0




ln(k + 2)

,

(7)

where i ≥ 2; 0 ≤ k ≤ r, ri —is a polynomial rank; Ti (xk )—is a EP coefficient; xk —is a bush with k-branchiness at i-level; ln iΣ 10 —is a ultraheight for 10-th TCG level. The total sum of independent outgoing bushes at any TCG level we consider as a tangential phrase. Symbols of such phrases are bushes with corresponding branchiness degrees. The linguistic approach is useful for considering the multifractal objects. Than under the summation symbol of Eq. (7) stands the ratio of entropy of bush reiteration to the entropy of bushes itself. The summation process is going by k up to EP rank value. In such case, the tangential phrase is the sum of normalized entropy functionals. Alternatively, the number of bush quantums expresses the quantity of diversity in the tangential phrase. Normalizations by EP rank and ultraheight of TCG are performed further. Numerical estimate for our TCG of Penrose tiling for TΣ 10 (xk ) according to abovementioned assumptions yields: dtg TΣ 10 (xk ) = (16.118 · · ·)−1 · 22.19 · · · ≈ 1.38.




(8)

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Fig. 4. The percolation of tangential fractal dimension on the tree Cayley graph of the Penrose tiling.

Obtained estimate (Eq. (8)) for tangential fractal dimension of TCG of Penrose tiling is interpreted as a capacity of percolation. This estimate is the supremum one, since is obtained in the approaching of outgoing independent bushes for each level of TCG hierarchy. In our papers [9–11] and in paper [29] the fractal dimension of Fibonacci system, which yields 0.6909 was calculated. If in paper [29] the fractal dimension was calculated by the Hausdorff–Besicovitch methodology, then in our studies it was calculated by the informodynamical method. The Fibonacci symmetry is essentially inherent to the pentagonal Penrose tiling. The percolation task of tangential fractality on the TCG of the Penrose tiling has been also analyzed on the level of tangential fractality. Fig. 4 shows a quite confident wavelike oscillatory percolation dependence of the tangential fractality. This supremum capacitive feature is quite important, therefore, it is necessary to calculate the estimate of fractal dimension by one additional technique. According to the enumerative tree graph theory [18], we will assume normalized EP or probabilistic EP and corresponding entropy functionals in averaged or asymptotic versions as the methodological basis. It is possible to obtain weaker supremum estimates under such conditions. The final equation is:


i

h

dtg tΣ 10 (xk ); K∞ ∨ K 10 =




sup H tΣ 10 (xk ) ln

h




i,

(9)

K∞ ∨ K 10 + 1

dtg tΣ 10 (xk ); K 10 = 1.4078 . . . ,




(10)

where tΣ 10 (x )—is a probabilistic cumulative EP (or bushes distribution by branchiness factor) of a 10-th level TCG; K∞ ∨ K 10 —are the asymptotic or normalized average branchiness on the TCG. In the numerator of Eq. (9) the upper limit of entropy is located, which in turn is divided to the entropy of average or asymptotic branchiness. Notice that this equation is quite different from the Eq. (7), but construction logic is the same for both equations. In this case the information-theoretical interpretation may be given, which is free from the service specific normalizations in Eq. (7). The fractal dimension is obtained as a ratio of entropy to the entropy of bush branchiness degrees either in asymptotic or in average. These upper fractal estimates may be interpreted as a linearized form of the derivative in the Radon–Nikodym sense [27]. k

