Kauffman cellular automata on quasicrystal topology
aa r X i v : . [ n li n . C G ] J un Kau ff man cellular automata on quasicrystal topology Carlos Handrey Araujo Ferraz a, ∗ , Jos´e Luiz Sousa Lima a a Exact and Natural Sciences Center, Universidade Federal Rural do Semi- ´Arido-UFERSA, PO Box 0137, CEP 59625-900, Mossor´o, RN, Brazil
Abstract
In this paper we perform numerical simulations to study Kau ff man cellular automata (KCA) on quasiperiod lattices.In particular, we investigate phase transition, magnetic entropy and propagation speed of the damage on these lattices.Both the critical threshold parameter p c and the critical exponents are estimated with good precision. In order toinvestigate the increase of statistical fluctuations and the onset of chaos in the critical region of the model, we havealso defined a magnetic entropy to these systems. It is seen that the magnetic entropy behaves in a di ff erent way whenone passes from the frozen regime ( p < p c ) to the chaotic regime ( p > p c ). For a further analysis, the robustness of thepropagation of failures is checked by introducing a quenched site dilution probability q on the lattices. It is seen thatthe damage spreading is quite sensitive when a small fraction of the lattice sites are disconnected. A finite-size scalinganalysis is employed to estimate the critical exponents. From these numerical estimates, we claim that on both pure( q =
0) and diluted ( q = .
05) quasiperiodic lattices, the KCA model belongs to the same universality class than onsquare lattices. Furthermore, with the aim of comparing the dynamical behavior between periodic and quasiperiodicsystems, the propagation speed of the damage is also calculated for the square lattice assuming the same conditions.It is found that on square lattices the propagation speed of the damage obeys a power law as v ∼ ( p − p c ) α , whereason quasiperiod lattices it follows a logarithmic law as v ∼ ln( p − p c ) α . Keywords:
Kau ff man cellular automata, quasiperiodic lattice, critical exponents, magnetic entropy, propagation speed of thedamage
1. Introduction
Kau ff man cellular automata (KCA) [1] or more gen-erally random Boolean networks have been studied inthe past to describe genetic regulatory networks but dueto their general features since they do not assume anyparticular function of the nodes these can also be usedto study a large variety of important issues concerningsychronization [2], stability [3], robustness [4] and con-trol of chaos [5], just to mention a few examples. In par-ticular, we are interested in studying how small failures(damages) produced on complex structures of automatapropagate throughout the entire system. These failuresmay stand for genetic mutations, fractures, infectiousdisease spreading and virus propagation on computernetworks. Earlier studies related to failure propagationon complex structures of automata have focused either ∗ Corresponding author
Email addresses: [email protected] (Carlos HandreyAraujo Ferraz), [email protected] (Jos´e Luiz Sousa Lima) on periodic lattices [6–9], which present short-rangeinteractions, or on random graphs [10, 11] in whichlong-range interactions take place. However, there havebeen few studies addressing the dynamics of the failurepropagation in systems that possess short-range inter-actions with breaking of translational symmetry. Thelack of this symmetry could influence the propagationof the damage cloud as well as change both the crit-ical threshold parameter p c and the universality classof these systems. The quasicrystals’ topology is quitesuitable for such a study since it does not present nei-ther periodic translational nor close orientational or-der. In fact, such systems can exhibit rotational sym-metries otherwise forbidden to crystals [12]. Further-more, given the lack of periodicity of these systems,only numerical approaches can be performed. Under apoint of view purely geometrical, quasicrytals can bethought as quasiperiodic lattices. Recently, a montecarlo study [13] has confirmed that both the periodic lat-tices and the quasiperiodic lattices belong to the sameuniversality class [14], despite the critical temperature Preprint submitted to Brazilian Journal of Physics June 28, 2018 igure 1: Quasiperiodic lattice generated by the strip projectionmethod [13]. The lattice is shown inside a square projection win-dow. The periodic boundary conditions are imposed at the lattice sitescloser to the projection window. of these lattices being di ff erent. It is the purpose of thispaper to present some numerical results concerning bothcriticality and the dynamics of the failure propagationon such quasiperiodic lattices and thereby understand-ing how this topology can influence the time evolutionof these systems.The phase transition of the KCA model is defined bycalculating the Hamming distance (see section 3) be-tween almost identical lattices (only a small number ofsites have di ff erent states). We denote this Hammingdistance as the damage; if it remains localized or even-tually vanishes, then one says that