Fractals are those concepts in mathematics which are quite interesting because it uses the concept of algebra and the concept of complex numbers. Fractals are always made up by applying some functions. Thus, Fractals are those set of numbers which represent some self repeating properties.

Equation

There are two important types of fractal equation which are described by the two most widely used fractal sets which are:
1) The Julia Set
2) The Mandelbrot Set:
We know that in the field of complex numbers, there are real numbers along with some imaginary numbers. Any complex number is made up of a real number along with some imaginary number.
For example: (4 + i5) and (6 + i7) are some examples of complex numbers.
In the field of fractals, the real numbers are used for representing the x coordinate systems and the complex numbers are used for representing the y coordinate system. Therefore, if we take a pair of numbers as (5, 7), then in the field of fractals we will consider this number as (5 + i7).

Note that all the other properties of complex numbers like associativity and the distributivity will remain the same for the fractals also and thus can applied on the same basis as in case of complex numbers.
Basic Equation of Fractals: Now we will define a Julia set as a function which takes the square of itself plus some complex number. That is, a function $f(x)$ = $x$2 + $c$, here c and take any value. Different values of c create different types of Julia sets.For Example: consider $c$ = (1 + i1) and $x$ = (3 + i 1), then the function $f(x)$
= $f$ (3 + i1)
= (3 + i1)2 + (1 + i1)
= 3 $\times$ 3 + 3 $\times$ i1 + 3 $\times$ i1 + i1 $\times$ i1 + 1 + i1
= 9 + i3 + i3 + 1(-1) + 1 + i1
= 10 + i7 - 1
= 9 + i7.
Thus, in the first loop we get the number as (9 + i7) and the same step can be repeated again to get the next second loops answer.
Now for the second loop we will consider $c$ = (1 + i1) and $x$ = (9 + i 7), then the function $f(x)$
= $f$ (9 + i7)
= (9 + i7)2 + (1 + i1)
= 9 $\times$ 9 + 9 $\times$ i7 + 9 $\times$ i7 + i7 $\times$ i7 + 1 + i1
= 81 + i63 + i63 + 1(-49) + 1 + i1
= 82 + i127 - 49
= 33 + i127
The repeated answers of the above loops would show us the self similarity patterns of the fractals.

Types

There are different types of fractals in the field of mathematics and in nature too. Some of these are very easily created by using the concept of algebraic mathematical equations or by using complex numbers.
There are some other fractals which are created by nature. These are listed below:
The Julia Sets: These are those sets of fractals which are created by using the exact same formula as in case of the Mandelbrot set, except that the initial or the starting points are different every time. This implies, c is any constant with z1 as the initial or starting point on a plane. This definition implies that there can be infinite number of Julia Sets, having infinite number of values of c. In general, each and every point on a complex plane gives result to its corresponding Julia set. The following is an example of the fractals created by the Julia Set:

The Mandelbrot Set: This set is a subset of a complex plane which always consist of those variables or parameters from which any Julia set is joined with. In simpler words, Any Mandelbrot set is a set of those values for which the set has always one and only one finite upper bound.
The relationship between the Julia set and the Mandelbrot set is that the Mandelbrot set acts as an index set for the Julia sets. All the values of c, which are inside any Mandelbrot set, we will always be joined with the Julia sets, which will be connected. And conversely, the values of c that are outside the Mandelbrot set, we will always get unconnected sets.

The Kleinian Group Fractals:
This group creates those types of fractals which depends on two pairs of Mobius transformations and facilitates to create quasifuchsian, Single and double Cusp, etc. The following is an example of the fractals created by the Kleinian Group Set:

The Newton Method Fractals:
This method was discovered with Isaac Newton which was created by using the calculus applications, and gives us different kinds of interesting self similar patterns. The following is an example of the fractals created by the Newton Method Set:

The Quaternion Fractals (3D):
The quaternion fractals which are in three dimensional are made by the exact principle, in which the old Julia set is created with the exception that this quaternion fractals uses a four dimensional complex number in spite of the use of two dimensional complex numbers. The following is an example of the fractals created by the Quaternion Fractals Set:

There are various other types of fractals which can be visualize by making a link between a 2 dimensional fractals in a three dimensional or more than 3 dimensions. As in case of the discovery of Mandelbulb, the following is an example of some hyper complex and the other three dimensional fractals:

The above are some of the most common fractals that are being used by various researchers for creating various design patterns.

