On a new class of planar fractals: the Pascal-Sierpinski gaskets

Fractals related to Pascal's triangle

“…The variety of applications of the Sierpinski gasket in physics made Holter, Lakhatkia et al [7,9,10] propose the Pascal--Sierpinski gaskets for theoretical models of physical phenomena and start off their study by numerical methods considering them like aggregates of particles. For the fractal dimension of Fp 1, calculated by means of numerical methods (mass-radius dimension) good approximations were obtained, but it was not so for Fpl~, r/> 2, due to the slow convergence of the sequences experimentally obtained.…”

“…may lead to not very good estimates for its fractal dimension. For example, in [7] the value D4 --1.6688 is given as an approximation to the mass-radius dimension of F2, using the subdivision p20OO. In that paper, the authors only assign a fractal dimension to T-P fractals of prime modulus.…”

“…Note that the fractal dimension is much less, a2 = log2 3 = In 3/In 2 -1.584962501. As we know the exact number of squares in each division, we can use the box counting method (which is essentially equivalent to that in [7]) to calculate approximate values for fractal dimensions. For example, from Theorem 3.1 for p = 2, r = 2 we have: N2~ = 3 s + (s -1)3 s-2.…”

Novel combined MOD‐P multiband antenna structures inspired by fractal geometries

“…In addition, by modifying the Sierpinski monopole flare angle, the antenna pattern at the different resonances can be tailored, as described in [4]. Further investigations on geometries inspired by fractal geometries show that the classical Sierpinski gasket is in fact a special case of a wider set of structures, which can be referred to as Pascal-Sierpinski gaskets [5]. In [6], the authors showed how the so-called mod-p Sierpinski gaskets can be used as a very efficient solution for the design of multiband antennas.…”

“…Recently, an intensive study of the propagation of electromagnetic waves in media with negative permittivity and permeability has been presented in the literature [1][2][3][4][5][6]. Some unique properties, such as a negative index of refraction, supporting backward waves, and so forth, have been shown.…”