the system is in afrozen phase. Otherwise, if the damage spreads out overa considerable part of the system, then one says that thesystem is in a chaotic phase. Usually, the control pa-rameter of the model [7, 11, 15] is the probability p forthe boolean function, which rules the time evolution ofa given site, yields as output the value 1. FollowingRef. [5], here ‘chaos’ is not the usual low-dimensionaldeterministic chaos but a phase where damage spread-ing takes place.In this paper we perform numerical simulations tostudy KCA on quasiperiodic lattices. In particular, weinvestigate phase transition, magnetic entropy and prop-agation speed of the damage [15] on these lattices. Boththe critical threshold parameter p c and the critical expo-nents were estimated for di ff erent treated cases.In order to investigate the increase of the statisticalfluctuations and the onset of chaos in the critical regionof the model, we have also defined a magnetic entropyto these systems. It is seen that the magnetic entropybehaves in a di ff erent way when the system changesfrom the frozen regime ( p < p c ) to the chaotic regime ( p > p c ). For a further analysis, the robustness of thepropagation of failures [15] was checked by introduc-ing a quenched site dilution probability q on the lattices.It is found that the damage spreading is quite sensitivewhen a small fraction of sites are disconnected fromthese lattices. The quasiperiodic lattices analyzed herewere generated using the strip projection method [13]with each automaton placed in the vertices of the rhombithat make up the lattice (Fig. 1). For this type of lat-tice, the number of nearest neighbors at a given site canvary from K = K =
10 with a mean coordina-tion number equal to < z > = .
98. A finite-size scalinganalysis was used to estimate the critical exponents andperiodic boundary conditions were imposed on the gen-erated lattices in order to reduce finite-size e ff ect in thesimulations. We calculate several quantities includingthe order parameter, logarithmic derivative of the orderparameter, propagation speed of the damage and mag-netic entropy. Moreover, with the aim of comparing thedynamical behavior between periodic and quasiperiodicsystems, the propagation speed of the damage was alsocalculated for square lattices assuming the same condi-tions.This paper is organized as follows. In section 2, wegive a brief review about the Kau ff man model. Next,we describe the computational procedure used to im-plement the model on quasiperiodic lattices as well as toaccomplish a site dilution on these lattices. After that,in section 4, we define the absolute magnetic entropyon KCA. In section 5, we present our results concern-ing the phase transition, entropy and speed propagationof the damage. In section 6, we conclude by summariz-ing the main results and providing recommendations forfurther research.
2. The Kau ff man Model Kau ff man originally introduced networks of Booleanautomata in order to study the behaviour of generic reg-ulatory systems. The basic idea of the Kau ff man modelis to consider a mixture of all possible binary cellularautomata. The Kau ff man model can be realized on alattice, by choosing Boolean rules individually for eachsite. Each of N lattice sites hosts a Boolean variable σ i (spin up or down) which is either zero or unity. Thetime evolution of this model is determined by N func-tions f i (rules) which are randomly chosen for each siteindependently, and by the choice of K input sites { j K ( i ) } for each site i . Thus the value σ i at site i for time t + σ i ( t + = f i ( σ j ( t ) , . . . , σ j K ( t )) i = , , . . . , N . (1)2ach Boolean function f i is specified, once a value isgiven for each one of the 2 K possible neighbour config-urations. A variable σ i is called relevant for the spread-ing damage process if it is unstable i.e. the state of othervariables { σ j } depend on σ i . If one imposes that the in-puts and the chosen Boolean functions do not changewith time, we have the quenched Kau ff man model. Onthe other hand, if one admits that both change with time,we have the annealed Kau ff man model. A big di ff er-ence between the two cases is that in the quenched casethere are limit cycles and in the annealed case not. Herewe consider only the quenched case. In this case, asthe time development is totally deterministic, and since N di ff erent Boolean variables can produce 2 N di ff erentlattice configurations, we must return after at most 2 N time-steps to the previous initial configuration. Thenthe system will repeat the same configurations, stayingwithin this limit cycle. For the nearest-neighbour Kau ff -man model on the square lattice, the number of relevantlimit cycles increases exponentially with system size inthe non-chaotic phase [16]. Kau ff man identified thesedi ff erent limit cycles with the di ff erent cell types in ourbody and found that their number grows as √ N for N interacting genes. The annealed case can be solved ana-lytically, whereas for the quenched case only computersimulations were performed up to now.
