Sierpinski Gasket Research Articles

An IFS for the Stretched Level-n Sierpinski Gasket

Broadly, fractals are sets that exhibit a repeating pattern at multiple scales. One important fractal set is the Sierpinski Gasket (SG) which is made up of nested equilateral triangles. A variation of the classic Sierpinski gasket is to create n-levels with the equilateral triangles. Another variation is to stretch the points of intersection for the triangles in SG into line segments of length 0 < α < 1/3. When one combines these variations, one arrives at the stretched level-n Sierpinski gaskets (SSGn) which are the focus of this work. We give an introduction to iterated function systems (IFS) and determine an IFS which generates SSG3. We then describe how one can acquire the IFS for SSGn in general, and conclude with a theorem which determines the Hausdorff dimension for SSGn.

Stochastic modelling of fractal diffusion and dimension estimation

The revision of classical methods for spectral and walk dimension estimates is the main aim of the paper. Being focused on the unbiased estimation of the walk and spectral dimensions, we aim to construct the estimates with the minimal mean square error. Accompanied simulation experiments are performed on finite substrates, spacial structures serving as a good model of both continuum and fractal sets. We compare the classical approach based on the log–log transform of asymptotic models of returning probabilities and the second moments, and we also develop a weighted approach to improve the statistical properties of dimension estimates. The other discussed aspect is whether to simulate diffusion using the classical graph diffusion model with zero probability of staying in the same vertex or to prefer the physically motivated model of diffusion over edges with the optimal value of jump probability. Finally, we present the results of simulation experiments on two-dimensional finite substrates which approximate the continuum and selected Sierpinski gaskets and carpets. The paper also summarises general suggestions based on the obtained results from the simulation experiments.

Spin-dependent transport in a driven non-collinear antiferromagnetic fractal network

Non-collinear magnetic texture breaks the spin-sublattice symmetry which gives rise to a spin-splitting effect. Inspired by this, we study the spin-dependent transport properties in a non-collinear antiferromagnetic fractal structure, namely, the Sierpinski Gasket (SPG) triangle. We find that though the spin-up and spin-down currents are different, the degree of spin polarization is too weak. Finally, we come up with a proposal, where the degree of spin polarization can be enhanced significantly in the presence of a time-periodic driving field. Such a prescription of getting spin-filtering effect from an unpolarized source in a fractal network is completely new to the best of our knowledge. Starting from a higher generation of SPG to smaller ones, the precise dependencies of driving field parameters, spin-dependent scattering strength, interface sensitivity on spin polarization are critically investigated. The spatial distribution of spin-resolved bond current density is also explored. Interestingly, our proposed setup exhibits finite spin polarization for different spin-quantization axes. Arbitrarily polarized light is considered and its effect is incorporated through Floquet–Bloch ansatz. All the spin-resolved transport quantities are computed using Green’s function formalism following the Landauer–Büttiker prescription. In light of the experimental feasibility of such fractal structures and manipulation of magnetic textures, the present work brings forth new insights into spintronic properties of non-collinear antiferromagnetic SPG. This should also entice the AFM spintronic community to explore other fractal structures with the possibility of unconventional features.

A noncommutative Sierpinski gasket

A quantized version of the Sierpinski gasket is proposed, on purely topological grounds, as a C⁎-algebra A∞ with a suitable form of self-similarity. Several properties of A∞ are studied, in particular its nuclearity, the structure of ideals as well as the description of irreducible representations and extremal traces. A harmonic structure is introduced, giving rise to a self-similar Dirichlet form E. A spectral triple is also constructed, extending the one already known for the classical gasket, from which E can be reconstructed. Moreover we show that A∞ is a compact quantum metric space.