4. Fractality of a Mandelbrot’s shell on Cayley’s tree graph of the Penrose tiling In contrast to the approach from the previous section, now we will change the point of view by analyzing the tree Cayley graph in the approaching of ingoing bushes. In this way we have an opportunity to treat the TCG in terms of Markovian ‘‘jungles’’ [9,12]. The connectivity of TCG provides the existence of geodesic fronts, which may be outlined on each level of the TCG. Fig. 5 clearly shows that these continuous fronts have the star-shaped structure, which possesses topological scaling property. Star-shapedness or saw-shapedness degree of fronts reflects the adjacency degree in the abovementioned sense. Such fronts are known as Mandelbrot’s shells (MSh) [30] and may be called ‘‘fat’’ fractals. It is natural to put in the ‘‘fatness’’ measure of the fractal here, which is interpreted as a ratio of MSh length of the corresponding TCG level to the typical circle length at the same hierarchy level. The length of ‘‘fat’’ fractal is a sum of incidences at the corresponding TCG hierarchy levels. In addition, the minimal number of edges on the MSh corresponds to its lowest overall length. The Mandelbrot’s shells should be treated as the fractal generalization of smoothened Huygens fronts. In this case, considering Fig. 5, one may analyze the MSh percolation and interpret shells as the expansion of fractal waves. From the information-theoretical point of view, the Mandelbrot shells have another interpretation based on adjacency and intersection conceptions. It points to

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Fig. 5. The fragment of Cayley’s tree graph for the Penrose tiling with outlined Mandelbrot’s shells.

Fig. 6. The percolation of Mandelbrot’s shell ‘‘fatness’’ on the tree Cayley graph of the Penrose tiling.

some coordination redundancy in the lattice systems, which explicitly presents in coordination the TCG. Just the adjacency degree makes a contribution to one or another type of structure ordering. If to consider the quasi-crystalline tilings as twodimensional texts [15] with offered tree grammar [10,11] than the ensemble of MSh reflects the semantic of quasi-crystalline text [28]. A quite simple estimate of MSh fractal dimension in the terms of ‘‘fatness’’ may be easily expressed as: dMSh (ξi ) = 1 +

ln ξi

(11)

ln(i + 1)

Pm

i where ξi = Li /(2π i)—is the ‘‘fatness’’ of MSh; Li = ki =0 (lki )—is the MSh length, expressed by the sum of incidences; the summation process is going by k-index at the fixed i-level of the TCG hierarchy, mi —is the minimal number of edges on the MSh front which is the requirement for geodesic property; ln(i + 1)—is the ultrametric height of TCG at the i-level. A ‘‘quick’’ version of Eq. (11) may be obtained, if to substitute the average or asymptotic ‘‘fatness’’ at the each hierarchy level of TCG for the ‘‘fatness’’ property of Msh. One may study the percolation of ‘‘fatness’’ feature over hierarchy levels of TCG itself before addressing to Eq. (11). Fig. 6 clearly shows the periodical percolation of MSh ‘‘fatness’’. We have already encountered similar wave-like behavior of tangential fractal dimension percolation (Fig. 4). This fact leads to the conclusion that such type of percolation belongs to characteristics of the Penrose tiling quasicrystallinity. Fig. 7 shows the percolation dependence of fractal dimension of MSh via ‘‘fatness’’ property (Fig. 7(b)) and analogous typical estimate (Fig. 7(a)), respectively. Estimation by the typical method yields:

dMSh (i = 10) = 1.2085 . . . ,

(12)

and estimation by the ‘‘fatness’’ property yields: dMSh (ξi=10 ) = 1.2637 . . . .

(13)

All introduced methodics allow a higher level of accuracy evaluation, since they are based on the ordinary calculation of TCG branches number and defining of corresponding EP and probabilistic EP.

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a

b

Fig. 7. The Mandelbrot’s shell length in double logarithmic axis (a), and fractal dimension of the MSh by ‘‘fatness’’ property (b).

Let us compare the fractal dimensions of tangential phrases and MSh. Eqs. (7)–(11), (13) show: dMSh (TCGP ) < dtg (TCGP ).

(14)

Such an inequality is quite typical for the information-theoretical analysis. The intersection, adjacency of the TCG may be interpreted as redundancy. The presence of any redundancy must decrease the fractal capacity of tangential phrase. It is methodically significant to draw the parallels between the redundancy of the ‘‘language system’’ and the organization of the Penrose tiling itself. Then the redundancy coefficient may be introduced:

η(TCGP ) = 1 −

dMSh (TCGP ) dtg (TCGP )

=1−

1.264 1.408

≈ 10.2%.