3. Computational Procedure
A standard way to implement the Kau ff man modelis introducing a parameter p such that for each site onthe lattice we select among the 2 K rules one which foreach outcome will have spin up with probability p . In acomputer simulation, first one goes through all N sitesof the system, and for each site one goes through all2 K neighbour configurations, and for each such config-uration one determines by drawing a random numberif its spin will be up or down; if the random numberis smaller than p then its spin will be up, otherwise itwill be down. Once one has gone through all neighbourconfigurations of that site, then one has fixed the rulefor that site, and one can go to the next site. After thatone selects an initial configuration of the Boolean vari-ables by randomly assigning to each lattice site a spinup or down with equal probability. We will considertwo systems (replicas), identical in the connections andrules, and also identical in the initial configuration ofthe Boolean variables, except that on one of them weflip the most central sites of the lattice (around 0.5% ofthe lattice sites) at every time-step along the simulation.The number of spins which at time t is di ff erent between the two replicas is called the Hamming distance d ( t ) orsimply the failure. For two lattice configurations { σ i ( t ) } and { ρ i ( t ) } , we have d ( t ) = N X i | σ i ( t ) − ρ i ( t ) | , (2)and we can define an order parameter ψ for the systemtaking in Eq. 2 the limit t → ∞ , namely Ψ = lim d (0) → d ( ∞ ) . (3)Computationally, convergence is typically reached aftera few thousand time steps. In this way we can studythe phase transition, entropy and the propagation speedof the damage cloud varying the value p . It has beenobserved that Ψ goes to zero at the critical threshold pa-rameter p c in systems with dimensions greater than onefor the short-range case of the Kau ff man model, in asimilar way to the para-ferromagnetic phase transition.In other words, for all p p c , a small initial damagevanishes or remains small, i.e. it belongs to a smallcluster of ‘damaged spins’ after a su ffi ciently long time.One says that the system is in the frozen phase. On theother hand, for all p > p c , a small initial damage spreadsthroughout a considerable part of the system. Then onesays that the system is in the chaotic phase. Of partic-ular interest is, however, the border case p = p c wherefractal properties appear [17]. Obviously, p and 1 − p are statistically equivalent, so that we do not consider p > . q on the lattices. Thus weconsider the Kau ff man model on quenched site-dilutedquasiperiodic lattices. The dilution procedure is as fol-lows: at the beginning of each simulation run, we gener-ate a new configuration lattice starting from a pure lat-tice ( q =
0) by disconnecting each lattice site with aprobability q ,
0. After this procedure, we are left witha new lattice with a density 1 − q of linked sites.In the transient regime, the propagation speed v re-quired for the damage to spread throughout the entiresystem was calculated by measuring the time it takesto touch the lattice boundaries. We perform severalcalculations of the propagation speed of the damagefor both square and quasiperiodic lattices assuming thesame conditions, it means that the most central siteswere flipped at every iteration along the simulation (i.e.,persistently disturbed sites). We wait up to 10 latticesweeps so that the damage cloud could reach the latticeboundaries. Only succeeded runs in which the damage3loud reached the lattice boundaries were considered inthe averages. We average over up to 600 independentruns. Di ff erent lattice sizes were considered for eachvalue of p . Besides, an extrapolation technique was alsoused to take into account the thermodynamic limit. Thisextrapolation was achieved by analyzing how the prop-agation speed of the damage v depends on the recipro-cal of the lattice size (1 / N ) when one takes the limit N → ∞ .
