Topology in fractals

Complex Pattern Formation

“…Multi-fractal measures are one of the hottest subjects in the current scientific discussion of fractals. The concept of topology is relevant for the analysis and synthesis of models, which attempt to explain the continuous distortion of forms, represented by fractal geometry [10]. Physiologists quantified observations of the branching pattern and discovered that lung tree networks of blood vessels, nerves and ducts have fractal-like structures.…”

“…10. Experimental setup for observing the development of fractal geometry during crystallization[22].…”

Fractal Deformation Using Displacement Vectors Based on Extended Iterated Shuffle Transformation

“…However, these techniques cannot achieve a local deformation because each contraction mapping of the IFS affects the whole shape. Similarly, the technique in [14] can achieve only a global deformation because of the global influence of a functional equation. Moreover, these techniques are manipulated by treating parameter values such as the coefficients of contraction mappings or the strength of attraction of fixed points, and a user cannot control a shape directly.…”

“…The study in [5] proposed a method for maintaining the connectedness of a deformed IFS attractor during the deformation process. Besides, in [14], a fractal shape is defined based on a recursive functional equation on the complex plane and is analyzed topologically; using this definition form, a fractal metamorphosis of the shape is achieved by changing the functional equation continuously. In [12], a simple procedural technique for creating a moving fractal tree was proposed; the shape of a tree is first defined using a simple recursive procedure that gives proper scale factors to the branch length, and then is given motions by rotating the branches.…”

“…The former produces a natural continuous transition between two objects, and the latter deforms a single object. Besides, as a kind of shape deformation, some fractal deformation techniques have been proposed [4,5,12,14,15,17,18,19]. These techniques * Presently, Tohoku Institute of Technology, muraoka@tohtech.ac.jp treat fractal shapes and deform them (cf.…”