Precessional dynamics of geometrically scaled magnetostatic spin waves in two-dimensional magnonic fractals

The control of spin waves in periodic magnetic structures has facilitated the realization of many functional magnonic devices, such as band stop filters and magnonic transistors, where the geometry of the crystal structure plays an important role. Here, we report on the magnetostatic mode formation in an artificial magnetic structure, going beyond the crystal geometry to a fractal structure, where the mode formation is related to the geometric scaling of the fractal structure. Specifically, the precessional dynamics was measured in samples with structures going from simple geometric structures toward a Sierpinski carpet and a Sierpinski triangle. The experimentally observed evolution of the precessional motion could be linked to the progression in the geometric structures that results in a modification of the demagnetizing field. Furthermore, we have found sets of modes at the ferromagnetic resonance frequency that form a scaled spatial distribution following the geometric scaling. Based on this, we have determined the two conditions for such mode formation to occur. One condition is that the associated magnetic boundaries must scale accordingly, and the other condition is that the region where the mode occurs must not coincide with the regions for the edge modes. This established relationship between the fractal geometry and the mode formation in magnetic fractals provides guiding principles for their use in magnonics applications.

Squeeze: Efficient compact fractals for tensor core GPUs

This work presents Squeeze, an efficient compact fractal processing scheme for tensor core GPUs. By combining discrete-space transformations between compact and expanded forms, one can do data-parallel computation on a fractal with neighborhood access without needing to expand the fractal in memory. The space transformations are formulated as two GPU tensor-core accelerated thread maps, λ(ω) and ν(ω), which act as compact-to-expanded and expanded-to-compact space functions, respectively. The cost of the maps is O(log2logs(n)) time, with n being the side of a n×n embedding for the fractal in its expanded form, and s the linear scaling factor. The proposed approach works for any fractal that belongs to the Non-overlapping-Bounding-Boxes (NBB) class of discrete fractals, and can be extended to three dimensions as well. Experimental results using a discrete Sierpinski Triangle as a case study shows up to ∼12× of speedup and a memory reduction factor of up to ∼315× with respect to a GPU-based expanded-space bounding box approach. These results show that the proposed compact approach will allow the scientific community to efficiently tackle problems that up to now could not fit into GPU memory.

Quantum transport and microwave scattering on fractal lattices

Abstract Studying the wave-particle nature of electrons in different ways has lead to many fundamental discoveries. Particularly, the dimensionality dependent electronic behavior in the Luttinger Liquid (1D), Quantum Hall (2D) and non-interacting Fermi Liquid (3D) regimes have already revolutionized our understanding of the mechanisms behind quantum electronics. In this work, the theoretical and experimental studies focus on the non-integer dimension represented by an sp2-carbon-based Sierpinski triangular structure with a 1.58D space occupancy. In the tight-binding approach, the spectral distribution of electronic states of such a structure exhibits distinct peak patterns, which are well-separated by gaps. Through quantum transport simulation, the conductance of electrons in 1.58D was studied. Both delocalized, conducting and localized, non-conducting states identified, which differ from the established features of both the fully 2D graphene sheet and 1D carbon nanotubes. In microwave scattering measurements on an adequate experimental setting and the respective simulations on the Sierpinski triangle, the obtained diffraction patterns showed interesting peculiarities such as a reduced number of minima and magic angle, next to diffraction regions of high and low intensity, as well as forbidden regions. The fractal geometry of the structure affects the propagation of waves by manipulating the way they interact with each other which results in structural metamaterial-like interference characteristics, decreasing or amplifying the transmitted or reflected signals, or blocking the transport completely.

DIRICHLET PROBLEM OF POISSON EQUATIONS ON A TYPE OF HIGHER-DIMENSIONAL FRACTAL SETS

Poisson equation is a partial differential equation with broad utility in theoretical physics. Dirichlet problem of Poisson Equations can be solved by Green’s function. It is a very attractive problem to look for analogous results of the above problem in the fractal context. This paper studies the network set Higher-Dimensional Sierpinski Gaskets, solves Dirichlet problem of Poisson Equations on them by expressing Green’s function explicitly.

Pascal's Triangle Fractal Symmetries.