4. Magnetic Entropy
A ‘magnetic’ bath can be associated to the system,where the parameter p will be the intensive thermody-namic variable in this case. Defining the hamiltonian ofthe system as H = − J N X i = δ ( σ i ⊕ ρ i , − δ ( σ i ⊕ ρ i , , (4)where J is the energetic coupling constant, δ is the Kro-necker delta function, ⊕ is the modulo-2 addition of σ i and ρ i (spin variables of the i th site in interacting repli-cas). From the fluctuation-dissipation theorem, we cancalculate the energy dispersion per spin as Ω ( p ) = K N ( < E > − < E > ) , (5)where K = J / p is the reciprocal of the parameter p (herewe assume J = E = U / N is the energy per spin (seeEq. 4) and < ∗ > denotes an average value.Based on statistical mechanics, one can define a en-ergy function associated to the empirical probability of agiven configuration µ ≡ { σ i ⊕ ρ i } (again ⊕ is the modulo-2 addition) of the system [18]: U µ = − ln( P µ ) , (6)The most probable states have low energy, while theless probable states correspond to high energies. On theother hand, the probability of the system to be in a cer-tain configuration of its microscopic states is computedfrom the energy function. So, we can write the canoni-cal partition function Z ( p ) as Z ( p ) = X µ exp ( − p U µ ) , (7)where p would be to the temperature in a thermody-namic system, here it is our ‘magnetic’ parameter. Us-ing Eqs. 6, we can rewrite Eq.7 as Z ( p ) = X µ P / p µ , (8) which allows to define a p -dependent probability distri-bution P { p } ( µ ) as P { p } ( µ ) = Z ( p ) P / p µ . (9)Once the canonical partition function Z ( p ) is defined,the magnetic entropy S ( p ) [18–20] and the energy dis-persion Ω ( p ) [18] can be respectively derived from it,i.e., S ( p ) = − X µ P { p } ( µ ) log( P { p } ( µ )) , (10)and Ω = p dSd p . (11)By integrating the above relation, one can obtain theentropy per spin at the parameter p as S ( p ) = S ( p ) + p Z p Ω p d p , (12)and next assuming the limit lim p → S ( p ) =
0, we finallyhave the absolute entropy at p : S ( p ) = p Z Ω p d p . (13)Eq. 13 above was evaluated here through numerical in-tegration [21].
5. Numerical Results
In order to determine the critical threshold parameter p c for the case q =
0, we calculate the order parameter Ψ in a wide range of values of p . Fig. 2 shows the orderparameter as a function of p for three di ff erent latticesizes ( N = , ff erent runs with an overall time of5 . × time-steps for the system achieves its asymp-totic regime. Looking at the inflection point of thosecurves, it can be noticed a typical second-order phasetransition around p c = . q , Ψ against p . Fig. 3 displays the phase transition on a latticeof 65391 sites for three di ff erent value of the site dilu-tion rate q . The damage spreading on such quasiperi-odic lattices is quite sensitive to the removal of active4 .15 0.20 0.25 0.30 0.35 0.40 0.450.00.10.20.30.40.5 N=10445 N=23543
N=65391 () ( p ) q=0 Figure 2: Plot of the order parameter Ψ as a function of p for threedi ff erent lattice sizes for the case q =
0. One can notice a typicalsecond-order phase transition around p c = . q=0 q=0.05 q=0.10 () ( p ) Figure 3: Phase transition on a quasiperiodic lattice of size N = ff erent value of the site dilution rate q . For q = .
10 thereis not a phase transition anymore. sites. That is likely due to the existence of unstable re-gions (unstable cores) [23]. From Fig. 3, we see thateven taking a small dilution rate q = .
05, the asymp-totic damage mass is quite diminished and the value ofthe critical point on the lattice is shifted to p c = . q = .
10 and above we nolonger observe a chaotic phase.The statistical fluctuations play an important rolefor understanding the phase transition exhibited in thismodel. Figs. 4 and 5 show the energy dispersion perspin Ω as a function of p (Eq. 5) for three di ff erent lat-tice sizes, respectively, for the case q = q = . N=10445 N=23543
N=65391 q=0 ( p ) ( p ) Figure 4: Plot of the energy dispersion per spin Ω as a function of p for three di ff erent lattice sizes for the pure case q =
0. An abruptincrease of the statical fluctuations can be seen around p c = . q=0.05 N=10445 N=23543
N=65391 ( p ) ( p ) Figure 5: Plot of the energy dispersion per spin Ω as a function of p for the diluted case q = .