(15)

It is seen that the redundancy level (Eq. (15)) of two-dimensional text of the Penrose tiling in the tree-graph representation is about 10%, which supports the corresponding level of semantics or organization of the Penrose tiling. Since this object is quasidetermined, such a system is almost 90% chaotic, but not stochastic. 5. Streamer’s fractality of the Cayley’s tree graph of the Penrose tiling This section is dedicated to another approach for estimation of TCG fractal characteristics of the Penrose tiling (TCGP). We consider the dual to Mandelbrot’s shells form of percolation on the TCGP. Here is an analogy of the wave propagation by Huygens fronts and their orthogonal rays. In our case, Mandelbrot’s shells (‘‘fat’’ fractals) substitute a locally convex Huygens surface in the planar case. Streamers (the ‘‘bushy’’ fractals) represent rays in the dual component. Thereby if talking about the percolation of coordinations than it is necessary to consider both the frontal percolation (MSh) and radial (streamer) percolation. Proceeding to the streamer’s representation it is a proceeding to the new type of simplicial decomposition of superTCGP (Fig. 8). Now the ensemble of streamers is chosen as a subsystem. This ensemble must satisfy the completeness property and possesses the σ -algebra properties. TCGP is an ensemble of streamers (Fig. 8), averaging process over which will give an estimate of streamer fractality of the TCGP. An algorithm for streamer’s fractal dimension estimation is as follows: (i) It is proposed that streamer ensemble on the TCGP has been already built and the hierarchy of branchiness for each streamer has been found. According to these findings a divisor is calculated: Dl_str (m) =

Km Y

Kl .

(16)

K1

(ii) Further, the partial entropy of the divisor for the l-streamer of m-height is built: h Dl_str (m) = ln Dl_str (m).




(17)

(iii) The geometrical entropy of TCGP is formed: S (m) = ln m.

(18)

(iiii) Partial entropy of the streamer’s branchiness may be considered as a certain function of geometrical entropy in the Radon sense: hD [S (m)], where both functions are entropies of the same class, and in fact they may be recognized as the Radon measures.

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1 2 8 3

7 4

6

5

Fig. 8. Ensemble of streamers on the tree Cayley graphs of the Penrose tiling.

a b

Fig. 9. Two streamers from the σ -ensemble (a), and the verification of statistical hypothesis for Pareto distribution (1—initial and, 2—asymptotic distributions).

(iiiii) Function introduced in the Radon sense has a Radon–Nikodym derivative, which in the particular case will be a linear dependence: dl_str (m) =

ln ln Dl_str (m) ln m

.

(19)

In Eq. (19) the only topological aspect of streamer fractality is considered. The upper tangential fractal estimate was calculated similarly. The final result of streamer’s ensemble (Fig. 8) fractal dimension yields dl_str (m) = 1.3855. This value quite conforms our capacity percolation value on the TCG for tangential fractal dimension. The fact of matching of these characteristics points to the consistency of Penrose tiling in the tree-graph representation. The next analyzing stage is to take into account the probabilistic alphabet [2q × 2p] structure. As Fig. 9(b) shows, this quartet-type alphabet conforms the hyperbolic statistic with unit critical index. It is a well-known law of Zipf–Pareto–Mandelbrot (ZPM) which points to existing of linguistic model for Penrose tiling, which in fact is the twodimensional text, the grammatics of which is realized by the TCGP. Fig. 9(a) shows two real streamers in the alphabet [2q × 2p] with corresponding probabilistic distribution shown in Fig. 9(b). In our opinion besides probabilistic the entropy structure of streamers must be accounted also. Eq. (20) intended for the quartet protodecomposition of superTCGP is the generalization of Eq. (19): ln dhstr (m) =

m P i =1

ln

4 P

Vα h p(Vα )

Vα =1

ln m



 .