We introduce a model of interacting bosons exhibiting an infinite collection of fractal symmetries-termed "Pascal's triangle symmetries"-which provides a natural U(1) generalization of a spin-(1/2) system with Sierpinski triangle fractal symmetries introduced in Newman etal.,[Phys. Rev. E 60, 5068 (1999).PLEEE81063-651X10.1103/PhysRevE.60.5068]. The Pascal's triangle symmetry gives rise to exact degeneracies, as well as a manifold of low-energy states which are absent in the Sierpinski triangle model. Breaking the U(1) symmetry of this model to Z_{p}, with prime integer p, yields a lattice model with a unique fractal symmetry which is generated by an operator supported on a fractal subsystem with Hausdorff dimension d_{H}=ln(p(p+1)/2)/lnp. The Hausdorff dimension of the fractal can be probed through correlation functions at finite temperature. The phase diagram of these models at zero temperature in the presence of quantum fluctuations, as well as the potential physical construction of the U(1) model, is discussed.

Average trapping time on a type of horizontally segmented three dimensional Sierpinski gasket network with two types of locally self-similar structures

As a classic self-similar network model, Sierpinski gasket network has been used many times to study the characteristics of self-similar structure and its influence on the dynamic properties of the network. However, the network models studied in these problems only contain a single self-similar structure, which is inconsistent with the structural characteristics of the actual network models. In this paper, a type of horizontally segmented three dimensional Sierpinski gasket network is constructed, whose main feature is that it contains the locally self-similar structures of the two dimensional Sierpinski gasket network and the three dimensional Sierpinski gasket network at the same time, and the scale transformation between the two kinds of self-similar structures can be controlled by adjusting the crosscutting coefficient. The analytical expression of the average trapping time of a random walker by a particular site on the network model is solved, which used to analyze the effect of two types of self-similar structures on the properties of random walks. Finally, we conclude that the dominant self-similar structure will exert a greater influence on the random walk process on the network.

Laughlin topology on fractal lattices without area law entanglement

Laughlin states have recently been constructed on fractal lattices, and the charge and braiding statistics of the quasiholes were used to confirm that these states have Laughlin type topology. Here, we investigate density, correlation, and entanglement properties of the states on a fractal lattice derived from a Sierpinski triangle with the purpose of identifying similarities and differences compared to two-dimensional systems and with the purpose of investigating whether various probes of topology work for fractal lattices. Similarly to two-dimensional systems, we find that the connected particle-particle correlation function decays roughly exponentially with the distance between the lattice sites measured in the two-dimensional plane, but the values also depend on the local environment. Contrary to two-dimensional systems, we find that the entanglement entropy does not follow the area law if one defines the area to be the number of nearest neighbor bonds that cross the edge of the selected subsystem. Considering bipartitions with two bonds crossing the edge, we find a close to logarithmic scaling of the entanglement entropy with the number of sites in the subsystem. This also means that the topological entanglement entropy cannot be extracted using the Kitaev-Preskill or the Levin-Wen methods. Studying the entanglement spectrum for different bipartitions, we find that the number of states below the entanglement gap is robust and the same as for Laughlin states on two-dimensional lattices.

A copper pyramidal fractal antenna fabricated with green-laser powder bed fusion

Recent advances in additive manufacturing have enabled a new generation of electromagnetic applications to flourish. Complex geometries for dielectrics and conductors can now be simulated and rapidly fabricated from digital data. Powder bed fusion of metals is arguably the most widely adopted additive process by industry and can provide intricately-detailed structures in a wide range of high performance alloys. Copper and copper alloys have remained a challenge in this additive process, as the typical laser wavelength (approximately 1070 nm) used fails to provide sufficient absorption. Moreover, the high thermal conductivity of copper does not allow for the required heat generation for a stable melt pool. However, the recent commercial introduction of the green laser (515 nm wavelength) is enabling the printing of copper, which is particularly interesting for electrical and electromagnetic applications due to the high electrical conductivity and solderability. This paper describes the use of a green laser powder bed fusion system used to fabricate a complex fractal Sierpinski gasket ground structure with an isolated internal pyramid antenna built simultaneously—within and dielectrically isolated from the external ground element: a ship-in-the-bottle design paradigm. The electromagnetic performance, surface finish, dimensional compliance, and conductivity were measured and reported to inform the design of freestanding, geometrically-complex antennas.

