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An irregular Sierpinski triangle is a non-empty compact set F satisfying where each T_i is a contracting affine transformation. The irregular Sierpinski triangle is not generated with similarity transformations. Therefore, the calculation of its fractal dimension is not straightforward. We associate with each irregular Sierpinski triangle a self-similar regularization. The dimension of the regularization dictates the shape of the irregular Sierpinski triangle. We also describe properties of the regularizations and show their relationship to the classic Sierpinski triangle.

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by . In the literature the best existing estimate is

The Sierpinski Triangle (ST) is a fractal which has Haussdorf dimension ${log}_{2}3\approx 1.585$ that has been studied extensively. In this paper, we introduce the Sierpinski Triangle Plane (STP), an infinite extension of the ST that spans the entire real plane but is not a vector subspace or a tiling of the plane with a finite set of STs. STP is shown to be a radial fractal with many interesting and surprising properties.

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.

A new neural network model is introduced in this paper. The aim of the proposed Sierpinski neural networks is to provide a simple and biologically plausible neural network architecture that produces emergent complex spatio-temporal patterns through the activity of the output neurons of the network and is able to perform computational tasks. Such networks may play an important role in the analysis and understanding of complex dynamic activity observed at various levels of biological neural systems. The proposed Sierpinski neural networks are described in detail and their functioning is analyzed. We discuss about emerging neural activity patterns and their interpretations, neuro-computation with such emerging activity patterns, and also possible implications for computational neuroscience.

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.