05. An abrupt increase of the staticalfluctuations can be seen around p c = . crease of the statical fluctuations on the energy values asdefined by Eq. 4 around their according critical points.However, it is interesting to observe that the maximumvalue of the dispersion is situated a little above p c . Fur-ther, the energy dispersion diminishes in a di ff erent waywhen one moves away from the critical region towardsthe frozen region ( p < p c ) than towards the chaotic re-gion ( p > p c ). To better characterize this asymmetricbehaviour, we calculate the magnetic entropy S (Eq. 13)by integrating the curves from Figs. 4 and 5. The abso-lute magnetic entropy S as a function of the parameter p considering a system of size N = .10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.5002468 E n t r op y ( S ) (p) q =0 q =0.05 (A=9.94, k=7.24, p =0.26) (A=8.58, k=6.29, p =0.28) Figure 6: The absolute magnetic entropy S as a function of the pa-rameter p considering a system with N = q = q = .
05 cases. In the frozen phase, the entropy exhibitsa clear linear dependence on p while in the chaotic phase it increasesnon-linearly as p increases. Lines are the best non-linear fits of theform S = A (1 − exp( − k ( p − p )) to the data points above p c . -3 -2 -1 0 1 2-1.2-1.0-0.8-0.6-0.4-0.20.00.20.4 N= 10445 N= 23543 N= 65391 l n ( L ) ln( p-p c ) L ) Figure 7: Log-log plot of Ψ L β/ν versus ( p − p c ) L /ν for the case q = Fig. 6 for both cases q = q = .
05. In the frozenphase, the entropy exhibits a clear linear dependence on p while in the chaotic phase it seems to increase in non-linear way as p increases. A non-linear curve fitting tothe data points above p c was performed for both cases q = q = .
05. It is found that the best fits to thedata are obtained by using the so-called monomolecularmodel, i.e., S = A (1 − exp( − k ( p − p )) , (14)where the fit parameters A , k and p for both cases aregiven in Fig. 6. Such a change in the entropy behaviour l n () ln(L) q=0 Figure 8: Log-log plot of Ψ ∗ versus the linear size L of the system forthe case q =
0. The red straight line is the best linear fit to the data( χ r = .
14 and a goodness-of-fit probability Q ( χ r ) = . l n () ln(L) q=0 Figure 9: Log-log plot of φ (calculated at p c = . L of the system for the case q =
0. The red straight line is thebest linear fit to the data ( χ r = .
08 with a goodness-of-fit probability Q ( χ r ) = . is useful to characterize both phases with respect to thedisorder degree present in these systems.After the critical region and the critical parameter p c have been determined for each case and knowing thatthe final damage mass vanishes as ( p − p c ) β inside thatregion, we can estimate the critical exponents for bothcases q = q = .
05 by making a collapse of thedata of Ψ through the scaling law: ψ ( L , p ) = L − β/ν F (( p − p c ) L /ν ) , (15)where L = √ N is the linear dimension of the lattice and ν is the exponent describing the divergence of the corre-6 .30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.460.51.01.52.02.53.03.54.0 N=153 N=255 N=300 N=400 N P r op a g a t i on s p ee d () (p) Figure 10: Propagation speed of the damage ( v ) on square lattices, forseveral lattice sizes along with an extrapolation of these data ( N →∞ ). N=10445 N=23543 N=65391 N=89223 N P r op a g a t i on s p ee d () (p) Figure 11: Propagation speed of the damage ( v ) on pure quasiperiodiclattices (case q = N → ∞ ). lation length at p c . Fig. 7 shows the best data collapsefor the case q = p c = . β = . ± .
05 and ν = . ± . Ψ ∗ at p c . Fig. 8 shows the log-log plot of Ψ ∗ versus thelinear size L of the system. The slope of the linear fit tothe data of Ψ ∗ is β/ν = . ± .
01. In this linear fit, thereduced chi-square χ r was 1 .
14 with a goodness-of-fitprobability Q ( χ r ) equals 33 . χ r would exceed the observed value, assumingthat the underlying statistical model is correct. A typical -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5-0.8-0.40.00.40.81.21.6 l n () ln ( p-pc ) Figure 12: Log-log plot of v versus ( p − p c ) for the extrapolated datafrom Fig. 10. The red straight line is the best linear fit to Eq. 17 onlog-log scale. -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.00.000.050.100.150.200.250.300.35 () ln ( p-pc ) Figure 13: Semi-log plot of v versus ( p − p c ) for the extrapolated datafrom Fig. 11. The red straight line is the best linear fit to Eq. 18. confidence level is in accepting fits with Q ( χ r ) > β/ν yields a valueof fractal dimension D = d − βν ≈ .