Modelling of Electron and Thermal Transport in Quasi-Fractal Carbon Nitride Nanoribbons

In this work, using calculations based on the density functional theory, molecular dynamics, non-equilibrium Green functions method, and Monte Carlo simulation, we study electronic and phonon transport in a device based on quasi-fractal carbon nitride nanoribbons with Sierpinski triangle blocks. Modifications of electronic and thermal conductance with increase in generation g of quasi-fractal segments are estimated. Introducing energetic disorder, we study hopping electron transport in the quasi-fractal nanoribbons by Monte Carlo simulation of a biased random walk with generalized Miller–Abrahams transfer rates. Calculated time dependencies of the mean square displacement bear evidence of transient anomalous diffusion. Variations of anomalous drift-diffusion parameters with localization radius, temperature, electric field intensity, and energy disorder level are estimated. The hopping in quasi-fractal nanoribbons can serve as an explicit physical implementation of the generalized comb model.

THE AVERAGE TRAPPING TIME WITH NON-NEAREST-NEIGHBOR JUMPS ON THE LEVEL-3 SIERPINSKI GASKET

In this paper, we consider the trapping problem on the level-3 Sierpinski gasket [Formula: see text] when both nearest-neighbor (NN) and non-nearest-neighbor (NNN) jumps are included. Based on the topological structure of network and the method of probability generation function, we get the analytical expression of the average trapping time (ATT). Therefore, compared with the case where only NN jumps are allowed, we modify the specific value of ATT. Our results prove the exponent of the scaling expression has nothing to do with the NNN jump probability [Formula: see text], i.e. the scaling expression of ATT still scales superlinearly with the large network size. According to the analytical expression, we do numerical simulations of ATT with respect to parameters [Formula: see text] and [Formula: see text] (the number of generations of the network). The results show that the ATT will decrease with the increase of [Formula: see text]. These also indicate that NNN jump is helpful to improve the efficiency of random walk on network [Formula: see text].

Sierpinski-Takagi combination for a uniform and optimal point-surface load transmission

The application of large loads on poor soils requires designs that optimize the transmission of a point load to a surface and distribute it uniformly throughout the structure. In this paper, this optimal transmission is achieved by using two fractals: a Sierpinski triangle that reproduces the mechanical structure and the Takagi function that determines the vertical deformation of its supports. This result is obtained by applying the Principle of Virtual Work to a finite approximation of the Sierpinski triangle and its supports and the subsequent infinite extension. The extension by continuity ensures that the load is uniform on the entire base and not only on the discrete supports.

Asymptotic analysis of an elastic material reinforced with thin fractal strips

<p style='text-indent:20px;'>We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.</p>

The localization of quantum random walks on Sierpinski gaskets

We consider discrete time quantum random walks on a Sierpinski gasket. We study the hitting probability as the level of the fractal graph goes to infinity in terms of their localization exponents βw, total variation exponents δw, and relative entropy exponents ηw. We define and solve the amplitude Green functions recursively when the level of the fractal graph goes to infinity. We obtain exact recursive formulas for the amplitude Green functions, on which the hitting probabilities and expectation of the first-passage time are calculated, and using the recursive formula with the aid of Monte Carlo integration, we evaluate their numerical values. We also show that when the level of the fractal graph goes to infinity, with probability 1, the quantum random walks will return to the origin, i.e., the quantum walks on a Sierpinski gasket are recurrent.

Electronic fractal patterns in building Sierpinski-triangle molecular systems

Fractal features in the electronic properties of Sierpinski triangle flakes and molecular chains are explored simulating experimental synthesized fractal nanostructures.

Spectral triples on irreversible C∗-dynamical systems

Given a spectral triple on a [Formula: see text]-algebra [Formula: see text] together with a unital injective endomorphism [Formula: see text], the problem of defining a suitable crossed product [Formula: see text]-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of [Formula: see text] in [Formula: see text] can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on [Formula: see text] and [Formula: see text].

Mean geodesic distance of the level-n Sierpinski gasket

In this paper, we obtain the mean geodesic distance of the level-n Sierpinski gasket by investigating the integral of geodesic distance with respect to self-similar measure by means of the finite pattern phenomenon of integrals.