Sierpinski Curve Research Articles

Online Evasive Strategy for Aerial Survey using Sierpinski curve

This paper deals with the aerial survey of a closed region using the Space-Filling curve, particularly Sierpinski curve. The specified region is triangulated, and the Sierpinski curve is used to explore each triangular region. The entire region may have one or more obstacles. An algorithm is presented which suggests evasive maneuver if an obstacle is detected. The algorithm is online; that is, it does not require prior knowledge of the location of obstacles and can be used while the robotic system traverses the designated path. The fractal nature of the Sierpinski curve and simple geometric observations were used to formulate and validate the algorithm. The non-uniform coverage and multiple obstacle problems are also dealt with towards the end.

Application of Selected Fractal Geometry Resonators in Microstrip Strain Sensors

Until now, strain gauges, fiber optic sensors, magnetic sensors, and piezoelectric sensors have been used for strain measurements. In recent years, the idea of using microstrip antennas in strain measurement applications has emerged. One of the key requirements for deformation sensors is the ability to mount them on small-size elements (therefore the size of these sensors is crucial). The possible way to miniaturize microstrip sensors is the application of appropriate patch geometry, e.g. fractal geometry. In this article application of various fractal patch geometries in microstrip strain sensors were investigated. All sensors were designed for the same operating frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${f}_{\textbf {r}} =2.725$ </tex-math></inline-formula> GHz and on the same laminate. This approach showed how the influence of a specific geometry affects the sensitivity and the transducer size. The Sierpinski carpet, the Sierpinski triangle, the Sierpinski curve, and the combination of the Koch curve and the Sierpinski carpet were used to obtain the fractal geometry and miniaturize the sensor dimensions. They have been compared with the widely used rectangular patch. Numerical and experimental analyzes for the proposed sensors were carried out. The best solution to this problem was to use a combination of the Koch curve and the Sierpinski carpet. This fractal geometry, compared to a rectangular patch, enables the reduction of the patch size by 70% while reducing the sensitivity by 26%.

Julia Sets of Cubic Rational Maps with Escaping Critical Points

It is known that the Julia set of a quadratic rational map is either connected or a Cantor set. In this paper, we explore this dichotomy for the maps in a type of three-dimensional space of cubic rational maps. We show that for a cubic rational map f, if f has an attracting fixed point p and all critical points are attracted to p under the iteration of f, then the Julia set J(f) is either a Cantor set or a connected set (and locally connected) with one possible exception; we also give a necessary and sufficient condition for J(f) to be a Sierpinski curve when it is connected.

Microstrip Patch Strain Sensor Miniaturization Using Sierpinski Curve Fractal Geometry.

In this paper miniaturization of a microstrip patch strain sensor (MPSS) using fractal geometry was proposed and analyzed. For this purpose, the transducer of Sierpinski curve geometry was utilized and compared with the most commonly utilized rectangular resonator-based one. Both sensors were designed for the same resonant frequency value (2.725 GHz). This fact allows analysis of the influence of the patch (resonator) shape and size on the resonant frequency shift. This is very important as the sensors with the same resonator shape but designed on various operating frequencies have various resonant frequency shifts. Simulation and experimental analysis for all sensors were carried out. A good convergence between results of simulation and measurements was achieved. The obtained results proved the possibility of microstrip strain sensor dimensions reduction using Sierpinski curve fractal geometry. Additionally, an influence of microstrip line deformation for proposed sensors was studied.

Low-Frequency Metamaterial Absorber Using Space-Filling Curve

The extensive use of metamaterials and metamaterial absorbers increases the demand for compact structures in various frequencies. Designing electrically small absorbers for lower frequencies, especially sub-gigahertz applications, is one of the open issues in this field. In this paper, a space filling curve is used to design an absorber operating on low frequencies. The unit cell design is based on a Sierpinski curve with the size of $$25\times 25\times 1.6\,{\text {mm}}^3$$ and air-gap of 10 mm. The structure shows 99.9% absorption at 900 MHz on the third step. The system also shows multiple resonances due to its structure. The proposed structure is fabricated and tested and shows a good agreement with simulation results.

Miniature tri‐wideband Sierpinski–Minkowski fractals metamaterial perfect absorber

With rapidly growing adoption of wireless technologies, requirements for the design of a miniature wideband multi-resonators are increasing. In this study, a compact fractal-based metamaterial structure with lumped resistors is described. The structure of the authors proposed absorber is a combination of Sierpinski curve and Minkowski fractal. The new combination provides larger capacitance and inductance in the system enabling perfect absorption at lower frequencies. The final structure with dimensions of 20 × 20 × 1.6 mm 3 and an air gap of 12.5 mm provides three main resonances at frequencies of 2.1, 5.1, and 12.8 GHz with bandwidth (absorption ratio over 90%) of 840 MHz, 1.05 GHz, and 910 MHz, respectively.

