Introduction to Fractal Geometry .................................................................. 1 1. A few Random Quotes ......................................................................... 1 2. Introduction ....................................................................................... 1 3. Types of fractals ........................................................................

In this paper, we firstly devise a new and general p-ary subdivision scheme based on normal vectors with multi-parameters to generate fractals. Rich and colorful fractals including some known fractals and a lot of unknown ones can be generated directly and conveniently by using it uniformly. The method is easy to use and effective in generating fractals since the values of the parameters and th...

Biological systems are by nature complex and this complexity has been shown to be important in maintaining homeostasis. The plant microtubule cytoskeleton is a highly complex system, with contributing factors through interactions with microtubule-associated proteins (MAPs), expression of multiple tubulin isoforms, and post-translational modification of tubulin and MAPs. Some of this complexity ...

Fractals with different levels of self-similarity and magnification are defined as reduced fractals. It is shown that spectra these fractals can be constructed used to describe complexity natural phenomena. Specific applications biological systems, such green algae, performed, it suggested the obtained classify considered algae by identifying associated them. The ranges for determined their ext...

Riesz potentials of fractal measures μ in metric spaces and their inverses are introduced . They define self–adjoint operators in the Hilbert space L2(μ) and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operato...

In undergraduate classrooms, while teaching chaos and fractals, it is taught as if there no relation between these two. By using some non linear oscillators we demonstrate that a connection fractals. plotting the phase space diagrams of four nonlinear box counting method finding fractal dimension established chaotic nature oscillators. The awareness all systems are good fractals will add more i...

This paper discusses one method of producing fractals, namely that of iterated function systems. We first establish the tools of Hausdorff measure and Hausdorff dimension to analyze fractals, as well as some concepts in the theory of metric spaces. The latter allows us to prove the existence and uniqueness of fractals as fixed points of iterated function systems. We discuss the connection betwe...

V-variable fractals, where V is a positive integer, are intuitively fractals with at most V different “forms” or “shapes” at all levels of magnification. In this paper we describe how V-variable fractals can be used for the purpose of image compression.

This paper explores methods of generating fractals from the mapping z→z −n +c and discusses the structure of the resulting images. Lyapunov exponents and cycle periodicity show details of these fractals that are obscured by prior escape-time techniques. The resulting fractals are shown to have a hypocycloid shape. The associated Julia sets can be classified into a variety of types.