What is the difference between fractal geometry and euclidean geometry

Some of these shapes include clouds, vegetables, colour patterns, lightning, and snowflakes. Fractal can define images that are not otherwise can be defined by Euclidean. Trees, rivers, coastlines, mountains, clouds, lightning, snowflakes and hurricanes, earthquake are all displaying or obeying fractal rules.

A fractal description of many things is a story about how they grow. The fractal patterns are used by artists for a long time. The image created by a fractal is complex yet striking. Fractals are used to capture the complex organic structure and to analyze various biological processes or phenomena such as the growth pattern of bacteria.

In lungs and blood vessels fractal structure can be observed with larger passages branching into smaller passages which branch again into yet smaller ones. It is widely used in image processing. Fractals are related to chaos as they are complex systems that have definite properties. Many disordered media are fractal structures. A number of works are done on deterministic fractal lattices by investigating random walk on them. It has been used to study the scattering of electromagnetic wave or scalar wave by random variation of refractive index.

One of the interesting areas is the study of physics underlying irreversible growth phenomena that generates fractal structures or Kinetic Critical phenomena. New research has discovered that fractals can be put to good use in photonics by creating transparent ultrathin metallic electrodes with superior optoelectronic properties. Our world is fractal world.

Its uses ranges from the branching of tracheal tubes, leaves, trees, vegetables, veins in hand, water swirling and twisting out of a tap, a puffy cumulus cloud, turbulent flow, tiny oxygen molecule or the DNA molecule and finally film and stoke market too!!

There are many surprises in generating fractals and investigating their properties. It proves to be a useful tool despite all its intricacies. A new approach to microphysics started with Fractals. The particle distribution in High Energy collision often show self similarity and power law behaviour which in turn can be related to the fractal structure of hadrons.

Many works focuses on the path of the quantum particles and fundamental physics. The internal fractal structure of the path of quantum particle may be attributed to the non-differentiable space — time. In that respect laws nature can be replaced by scale relativity. Fractal geometry developed as a new geometry of nature [4] -[8] has been conceived with the aim to describe and evaluate the complexity and irregularity of shapes and processes in nature [3].

Up to now fractal geometry and fractal analysis are used in diverse research areas [9] -[12]. The goal of fractal geometry is to provide a platform for the demonstration of the importance of the fractal disciplines and to continuously explore the world of fractals not only in mathematics but also in sciences and technologies.

One of the aims of the present paper is to show how fractal geometry gives rise to the fractal analysis confining our consideration to fractal sets and mathematics. In Fractal geometry the geometrical fractal set should be considered as an infinite ordered series of geometrical objects defined on a metric space.

To determine a fractal set we need specify three things [2] [13] [14] : 1 the shape of a starting object; the initiator , 2 the iterate algorithm enabling its iterative application on the initiator and then, repeatedly, on all obtained geometrical objects the generators , and 3 the conditions which these generators should satisfy, before all the properties of geometrical similarity see Subsection 2.

In that case, such geometrical object generator is called a prefractal [2] [4] [6]. The final result of such infinite procedure is the limit fractal or simply fractal [5] [8]. The initiator, prefractals and fractal represent the geometrical fractal set [13] [15].

Basic definitions and laws of fractal planimetry can be demonstrated on some classical fractal models [3] [13] such as Cantor dust, Sierpinski gasket, Koch triadic curves, Mandelbrot set, etc. For that purpose we chose the triadic snowflake Koch curve set.

The sequential construction of the this set can begin with the initiator which is an equilateral triangle of the edges length r 0 Figure 1 a. Figure 1. Koch fractal set. For the Koch prefractals, the length of a segment at the z th stage of construction r z and the number of segments at the same stage N z are respectively,. Similarity is a typical property of fractal sets. To define this concept we introduce a generating element of a generator. A generator is usually made up of straight-line segments for example, see Figure 1.

For example, the drawing in Figure 1 b can be subdivided into 3 generating elements, that in Figure 1 c into 12 generating elements, and so on. In our example shown in Figure 1 the generators of the Koch prefractals, at the first and second stages of construction, have the generating elements made of four equal segments each, as shown as details below the drawings in Figure 1 b and Figure 1 c.

Two successive generating elements of two generators of a set of objects can be geometrically similar or not. Fractal dimension is a quantitative measure of morphological complexity of an object. There are many different definitions of fractal dimension [2]. The similarity dimension is basic dimension related to all similar fractal sets but this dimension is not the subject of our study. The most important definitions are the Hausdorff and capacity dimensions [2].

These two dimensions are quite similar [2] , but the Hausdorff dimension is rather sophisticated, being a subject of mathematical measure theory.

The capacity dimension can be given by [2]. If we submit Equation 1 into the last definition Equation 3 and put z to tend to infinity, one can see that the expression in Equation 3 gives :. Fractal dimension D is the main quantifier to measure complexity of a set of geometrical and natural objects. The larger the D, the higher the complexity of the set is. For example, since the fractal dimension of the Sierpinski set is , it means that the last set is more complex than the Koch set Equation 4.

Considering Equation 3 it would be interesting to analyze a relation between N z and r z of the Koch fractal set in a log-log coordinate system. The fitting power function is:.

Figure 2. Log-log method applied to the Koch construction the beginning of the construction is shown in Figure 1. The fitting graph of relationship between the number of segments and segment sizes in cm is a decreasing straight line with the high coefficient of determination R 2.

This procedure can be thought of as the log-log method. This method is also used in many other fractal techniques, particularly in computer-based ones. Following the fractal methodology, we inscribe an equilateral triangle as the initiator in a circle of unit radius Figure 3 a and state the following iterate algorithm: from the middle of each triangle side the normal is erected to the sections with the circle.

These three sections are connected to the nearest vertices of the triangle sides forming a hexagon inserted in the circle Figure 3 b. The result of the next iteration Figure 3 c is a dodecagon. The next polygons can be constructed using the same algorithm. Inspecting the details below the drawings in Figure 3 b and Figure 3 c , which are the generating elements for these polygons generators , it is obvious that they are not mutually similar since the angles between the corresponding segments of the generating elements are different.

The condition of side ratios constancy geometrical similarity also fails. Namely, it is known that for a circle of unit radius:. If n tends to infinity, the polygons tend to the circle, which is not a limit fractal.

Therefore this set of regular polygons inscribed in a circle cannot be a fractal set. Peitgen, J. In nature, the self-similarity ends after a few orders of magnitude of zooming in. But we can give mathematical definitions with extremely simple rules that generate objects that are infinitely self-similar.

Here's an introductory example:. 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. 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.

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. 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. 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. The first few iterations of this procedure are shown below.