Connected generalised Sierpiński carpets

Abstract

Generalised Sierpiński carpets are planar sets that generalise the well-known Sierpiński carpet and are defined by means of sequences of patterns. We study the structure of the sets at the kth iteration in the construction of the generalised carpet, for k ≥ 1. Subsequently, we show that certain families of patterns provide total disconnectedness of the resulting generalised carpets. Moreover, analogous results hold even in a more general setting. Finally, we apply the obtained results in order to give an example of a totally disconnected generalised carpet with box-counting dimension 2.

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

“…The recent interest in the study of complex systems has led to an increased focus on anomalous diffusion [4-7, 36, 42, 45], which has been described in biological systems [42][43][44], protostellar birth [47], complex fluids [48][49][50][51][52][53], electronic transportation [54][55][56][57][58][59], porous media and infiltration [60][61][62], drug delivery [63][64][65][66], fractal structures and networks [67][68][69][70][71], and in water's anomalous behavior [72][73][74], phase transitions in synchronizing oscillators [39,40,[75][76][77], to name a few.…”

On certain families of planar patterns and fractals

“…First, let us recall the definition of a pattern, as it is given in some of the above mentioned papers [9,10]. Let x, y, q ∈ [0, 1] such that…”

“…We let W 1 = A 1 , and call it the set of white squares of level 1. For n ≥ 2 we define the set of white squares of level n by Figure 1, that can also be viewed as a 20-pattern By defining three types of graphs associated to the patterns that generate the generalised Sierpiński carpets, necessary and sufficient conditions for the connectedness (with respect to the usual topology of the Euclidean plane) of these objects were proven [9].…”

“…A different approach, namely identifying certain families of patterns, was used [10] in order to study the structure of the sets obtained at the nth iteration in the construction of a generalised carpet, for n ≥ 1, and it was shown that certain families of patterns provide total disconnectedness of the resulting fractals. Moreover, analogous results hold even in a more general setting [10]. This approach of the carpets provides the possibility to construct disconnected carpets of box-counting dimension less than or even equal to 2, as it is shown in an example in the more extended arXiv-version [11] of the published paper [10].…”

Fractal analysis in particle dissolution: a review

“…The roots of mathematically rigorous treatment of fractals can be traced back to functions studied by Georg Cantor who represented the Cantor set in 1883 (Levenspiel 1972, Mandelbrot 1983, Mortimer and Taylor 2002, Waclaw Sierpinski who introduced the Sierpinski gasket and carpet in 1916 (Birdi 1993, Peitgen et al 2004, Stós 2006, Cristea and Steinsky 2010, and Helge von Koch who invented the Koch curve in 1904 (Birdi 1993, Peitgen et al 2004, Milosevic and Ristanovic 2007, Paramanathan and Uthayakumar 2010. Besides, there are some other basic fractal models, such as the Peano curve, the space-filling curves, discovered by Giuseppe Peano in 1890 (Kennedy 1974, Sprecher andDraghici 2002), the Hilbert curves and RBG curves presented by David Hilbert (Liu 2004, Chen andChang 2005), and the Julia sets proposed by Gaston Julia (Kameyama 1993).…”