Akhlaq Husain

The fractal rep-tiles of the Euclidean plane considered in this article are examples of rep-tiles (tilings) with fractal boundaries. Several new examples of fractal rep-tiles are constructed using reflection transformations and integer matrices. A new class of foldable fractal rep-tiles based on general reflection mappings is introduced, and it is shown that these rep-tiles tile the plane using similitudes, including reflections, if the iterated function system (IFS) satisfies lattice tiling conditions. We prove the existence of foldable fractal 2-rep and 4-rep tiles that exhibit novel properties (chirality and aperiodicity) caused by reflection mappings. Fractal variations of foldable rep-tiles are also constructed. The fractal rep-tiles and the foldable rep-tiles presented here are in one-to-one correspondence with finite reflection groups, and this novel class of foldable rep-tiles can be lifted to construct new classes of fractal rep-tiles with roots in classical reflection groups. The images of rep-tiles are rendered using the random iteration algorithm, which is one of the popular iterative methods to generate self-similar fractals and tilings.

Urban road networks possesses fractal characteristics due to ordered division and structural layout of the road networks which contain inherent self-similarity and scale invariance. We investigate urban planning of road networks in the Buraydah City, Al Qassim, Saudi Arabia using techniques from fractal geometry via fractal dimension (a non-integer number which measure geometric complexity of an object) calculations. The main objective is to study the fractal structure of the road network of the Buraydah city. To accomplish this goal we provide simulation and compare various kinds of fractal dimensions including box dimension ($D_\textrm{box}$), capacity dimension ($D_\textrm{cap}$), information dimension ($D_\textrm{inf}$), correlation dimension ($D_\textrm{cor}$), and probability dimension ($D_\textrm{prob}$) for the entire road network of the Buraydah city. The study reveals that a better planning and optimization of road networks can be obtained using these fractal indicators. The computational experiments are done using QGIS software and open source fractal dimension programs to understand fractal spatial patterns in the road network.

Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a series of two papers dedicated to an eclectic survey of fractals describing the infinite complexity and amazing beauty of fractals from historical, theoretical, mathematical, aesthetical and technological aspects, including their diverse applications in various fields. In this article, our focus is on engineering, industrial, commercial and futuristic applications of fractals, whereas in the first part, we discussed the basics of fractals, mathematical description, fractal dimension and artistic applications. Among many different applications of fractals, fractal landscape generation (fractal landscapes that can simulate and describe natural terrains and landscapes more precisely by mathematical models of fractal geometry), fractal antennas (fractal-shaped antennas that are designed and used in devices which operate on multiple and wider frequency bands) and fractal image compression (a fractal-based lossy compression method for digital and natural images which uses inherent self-similarity present in an image) are the most creative, engineering-driven, industry-oriented, commercial and emerging applications. We consider each of these applications in detail along with some innovative and future ready applications.

Fractals are geometric shapes and patterns that may repeat their geometry at smaller or larger scales. It is well established that fractals can describe shapes and surfaces that cannot be represented by the classical Euclidean geometry. An eclectic survey of fractals is presented in two parts encompassing applications of fractals in a variety of diverse and innovative fields. The goal of the first part is to focus on the glossary of fractals, their mathematical description, aesthetic, artistic, and architectural applications, while the second part is focused on engineering, industry, commercial, and futuristic applications of fractals.