This paper describes a simple but flexible family of Haskell programs for drawing pictures of fractals such as Mandelbrot and Julia sets. Its main goal is to showcase the elegance of a compositional approach to program construction, and the benefits of a clean separation between different aspects of program behavior. Aimed at readers with relatively little experience of functional programming, ...

T he fractal natu re of quantum paths contributing to Feynman path integral formul ation of Quantum mechanics is investigated. A comp ute r simula t ion of both oneand two-dimensional qu antum harmonic oscilla.tors yields resul ts in agreement wit h rigorou s results on the Hausdorff-Besicovitch dimension for Brownian motion .

Spatial variation in concentrations or flows within an organ and temporal variation in reaction rates or flows appear to broaden as one refines the scale of observation. How can we characterize heterogeneity independently of scale? Fractals come to our rescue! A system is fractal if its features adhere to the same rules through a succession of different scales. Fractals efficiently describe man...

This paper investigates an approach to the use of fractals in architectural design. Two major aspects are discussed. First, the effect of the direction of the line segments on the generated fractal, as well as, the proportions between the generator and the initiator. Secondly, the meanings attached to the lines segments that constitute the fractal are based on some known architectural organizat...

We give a sufficient condition to construct non-trivial μ-symmetric diffusion processes on a locally compact metric measure space (M,ρ, μ). These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of Γ-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that ...

The probability that a random walker returns to its origin for large times scales as t;{-d[over ]2} , where d[over ] is the spectral dimension. We calculate d[over ] for a class of tree structures using a renormalization technique on an infinite continued fraction. We consider a wide range of homogeneous networks based on replacing the branches of a self-similar tree with arbitrary fractals and...

1. Bourke, P.: An Introduction to fractals ,http://paulbourke.net/fractals/fracintro/ 2. Gangopadhyay, T. The effect of multiple rotations on Mobius transformations in generating IFS Fractalss, International journal of Computer Applications 142(6):18-22, May 2016 3. Julia, G. Mémoire sur l'itération des fonctions rationnelles, Journal de Mathématiques Pures et Appliquées, 1918 4. Krantz, S. G. ...

We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions, random walks, resistor networks, and Gaussian field theories. The mechanism is unique in that it is absent in uniform media, while universal in that it is observed in a wide class of fractals. 05.40+j, ...

The famous game Towers of Hanoi is related with a family of so–called Hanoi–graphs. We regard these non self–similar graphs as geometrical objects and obtain a sequence of fractals (HGα)α converging to the Sierpiński gasket which is one of the best studied fractals. It is shown that this convergence holds not only with respect to the Hausdorff distance, but that also Hausdorff dimension does co...

In the case of fractals and similarly complex objects, quantities such as length, area and volume may be infinite, or rather undefined. Hence it is necessary to consider other notions of measurement when comparing fractals. This report aims to motivate such a measure and show that it contains many of the intuitive properties of dimension. Furthermore, some methods for measuring this ‘fractal di...