We provide a review on the physics associated with phase transitions in which continuous scale invariance is broken into discrete invariance. The rich features of this transition characterized by abrupt formation geometric ladder eigenstates, low energy universality without fixed points, anomalies and Berezinskii-Kosterlitz-Thouless scaling described. important role various celebrated single ma...

Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the d...

A multiband double square loop antenna with fractal concept is proposed. The design is based on an initial model of fractal geometry and a double square slot antenna structure. The initial model of fractal geometry influences to the second frequency band and it will be carefully investigated. The radiation patterns of the proposed antenna are still similarly to bidirectional radiation pattern. ...

Since Euclid (300 BC) we have been used to perceiving nature with the concept of a threedimensional (3-D) geometry. We measure linear structures in one dimension, area-like structures in two dimensions, and volume-like structures in three dimensions. However, when we measure an object in terms of these three dimensions, we are aware that the geometric model describes a solid body, while a natur...

In this paper we consider the topological aspects of sets defined by fractal geometry. Topology, as a geometry, is extended to the definition of fractal structures by characterising equivalences in different levels of detail. A group of distinct topological fractals are defined by taking the equipotence and multiplicity properties of sets. Additionally, we define fractal metamorphosis as the se...

Fractal geometry has sparked considerable interest for analyzing remote sensing imageries. Fractal models have been used in several image processing and pattern recognition applications such as texture analysis and classification. Applications of fractal geometry in remote sensing rely heavily on estimation of the fractal dimension (D). When land areas are clustered into groups of similar land ...

Fractal geometry has proven to be a useful tool in quantifying the structure of a wide range of idealized and naturally occurring objects, from pure mathematics, through physics and chemistry, to biology and medicine. In the past few years fractal analysis techniques have gained increasing attention in signal and image processing, especially in medical sciences, e.g. in pathology, neuropsychiat...