An introduction to the importance of fractal geometry

What does self-similarity mean? If you look carefully at a fern leaf, you will notice that every little leaf - part of the bigger one - has the same shape as the whole fern leaf. You can say that the fern leaf is self-similar. The same is with fractals:

An introduction to the importance of fractal geometry

To create a 2D image using this technique each point in a partition of the plane is used as initial guess, zo, to the solution. The following shows the appearance of a small portion of the positive real and imaginary quadrant of the complex plane.

A trademark of chaotic systems is that very similar initial conditions can give rise to very different behaviour. In the image shown there are points very close together one of which converges to the solution very fast and the other converges very slowly.

Many attractive images can be generated using theory from areas of Chemistry and Physics. One such example is diffusion limited aggregation or DLA which describes, among other things, the diffusion and aggregation of zinc ions in an electrolytic solution onto electrodes.

Another more colourful description involves a city square surrounded by taverns. Drunks leave the taverns and stagger randomly around the square until they finally trip over one of their insensate companions at which time lulled by the sounds of peaceful snoring they lie down and fall asleep.

The tendril like structure is an aerial view of the sleeping crowd in the morning. Fractal Geometry Almost all geometric forms used for building man made objects belong to Euclidean geometry, they are comprised of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc.

These elements can be classified as belonging to an integer dimension, either 1, 2, or 3. This concept of dimension can be described both intuitively and mathematically. Intuitively we say that a line is one dimensional because it only takes 1 number to uniquely define any point on it.

That one number could be the distance from the start of the line. This applies equally well to the circumference of a circle, a curve, or the boundary of any object. A plane is two dimensional since in order to uniquely define any point on its surface we require two numbers.

There are many ways to arrange the definition of these two numbers but we normally create an orthogonal coordinate system.

Introduction to Fractal Geometry

Other examples of two dimensional objects are the surface of a sphere or an arbitrary twisted plane. The volume of some solid object is 3 dimensional on the same basis as above, it takes three numbers to uniquely define any point within the object.

1. Introduction to Fractals and IFSis an introduction to some basic geometry of fractal sets, with emphasis on the Iterated Function System (IFS) formalism for generating fractals. In addition, we explore the application of IFS to detect patterns, and also several examples of architectural fractals. 2. Fractal geometry offers almost unlimited waysof describing, measuring and predicting these natural phenomena. But is it possible to define the whole world using mathematical equations? This article describes how the four most famous fractals were created and explains the most important fractal properties, which make fractals useful for. Fractal Geometry. Almost all geometric forms used for building man made objects belong to Euclidean geometry, they are comprised of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc. These elements can be classified as belonging to an integer dimension, either 1, 2, or 3.

A more mathematical description of dimension is based on how the "size" of an object behaves as the linear dimension increases. In one dimension consider a line segment.

An introduction to the importance of fractal geometry

If the linear dimension of the line segment is doubled then obviously the length characteristic size of the line has doubled. In two dimensions, ff the linear dimensions of a rectangle for example is doubled then the characteristic size, the area, increases by a factor of 4.

In three dimensions if the linear dimension of a box are doubled then the volume increases by a factor of 8. This relationship between dimension D, linear scaling L and the resulting increase in size S can be generalised and written as This is just telling us mathematically what we know from everyday experience.

If we scale a two dimensional object for example then the area increases by the square of the scaling. If we scale a three dimensional object the volume increases by the cube of the scale factor. Rearranging the above gives an expression for dimension depending on how the size changes as a function of linear scaling, namely In the examples above the value of D is an integer, either 1, 2, or 3, depending on the dimension of the geometry.

This relationship holds for all Euclidean shapes.

An introduction to the importance of fractal geometry

There are however many shapes which do not conform to the integer based idea of dimension given above in both the intuitive and mathematical descriptions. That is, there are objects which appear to be curves for example but which a point on the curve cannot be uniquely described with just one number.

If the earlier scaling formulation for dimension is applied the formula does not yield an integer. There are shapes that lie in a plane but if they are linearly scaled by a factor L, the area does not increase by L squared but by some non integer amount. These geometries are called fractals!

One of the simpler fractal shapes is the von Koch snowflake. The method of creating this shape is to repeatedly replace each line segment with the following 4 line segments.

The process starts with a single line segment and continues for ever.Fractal World 1 Introduction to Fractal Definition of Fractal The formal mathematical definition of fractal is defined by Benoit Mandelbrot. It says that a fractal is a set for which the Hausdorff Besicovich dimension strictly exceeds the topological dimension.

However, this is a very abstract definition. Because fractal geometry is relatively new -- the term was coined in by the late Benoit Mandelbrot, -- it is a concept not well understood by a portion of the population. Fractal geometry offers almost unlimited waysof describing, measuring and predicting these natural phenomena.

But is it possible to define the whole world using mathematical equations? This article describes how the four most famous fractals were created and explains the most important fractal properties, which make fractals useful for different domain of science.

Fractal Geometry. Almost all geometric forms used for building man made objects belong to Euclidean geometry, they are comprised of lines, planes, rectangular volumes, arcs, cylinders, spheres, etc. These elements can be classified as belonging to an integer dimension, either 1, 2, or 3.

Keble Summer Essay: Introduction to Fractal Geometry Martin Churchill: Page 6 of 24 6. Further Analysis of the Gasket Let us consider a Sierpinksi Gasket whose axiom is a triangle, of unit area. Introduction. The word "fractal" often has different connotations for laypeople than for mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception.

The mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background.

THE FRACTAL GEOMETRY OF NATURE - Introduction