Narrow Results

There are several methods given for constructing the Sierpinski Triangle (ST). In particular, the ST can be created as a countably infinite intersection of contained figures, or as the countably infinite union of triangles. At first glance they both appear to create the same object. In this paper, we show that they are not the same. In addition, we analyze and compare with the previous two methods, a third method of constructing the ST which entails generating a countable number of points using an algorithm.

In this paper, we provide extensions to hyperbolic numbers plane of the classical Chaos game algorithm and the Shannon entropy. Both notions connected with that of probability with values in hyperbolic number, introduced by Alpay et al. [Kolmogorov’s axioms for probabilities with values in hyperbolic numbers, Adv. Appl. Clifford Algebras27(2) (2017) 913–929]. Within this context, particular attention has been paid to the interpretation of the hyperbolic valued probabilities and the hyperbolic extension of entropy as well.

The paper presents a novel method for evaluating the different natural processes. The method is based on the chaos game technique, and evaluation of the results is based on the measurements of the fractal dimension of the obtained Sierpinski triangles. Pathways of the natural rivers channels case study has been used for the demonstration of the methodology, despite the method can be applied to many natural phenomena. Authors would like to encourage others to make similar analysis, because more case studies are needed to draw more solid conclusions.