Generalized Rings Around the McMullen Domain

We consider the family of rational maps given by $$F_\lambda (z)=z^n+\lambda /z^d$$ where $$n, d \in \mathbb {N}$$ with $$1/n+1/d<1,$$ the variable $$z\in {\widehat{{\mathbb {C}}}}$$ and the parameter $$\lambda \in {\mathbb {C}}$$ . It is known that when $$n=d \ge 3$$ there are infinitely many rings $${\mathcal {S}}^k$$ with $$k\in \mathbb {N}$$ , around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter $$\lambda $$ -plane where the Julia sets of $$F_\lambda $$ are Cantor sets of simple closed curves. The rings $${\mathcal {S}}^k$$ converge to the boundary of the McMullen domain as $$k \rightarrow \infty $$ and contain parameter values that lie at the center of Sierpinski holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of $$F_\lambda $$ are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when $$1/n+1/d<1$$ where n is not necessarily equal to d. The number of Sierpinski holes and superstable parameters on $${\mathcal {S}}^1$$ is $$\tau _1^{n,d} = n-1,$$ and on $${\mathcal {S}}^k$$ for $$k> 1$$ is given by $$\tau _k^{n,d} = dn^{k-2}(n-1)-n^{k-1} + 1$$ .

The Behavior of CPW-Fed Sierpinski Curve Fractal Antenna

In this paper, the behavior of Coplanar Waveguide (CPW) fed SIERPINSKI curve fractal antenna is studied. The results show that there is a relationship between the iteration number and the resonance frequencies. With increase in the number of iteration the resonance frequency decreases with a constant ratio. The use of fractal structures to design antennas makes them more miniaturized. The simulated results obtained from CADFEKO a Method of Moments (MoM) model based Solver and measurement using Vector Network Analyzer Anritsu MS2026C are in good agreement

On homeomorphism of the Sierpiński curve

On homeomorphism of the Sierpiński curve

The Sierpinski curve viewed by numerical computations with infinities and infinitesimals

The Sierpinski curve is one of the most known space-filling curves and one with the highest number of applications. We present a recently proposed computational methodology based on the infinite quantity called grossone to investigate the behavior of two different constructions of the Sierpinski curve. We emphasize that, adopting this point of view, we have infinitely many Sierpinski curves depending, contrarily to traditional analysis, on each specific starting configuration. Of particular interest are some power series expansions in the new infinitesimal quantities emerging from the study of the considered curves.

A New Compact Dual-band Antenna Based on Sierpinski Curve Slotted Ground Plane and Current Distribution Analysis

A new design approach has been used to achieve a dual band response from a multi-band resonance. The design approach is wholly depending on current distribution analysis on the surface of a multi-band antenna. The proposed multi-band antenna consists of a slotted ground plane with a simple 50-ohm microstrip feed line on the other side of an FR4 substrate having 4.4 relative dielectric constant and 1.6 mm thickness. The geometry of the first iteration Sierpinski curve fractal has been employed to the slotted ground plane antenna structure. Two small squares have been inserted to both internal upper corners of the slotted ground plane as a technique to control the path of electrical current on the surface of the multi-band antenna. With this technique, the resulted antenna can offer two resonating bands with respect to -10 dB S11. The first band (2.28-2.6) GHz, while the second band (5-5.58) GHz. The proposed antennas have been modeled and simulated using two softwares (CST Microwave Studio and High-Frequency Structure Simulation HFSS) to verify the results of both antennas and the other antenna parameters have been studied by using CST only.

A Compact UWB Koch Fractal Antenna for UWB Antenna Array Applications

In this paper, compact ultrawideband (UWB) antennas (1-Element and 2-Element) are presented in order to demonstrate application of fractal geometry in UWB antenna design, which helps to improve the antenna characteristics in a given smaller area. Some features of Euclidean shape geometry are improved after the application of fractals at the edges of the polygon. It is demonstrated that the application of Koch fractal in the antenna design helps to achieve the desired miniaturization, compactness and wideband operability due to its self-similar and space filling characteristics. An analytical expression is provided to calculate the effective length and area of the structure with the increase in iteration's order. Its performance is further compared with other fractal geometries such as Minkowski and Sierpinski curve. It is observed that the proposed Koch fractal based antenna yields the widest bandwidth and better gain compared to other fractal geometry based antennas. In addition, the time-domain analysis of the antenna is performed in terms of fidelity factor and its values are better than 0.81. 1-Element antennas shows operational bandwidth from 2.3 to 13.2 GHz with a maximum gain value of 3.8 dB, whereas in case of 2-Element antenna operating bandwidth is 2.3?14 GHz with a maximum gain value of 7.2 dB. Moreover, the radiation patterns of the single element and 2-Element are nearly omnidirectional. These characteristics offer opportunities to explore the proposed UWB antenna for various applications such as WPAN, WBAN, etc.