(20)

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Fig. 10. Cellular structure of mesodefects of metallic glass Co-P (a) and quartz glass KUVI-1 (b).

Fig. 11. Tree Cayley graphs for metallic glass Co-P (a) and quartz glass KUVI-1 (b).

Protodecomposition is also the simplicial decomposition, but here the additive decomposition is executed into four types of branches (specified contact and edge type). Fractal estimation on the σ -ensemble yields dhstr (m) = 1.278. The structuredness degree is: dl_str (m) − dhstr (m) dl_str (m)

≈ 7.8%.

(21)

So, employing more sophisticated grammatics leads to fractal dimension increasing. By an analogous way the tangential fractal dimension in the alphabet [2q × 2p] is related to MSh fractal dimension. The ensemble of streamers possesses the semantic, which is manifested as a certain structure of the streamer’s bush distribution by the hierarchy levels (Fig. 9(a)). This figure shows certain quasi-periodicity of streamer’s branchiness in the radial direction. Here bushes with lesser branchiness succeed bushes with high branchiness with more probability, and vice versa. We called it an intermittency phenomena. Fig. 4 shows the same quasiperiodic percolation of tangential fractality. In our opinion the quasicrystalline Penrose tiling right possesses the typical quasiperiodicity of some functionals on TCGP, which may be because of a certain criteria of quasicrystallinity. 6. An application of informodynamical method to quartz and metallic glasses After consideration of the model structures, such as a pentagonal Penrose tiling, we will also show an application of our informodynamical method to certain real systems. In this paragraph, we will consider quartz glasses (QG) and thin amorphous films (AF) as such examples. From the whole variety of amorphous films rare-earth–transition metal, transition metal–metalloid — we selected the AF Co-P (Co-9 at.%P) (Fig. 10(a)), and its TCG is shown on the Fig. 11(a). From the four types of quartz glasses, we will introduce the mesostructure of KUVI-1 QG (Fig. 10(b)) and its tree Cayley graph (Fig. 11(b)). The electron microscope images (Fig. 10) clearly show the cellular structure of objects. The average size of AF Co-P cells yields 200–300 Å, with a thickness of about 300–500 Å. Quartz glass KUVI-1 presents the massive sample, whose image reveals the clearly defined granular structure. Note that, on the topological plan, we have dealt with either convex polygonal

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a

b

c

d

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Fig. 12. Tangential (a, b) and streamer (c, d) fractal characteristics for TCG’s of AF Co-P (a,c) and QG KUVI-1 (b, d).

cells or with singularities, which are contoured by the second-order curves. It is necessary to note that they are real, typical meso- and nanostructures of the above-mentioned objects. According to the main orientation of our paper – the fractal analysis of objects – we will give only the averaged estimates for fractal characteristics of corresponding tree Cayley graphs here (Fig. 12). We observe that fractal characteristics of QG KUVI-1 for tangential and streamer representations reach the values of the order of 1.68–1.70. The same characteristics for AF Co-P are around the order of 1.53–1.58. Note that, in both cases, both types of fractal characteristics are fully consistent within the accuracy of calculation. As it requires the Hausdorff–Besicovitch methodology, the values of these fractal dimensions exceed the unity, which itself corresponds to the edges of cells, singularities. It is difficult to reveal the property of fractality if we are to consider only the electron microscope images itself. But the tree-graph representation of these cellular structures, according to the concept of simplicial decomposition, allows us automatically to employ the fractal methodology. Also, if we are to use the same characteristics, then it is possible to track the thermal-radiation kinetics of the above-mentioned disordered mediums [22]. The results given in Fig. 12(a), (b) point to the presence of certain fractal percolation invariants for glassy and amorphous mediums. Finally, in conclusion of the present paper, we will show one more example of a real system, where the phenomenon of quasicrystallinity was detected. Fig. 13 shows the optical bright-field microphotographs of free surface of Fe70 B15 Cr15 ultrarapid quenched tape. This image presents the site-type spontaneous quasicrystallization of pentagonal morphology. Such quasicrystallization sites are quite rare on the ultrarapid quenched tape, and they appeared immediately after the finishing of the flow-turning process. It was observed that the rising of quasicrystalline sites is conditioned by the large tension gradients (on the edges of tape) that occurred there. Fig. 13(b) shows the effects of quasicrystalline stars’ growth from the liquid-phase basis of the ultrarapid quenched tape-free surface. The initial melt temperature was 1500 °C, the tape thickness is around 100–300 µm, and the centrifugal potential (the upper estimate) yields 1000 g. The stratified tape structure presents the compromise of Archimedean buoyancy forces, of centrifugal forces and thermal gradient equals 0.01 °C/nm. Fig. 13(a) shows that site size is around 70–100 µm, and the average size of five-fold star yields 10–15 µm. Such intense nonequilibrium state conditions the processes of spontaneous site-type quasicrystallization. The materials of the present section, which belong to such different objects, illustrate the introduced informodynamical method for analyzing complex systems constructively.