78 (where d = ν we considered the power-law dependence of the loga-rithmic derivative of the order parameter ( φ ) on the sys-tem size expressed by φ = d ln( Ψ ) d p ∝ L /ν . (16)Fig. 9 shows the log-log plot of φ (calculated at p c ) ver-sus the linear size L of the system. The slope of the lin-7ar fit to the data of φ is 1 /ν = . ± .
03. The reducedchi-square χ r was 2 .
08 with a goodness-of-fit probabil-ity Q ( χ r ) equals 10 . ν = . ± . β/ν and 1 /ν , we find that β = . ± . β obtained via collapse of the data.A similar analysis was also performed for the dilutedcase ( q = . β = . ± . ν = . ± .
05. Therefore, based on the criticalexponents hereby estimated we claim that on both pure( q =
0) and diluted quasiperiodic lattices ( q = . v as a function of p , respectively, on square andon quasiperiodic lattices. In these figures, several latticesizes are shown together with an extrapolation of thesedata for each value of p considered. As previously ex-plained in the section 3, this extrapolation was achievedby analyzing how v depends on the reciprocal of the lat-tice size (1 / N ) when one takes the limit N → ∞ . Fig. 12shows the log-log plot of v versus ( p − p c ) for the ex-trapolated data from Fig. 10, while Fig. 13 shows thesemi-log plot of v versus ( p − p c ) for the extrapolateddata from Fig. 11. Thus, we can observe that the av-erage propagation speed v on square lattices follows apower law as: v = v S ( p − p c ) α , (17)where v S is a constant term and α ≈ .
67 is the criti-cal exponent of the speed for the square lattice. Whilethe average propagation speed v on quasiperiodic lat-tices follows a logarithmic law as: v = v Q + ln( p − p c ) α , (18)where v Q is a constant term and α ≈ .
08 is the criti-cal exponent of the speed for the quasiperiodic lattice.Therefore, our results lead us to conclude that quasiperi-odic lattices are topologically more resistant than peri-odic lattices with respect to the propagation of failuresgenerated by persistently disturbed sites.
6. Summary and Conclusion
In summary, we have employed the KCA model tostudy the breaking e ff ects of the periodic translational symmetry on both the phase transition and the prop-agation dynamics of failures in quasiperiodic systems.These failures may mimic fractures, infectious diseasespreading and infection by computer virus on latticeswhich possess short-range interactions but lacking a pe-riodic translational symmetry as seen in quasicrystals.Concerning the critical properties of the model, wehave employed a finite-size scaling analysis to estimatethe critical threshold parameter p c and its critical ex-ponents β/ν and 1 /ν on both pure ( q =
0) and dilutedquasiperiodic lattices ( q = . ff erent critical threshold p c . We havealso defined an absolute magnetic entropy in order bet-ter to characterize the onset of chaos in these systems. Itwas seen that the value of the magnetic entropy linearlyincreases as the value of p increases in the non-chaoticregion; while in the chaotic region, it increases in a non-linear way as p increases.As for the robustness of the damage spreading, it wasobserved that such systems are quite sensitive when asmall fraction of sites are disconnected from these lat-tices. In particular, for dilution rates of q ≥ .
10, wedid not find a chaotic phase anymore. In addition, thepropagation dynamics of the damage was investigatedby performing calculations of the propagation speed v asa function of p on both square (periodic) and quasiperi-odic lattices. By using a data extrapolation procedurewas found that the propagation speed of the damage onsquare lattices obeys a power law as v ∼ ( p − p c ) α ,whereas on quasiperiodic lattices it follows a logarith-mic law as v ∼ ln( p − p c ) α . For the square lattice, theestimated critical exponent of the speed ( α ) was 0 . .
08. Therefore,we can conclude that quasiperiodic lattices are topolog-ically more resistant than periodic lattices with respectto the propagation of failures when these are generatedby persistently disturbed sites. Future work will con-cern numerical studies on 3 D quasiperiodic lattices sothat the more realistic structures found in alloys such as Al - Fe and Al - Mn can also be better investigated.
7. Acknowledgements
We wish to thank UFERSA for computational sup-port.
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