Exotic topology in complex dynamics

We describe three different exotic topological objects that arise as Julia sets for complex maps, namely, Cantor bouquets, indecomposable continua, and Sierpinski curves.

On the Hausdorff dimension of the Sierpiński Julia sets

We estimate the Hausdorff dimension of hyperbolic Julia sets of maps from the well-known family $F_{\lambda,n}(z) = z^n + \lambda/z^n$, $n \ge 2$, $\lambda \in \mathbb{C} \setminus \{0\}$. In particular, we show that $\dim_H J(F_{\lambda,n}) = \mathcal O (1/\ln |\lambda|)$ for large $|\lambda|$, and $\dim_H J(F_{\lambda,n}) = 1 + \mathcal O (1/\ln n)$ for large $n$ in the three cases: when $J(F_{\lambda,n})$ is a Cantor set, a Cantor set of quasicircles and a Sierpinski curve.

Sierpinski curve Julia sets for quadratic rational maps

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpi«ski curve.

Miniaturized and dual-band metamaterial absorber with fractal Sierpinski structure

We report on the design, characteristics, and measurements of a terahertz (THz) metamaterial absorber (MA) based on fractal Sierpinski curves. By applying the fractal structure as the top resonators array, a more compact unit cell with a size reduction of 42% and dual-frequency operation has been achieved as an advantage over the conventional square-shaped MA. In addition, due to the rotationally symmetric structure, the fractal absorber is polarization insensitive and can perform well at a wide range of incident angles. Both the effective medium theory and the multireflection interference theory have been employed to investigate the underlying physical mechanism of the proposed THz MA, and it is found that the latter theory is not applicable for explaining the absorption mechanism of our investigated structure. The THz MA was measured in the case of 30° oblique incidence under TE polarization, and two absorption peaks have been observed at 0.2 and 0.58 THz with absorptivities of 91% and 92.2%, respectively. A microwave MA based on the same Sierpinski structure has also been demonstrated to validate the performance of the fractal MA at various incident angles, and good agreements between the full-wave simulation and experimental results have been achieved.

Homeomorphisms of the Sierpinski curve with periodic properties

In this paper, we study the three following types of homeomorphisms of the Sierpinski curve of the two sphere : pointwise periodic, periodic, and almost periodic, and we prove that they are equivalent. We show that a subgroup of homeomorphisms whose orbits are all finite, is a finite subgroup.

A Wideband Minkowski Fractal Dielectric Resonator Antenna

A CPW slot loop fed Minkowski shaped fractal dielectric resonator antenna is proposed. Self similar property of fractal geometry is utilized to bring higher order modes close together to realize a wide impedance bandwidth. A comparative study of the resonant frequency behavior between the well established fractal electrical boundary antennas and fractal magnetic boundary antennas is presented to provide insight into the functionality of the proposed antenna. The comparison highlights the fact that electrical boundary lowers resonance frequency, whereas fractal magnetic boundary increases the resonant frequency. Minkowski fractal is also compared with Koch fractal and Sierpinski curve geometries. It is observed that the Minkowski fractal DRA yields the widest impedance bandwidth along with stable gain amongst the three proposed geometries. The proposed antenna exhibits a fractional bandwidth of 64% (5.52 -10.72 GHz) and a maximum gain of 4.9 dBi.

TERAHERTZ SENSING APPLICATION BY USING FRACTAL GEOMETRIES OF SPLIT-RING RESONATORS

In this study, we report the simulation, fabrication and characterization of a dual-band fractal metamaterial used for terahertz sensing application. By applying the fractal structures of square Sierpinski (SS) curve to the split-ring resonators (SRRs), more compact size and higher sensitivity can be achieved as privileges over conventional SRRs. The in∞uence of difierent geometrical parameters and the order of the fractal curve on the performances are investigated. Then overlayers are added to the fractal SRRs in order to explore the performance of the entire system in terms of sensing phenomenon. The changes in the transmission resonances are monitored upon variation of the overlayer thickness and permittivity. Measured results show good agreement with simulated data. At the second resonance of the second-order SS-SRRs, maximum frequency shifts of 19.8GHz, 26.3GHz and 37.8GHz were observed for a 2m, 4m and 10m thickness of photoresist. The results show good sensitivity of the sensors suggesting they can be used for a myriad of terahertz sensing applications in biology and chemistry.

A combinatorial invariant for escape time Sierpiński rational maps

An escape time Sierpinski map is a rational map drawn from the McMullen family z 7→ zn + λ/zn with escaping critical orbits and Julia set homeomorphic to the Sierpinski curve continuum. We address the problem of characterizing postcritically finite escape time Sierpinski maps in a combinatorial way. To accomplish this, we define a combinatorial model given by a planar tree whose vertices come with a pair of combinatorial data that encodes the dynamics of critical orbits. We show that each escape time Sierpinski map realizes a subgraph of the combinatorial tree and the combinatorial information is a complete conjugacy invariant. Dedicated to Joyce T. Macabea, in loving memory.