Theory

Fractals are made up of using very basic mathematical equations. The most important and basic fractal is known as The Mandelbrot Set, whose mathematical equation is $f (x)$ = $x$2 + $c$. This equation is also known as dynamic equation because here the result of the equation again has to put back into the above equation to get the next result. Hence, fractals work likes a cycle or a loop which works on its feedback. Because, the feedback or the result of the equation is again considered as the variable for the next equation.
Fractals are any image or set of numbers which shows self repeating properties. The most important and famous examples can be found in the nature itself.The method for creating any fractals using mathematical equations or functions is a very simple and yet interesting procedure. Whenever we see any picture which shows us some self similar pattern, this would be a fractal image and the screen on which the image is shown would be a plane which is made up of many points, known as pixels in the computer science field. Each point or pixel has some x coordinate along with its y coordinate, so as to identify its correct position on the plane. And as every point is situated at a different plane, therefore, every point has exactly different points of coordinates.
Thus fractals are those images that can appear on anything, be its nature, like lightening, rivers, shells, clouds, etc or be it computerized.

The theory behind fractals is that though fractals are made up in nature works almost in a similar manner as in case of fractals made up by the use of some algebraic mathematical equations using a self similar pattern.

The most important example of a fractal growth in the nature is the following leaf of a plant which shows how a fractal is growing on a leaf:

Thus we got to know that a fractal is any rough or fragmented shape or figure in any geometry which can be separated or again divided into its sub parts, in which each and every part is similar and exactly the same as to the whole image itself. Therefore, one of the most important and amazing property of fractal is that it has self repeating tendency in which the figure itself appears exactly in a same manner over the other figures. In the field of mathematics, the most important and widely used fractal structure is the sierpinsiki triangle, the Koch snowflake and the most famous of all is the Mandelbrot set.

Note that any fractal can be created by two factors. The first factor is that it can be created by following a self similar pattern, in which every curve of the figure follows an isotropic invariance of the scale. And the other which is the second factor is that, any fractal can be created by following a self affine pattern, in which every curve of the figure follows a dependent variance of the scale.

Examples

The following is a list of some of the common fractals which are widely applied in the area of mathematics and sciences:
1) Fractal as of Barnsley Fern: This fractal looks as the following which depicts its large extent of self similarity pattern:

2) The famous Box Fractal: This fractal takes the box shapes which are generated as follows:

3) The fractal made up by the Cantor Set: This fractal is made up by repeatedly cutting a line from its centre.
4) The fractal made up by the Cantor Comb: This fractal takes the following shape:

5) The fractal made up by the Cantor Curtains, Cantor Square and by the Cesaro Sweep.

6) The fractal made up by the Feigenbaum Fractal, by the Fudge flake, by the Henon Map and many more.

7) The fractal made up by the Hilbert Curve: The fractal made up by the famous Hilbert curve, which is being used by almost every researcher for further analysis is shown below:

8) The fractal made up by the Sierpinski Pyramid: This fractal takes the form of a triangle first and then results in a three dimensional space in the form of a pyramid as shown below:

9) The fractal made up by the Star Fractal: The following fractal shows how a star is talking its self similarity in a specific direction as shown below:

Analysis

The analysis of fractals can be understood, by first knowing that to create any type of fractal, we should use some algebraic mathematical function, wherein, that mathematical function is any given formula, showing the two coordinate systems, in which the input is inserted in one end, giving the new two coordinate systems as the output on the other end.
To create a Fractal, we start by taking a point on the plane, and then its coordinate are put in its one end of the equation, to get a new coordinate point from the other point of the equation. And then we select this new created point as our input for the equation. Now moving from the first point to the other point, we get another new coordinate point, and thus continuing the same pattern every time we get a new coordinate point.To create any fractal we should know about its scaling features. The nature itself is a more useful example of fractal geometry, as discovered by the famous scientist Euclidean. Most of the fractal objects which are found in the nature are only self repeating over various scales like the islands of coastline appearing on a large range. Similarly, in the field of mathematics there are various geometrical fractal objects which are self repeating on an infinite range of scales.
Similarly, in the field of geology, many geological circumstances like earthquakes, rock formation and fragmentation, volcanic eruptions, or in various oil fields, the concept of self similarity is analyzed, which shows that these things uses the concept of fractals, which can be further analyzed by various researchers using the concept of algebraic mathematical equations.
We define any set of fractal by the following notation:
$N$ = $\frac{C}{D}$,
Here we should know that $N$ denotes the number of objects or values in a given set,
$D$ denotes the dimension of the fractal, and $C$ is the constant of the proportionality.
Note that when the value of $D$, which is the fractal dimension, belongs to the set of integer, then this value of $D$ becomes equal to the dimension of Euclidean space.
For Example:
 If the value of $D$ is equal to zero then this fractal dimension will represent a point in a plane. If the value of $D$ is equal to one then this fractal dimension will represent a line in a plane. If the value of $D$ is equal to two then this fractal dimension will represent a square on a plane. If the value of $D$ is equal to three then this fractal dimension will represent a cube on a plane.

Thus, whenever $D$ represents an integer value then the dimension of the fractal always follows the Euclidean dimension. However, in reality, any fractal follows a dimension which is never an integer; in fact, most of the fractal follows fractional or rational dimensions.

Thus, the analysis of fractal requires an in depth knowledge of mathematics, calculus and other areas of science too.