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Fig. 13. The optical bright-field microphotographs of quasicrystals on the free surface of Fe70 B15 Cr15 ultrarapid quenched tape.

7. Conclusion We have offered an adequate and one-one mapping of lattice structures into coordination tree Cayley graphs. Such representation allows describing the coordination component, or the way of cell interaction of the cellular system in an explicit way. Complex systems have to be represented in the tree-graph hierarchy, which is realized by the generalized Bethe–Cayley graphs. We suggest studying graphs by the method of simplicial bush decomposition. Generally, the bush is similar to the supertree, which is the necessary fractality condition. After performing a simplicial bush decomposition of TCG, at each hierarchy level the enumerating polynomials and different functionals over them are built, among which we mark out the informational entropy. The symmetry of TCG and ordering of cellular structure are studied by analyzing percolation of functional dependencies by the hierarchy levels. If we consider quasicrystals, then the graph itself is determined. In analysis of internal percolation task on TCG the entropy invariants in average with oscillation-wave behavior were identified. According to tree-graph formalism, we have to deal with principle of topological tree scaling. This fact allow the introduction of fractal characteristics on TCG, which represent the percolation of functionals through the system of corresponding fronts and duals to those rays. Both these systems are expressed in corresponding fractal characteristics. The first of them corresponds to the MSh with fractal dimension value in asymptotic which equals 1.264. MSh is a geodesic fractal, whose existence is provided by the presence of the intralayer interbush local intersection property. In turn, the upper estimate of MSh fractal dimension is a tangential fractal dimension, which yields 1.380.The relative difference between these two characteristics points to a certain redundancy or structuredness degree of TCG. Second system—the radial component corresponds to streamers, and the σ -complete ensemble of them is characterized by the fractal dimension and equals 1.386. Thereby TCGP is a highly coherent structure. An estimation of fractal dimension in our method is performed via Radon functions, which are the coefficients of EP entropy as a function of TCG ultraheight. Thereby any fractal dimension is a derivative in a sense of Radon–Nikodym. The simplicial decomposition of TCGP may be generalized to protodecomposition on the quartet alphabet [2q × 2p]. The probabilities distribution of this alphabet obeys the Pareto function with a unit critical index. Thereby TCGs may be treated as linguistic planar systems. Formalism of tree graphs allows the introduction of general fractal percolation as a direct sum of MSh and streamer’s percolations which yields 2.53. Since this dimension exceeds the packing space dimension, such percolation is overdimensioned. This fact means that planar mediums may possess the overdimensional percolation properties. But it doesn’t exceed the topological dimension of minimal smooth space of embedding. Existence of planar mediums with overdimensional percolation in the tree-graph topology may reveal the opportunities for studying not only the low-dimensional quantum systems. The application of such a type of percolation to neuronet systems may give an explanation of the range of known integral system effects of brain functioning. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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