Concept# Fractal

Summary

In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.
One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to t

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In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at

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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These top

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In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such

In this study, we will discuss the engineering construction of a special sixth generation (6G) antenna, based on the fractal called Minkowski's loop. The antenna has the shape of this known fractal, set at four iterations, to obtain maximum performance. The frequency bands for which this 6G fractal antenna was designed in the current paper are 170 GHz to 260 GHz (WR-4) and 110 GHz to 170 GHz (WR-6), respectively. The three resonant frequencies, optimally used, are equal to 140 GHz (WR-6) for the first, 182 GHz (WR-4) for the second and 191 GHz (WR-4) for the third. For these frequencies the electromagnetic behaviors of fractal antennas and their graphical representations are highlighted.

The present work has been focused on the development of analytical and numerical techniques for the analysis of highly convoluted antennas and microwave devices including fractal shaped antennas. The accurate prediction of the frequency response of a high-iterated pre-fractal structure is frequently a very consuming task, in terms of computer resources. The techniques presented in this work try to ameliorate some of the bottlenecks in the solving process of fractal shaped or highly convoluted devices. First, in the frame of the Mixed Potential Integral Equation (MPIE) technique, a new set of basis functions for the discretization of the currents in the Method of Moments (MoM) solution is presented. The basis functions are defined over quadrangular domains and their aim is, on one hand, to allow a good representation of the current while preserving the main longitudinal direction existing in many practical surfaces, used as metalization in printed circuits, and on the other hand, to reduce the number of unknowns compared to a standard triangular mesh. The basis functions over quadrangular cells comprise as particular cases the classic rectangular and triangular rooftops. However, a new basis function over triangular domains is also included as a case derived from the general quadrangle, and has the particularity of being able to model the connection between two triangles with a common vertex, instead of the conventional attachment at the edge of the classic triangular pair. Second, different highly convoluted Euclidean structures have been studied in order to provide a benchmark for fractal devices performance. The considered structures are a meander line and a two-arm square spiral antenna. Both structures show miniaturization capabilities, the spiral being one of the outstanding shapes in terms of miniaturization keeping a reasonable frequency behavior. With the study of these two structures it has been shown that some properties, considered exclusive of the fractal shaped family, appear also in non-fractal shapes. Third, several analysis techniques based on a transmission line approach and specially suited to solve highly convoluted printed line devices have been developed. The aim is to have a simple and fast tool to allow a rapid analysis of complicated structures, providing reasonably accurate results taking into account the simplicity of the model. These methods have been applied to a set of fractal curves belonging to the family of the fractal tree. Some prototypes have been built in microstrip technology and measured to verify the validity of the method. Finally, in order to obtain a fast full-wave analysis of printed-line or wire-strip devices, a new technique is developed. The method takes advantage of the geometry of the structures presenting currents flowing mainly in the longitudinal direction. These 2D structures are then considered as 1D ones, thus, the cells are reduced to their axis or backbones. This approximation is valid for the limiting cases, namely, very narrow structures or interactions between far away cells. However, once the width compensation factor that is computed analytically is included, the approximation becomes valid for all the structures not having a width bigger than the standard mesh density limit in a 2D problem. Values of the error committed with respect to a classic 2D method are given, and the analysis of some line structures is performed, proving the validity of the proposed method.