Class Of Fractals Research Articles

Categorization of new fractal carpets

Sierpinski carpet is one of the very beautiful fractals from the historic gallery of classical fractals. Carpet designing is not only a fascinating activity in computer graphics, but it has real applications in carpet industry as well. One may find illusionary delighted carpets designed here, which are useful in real designing of carpets. In this paper, we attempt to systematize their generation and put them into categories. Each next category leads to a more generalized form of the fractal carpet.

Elementary density bounds for self-similar sets and application

Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H s -almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition (OSC), and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.

Modeling minimal foliated spaces with positive entropy

Using methods and results of decomposition theory we construct minimal actions of groups of homeomorphisms of some classical fractals (the Sierpi\'nski carpet and its generalizations, and the Menger curve) with positive entropy. Suspending these group actions we get minimal foliated spaces which have positive geometric entropy and are modelled on these fractals.

Connecting Hausdorff measure and upper convex density or [formula omitted]-a.e. covering

In this paper, we make precise the relationship between Hausdorff measure and upper convex density or H s -a.e. covering. We mainly study a class of s-sets, namely, the s-straight s-sets, especially the self-similar s-sets, which are the most important fractals in the study of fractal geometry. Sufficient and necessary conditions for the s-straight s-sets associated with H s -a.e. covering, for the upper convex density of the self-similar s-set at the simple-contracting-similarity fixed point less than 1, and for the existence of the best H s -a.e. covering of the self-similar s-set, are obtained. In addition, some results are applied to the famous classical fractals.

Growing a class of fractals based on combination of classical fractals and recursive mathematical series in L-systems

L-systems ([1]) can be used to generate fractals. In this paper L-systems with a standard turtle alphabet are combined with L-systems, which emulate simple number sequences. The combination of two such L-systems is achieved by concatenating their axioms as well as merging their rule sets. This can happen in several ways, depending in part on the sequence in which axioms and right-hand sides of rules are concatenated. The purpose of this paper is to combine these two applications of L-systems to generate novel fractals. A turtle interpretation is applied to the yield of the generated hybrid L-systems resulting in new fractal images.

Fractals in pixellated video feedback

Using experiments and computer simulations we show that video feedback with a pixel-based monitor can lead to self-similar stationary patterns. We found an astonishing variety of such patterns, which includes many 'classic fractals' such as the Sierpinski gasket and the von Koch snowflake. An important, previously unidentified parameter is the position of the centre of the camera zoom with respect to the individual pixels on the screen. This article is a detailed account of work recently published by the present authors.

Parametric 2-dimensional L systems and recursive fractal images: Mandelbrot set, Julia sets and biomorphs

L systems have proved their expressive power. They have been used to represent the class of the initiator/iterator fractal curves (such as Sierpinski's gasket and von Koch's snowflake curve). Parametric L systems, introduced by Prusinkiewicz and Lindenmayer, link real-valued parameters to the symbols. In this paper, parametric 0L systems are extended to n dimensions and used to represent a different class of classic fractals that includes objects such the Mandelbrot and Julia sets, or Pickover's biomorphs.

Differences Between Classical and Superconductive Percolation Fractals

The average mass of the backbone has fractal dimensionalities [Formula: see text] for superconductive percolation and theat of classical percolation [Formula: see text] in dimensions d = 3, 4, 5 and 6 respectively. A new fractal dimension of "dead ends" [Formula: see text] refers to the difference between two percolations, because if most of the mass of the backbone belong to the dead ends, the superconductive behavior becomes impossible. The definition of the minimum path (or the chemical distance) and its fractal dimensionality does not exist in the superconductive percolation. Instead of the "red" bonds in classical percolotion, one has the real physical phenomena — the "special" nodes, which can be found in the experimental investigations. The order of ramification R for superconductive fractals is equal to the critical exponent of the number of the "special" nodes N(p) ∝ (p - pc)dν, (R = dν). The fractal dimension of order of ramification equals to zero.

Mathematics: Classical continued fractals

Mathematics: Classical continued fractals

Computer rendering of fractal curves and surfaces

Fractals are a class of highly irregular shapes that have myriad counterparts in the real world, such as islands, river networks, turbulence, and snowflakes. Classic fractals include Brownian paths, Cantor sets, and plane-filling curves. Nearly all fractal sets are of fractional dimension and all are nowhere differentiable.Previously published procedures for calculating fractal curves employ shear displacement processes, modified Markov processes, and inverse Fourier transforms. They are either very expensive or very complex and do not easily generalize to surfaces. This paper presents a family of simple methods for generating and displaying a wide class of fractal curves and surfaces. In so doing, it introduces the concept of statistical subdivision in which a geometric entity is split into smaller entities while preserving certain statistical properties.

Computer rendering of fractal curves and surfaces

Fractals are a class of highly irregular shapes that have myriad counterparts in the real world, such as islands, river networks, turbulence, and snowflakes. Classic fractals include Brownian paths, Cantor sets, and plane-filling curves. Nearly all fractal sets are of fractional dimension and all are nowhere differentiable. Previously published procedures for calculating fractal curves employ shear displacement processes, modified Markov processes, and inverse Fourier transforms. They are either very expensive or very complex and do not easily generalize to surfaces. This paper presents a family of simple methods for generating and displaying a wide class of fractal curves and surfaces. In so doing, it introduces the concept of statistical subdivision in which a geometric entity is split into smaller entities while preserving certain statistical properties.