Types Of Fractals

As we said earlier, Fractals are generated using some kind of mathematical relationship(Algorithm). There are three common types of systems which are used to genetate fractals.

Escape-time fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the

Mandelbrot set



Julia set


Burning Ship fractal

*pics in courtesy of www.wikipedia.org


Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.


Sierpinski carpet



Sierpinski gasket



More interesting examples...

2-D tree generated using IFS



3-D trees generated using IFS









For more details on Iterated Function Systems,visit:
http://iteratedfunctions.blogspot.com/


Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.


await for more details!!!!

Fractals-History

Objects that are now described as fractals were discovered and described centuries ago. Some of the earliest man-made examples can be seen in indigenous African craft work. In 1525, the German artist Albrecht Dürer published The Painter's Manual, which contained a section on "Tile Patterns Formed by Pentagons". Dürer's Pentagon largely resembles the Sierpinski carpet, replacing squares with pentagons.


Albrecht Dürer


The mathematics behind fractals began to take shape in the 17th century when philosopher Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).


Leibniz

It took until 1872 before a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the nonintuitive property of being everywhere continuous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. In 1915, Waclaw Sierpinski constructed his triangle and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. The idea of self-similar curves was taken further by Paul Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve.

Georg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.

Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of many of the objects that they had discovered.


Benoît Mandelbrot

In the 1960s, Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".

Fractal Dimension

To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place. Obviously, a line has dimension 1, a plane dimension 2, and a cube dimension 3. But how is this happens to be ???

We may break a line segment into 4 self-similar intervals, each with the same length, and ecah of which can be magnified by a factor of 4 to yield the original segment. We can also break a line segment into 7 self-similar pieces, each with magnification factor 7, or 20 self-similar pieces with magnification factor 20. In general, we can break a line segment into N self-similar pieces, each with magnification factor N

A square is different. We can decompose a square into 4 self-similar sub-squares, and the magnification factor here is 2. Alternatively, we can break the square into 9 self-similar pieces with magnification factor 3, or 25 self-similar pieces with magnification factor 5. Clearly, the square may be broken into N^2 self-similar copies of itself, each of which must be magnified by a factor of N to yield the original figure


A square may be broken into N^2 self-similar pieces, each with magnification factor N


Finally, we can decompose a cube into N^3 self-similar pieces, each of which has magnification factor N.

So we can define Fractal dimension as...




for the square, we have N^2 self-similar pieces, each with magnification factor N. So we can write




However, many things in nature are described better with dimension being part of the way between two whole numbers. While a straight line has a dimension of exactly one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it curves and twists.

As an example let's consider Sierpinski triangle



Dimension


The more a fractal fills up a plane, the closer it approaches two dimensions. In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between two and three. Hence, a fractal landscape which consists of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with many average sized hills would be much closer to the third dimension.



Some Fractal Scenes

Fractals - An Introduction

Real fern leaf (left) and a fractal fern leaf (right), miracle in nature and a reality in fractal world

While the classical Euclidean geometry works with objects which exist in integer dimensions, fractal geometry deals with objects in non-integer dimensions. Euclidean geometry is a description lines, ellipses, circles, etc. Fractal geometry, however, is described in algorithms -- a set of instructions on how to create a fractal.

The world as we know it is made up of objects which exist in integer dimensions, single dimensional points, one dimensional lines and curves, two dimension plane figures like circles and squares, and three dimensional solid objects such as spheres and cubes.

Fractal Dimension

To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place. Obviously, a line has dimension 1, a plane dimension 2, and a cube dimension 3. But how is this happens to be ???

We may break a line segment into 4 self-similar intervals, each with the same length, and ecah of which can be magnified by a factor of 4 to yield the original segment. We can also break a line segment into 7 self-similar pieces, each with magnification factor 7, or 20 self-similar pieces with magnification factor 20. In general, we can break a line segment into N self-similar pieces, each with magnification factor N

A square is different. We can decompose a square into 4 self-similar sub-squares, and the magnification factor here is 2. Alternatively, we can break the square into 9 self-similar pieces with magnification factor 3, or 25 self-similar pieces with magnification factor 5. Clearly, the square may be broken into N^2 self-similar copies of itself, each of which must be magnified by a factor of N to yield the original figure


A square may be broken into N^2 self-similar pieces, each with magnification factor N


Finally, we can decompose a cube into N^3 self-similar pieces, each of which has magnification factor N.

So we can define Fractal dimension as...




for the square, we have N^2 self-similar pieces, each with magnification factor N. So we can write




However, many things in nature are described better with dimension being part of the way between two whole numbers. While a straight line has a dimension of exactly one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it curves and twists.

As an example let's consider Sierpinski triangle



Dimension


The more a fractal fills up a plane, the closer it approaches two dimensions. In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between two and three. Hence, a fractal landscape which consists of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with many average sized hills would be much closer to the third dimension.



Some Fractal Scenes