Fractals and it’s dimensions
Fractals are crazy shapes that show order and patterns in chaotic designs. It has a lot of fascinating curves. These interesting patterns have been individually studied due to it’s unique properties. One of them is the Sierpinski triangle.
The Sierpinski triangle is basically an equilateral triangle that is divided into four equilateral triangles (as shown in the image below) and the central triangle is removed. Then those sub-triangles are again similarly divided into four equilateral triangles and the central triangle is removed. This process is iterated infinitely and in the process, the complex triangle received is the Sierpinski triangle. Now, if in a Pascal’s triangle, all the odd numbers are coloured black and the even numbers are coloured white, then what you finally receive is the Sierpinski triangle. Unexpected, right?
Fractals was not only random shapes or patterns mathematically created. It was also seen in the population graph. It was observed that food was increasing linearly, but, population was increasing in an exponential manner. It was later discovered that population did not keep increasing in this manner. It increased for a few years, then due to deficiency of food and resources, it again went down. This population changes followed a simple function,
[Let the above equation be labelled (1).]
Where, X is the population of the present year and X_next is the population of the year after X and r is some constant which can be adjusted according to the population being modelled. For observing long-term behaviour of systems, this formula was repeated over and over and over again and to see what happens. This process is called iteration.
Equation (1) is plotted by taking ‘r’ as 3.5 and assuming with a hypothetical situation that the value of X is only between 0 and 1, and iterated infinitely. The following was the graph obtained:
This graph was considered as a fractal as it showed the property of self similarity in it. When you zoom into ‘window of order’ of the graph, which is the wide gap in the graph, you will notice the same original graph again present in that window. The more you zoom in, you find the same graph again and again in the window of chaos. This Fractal was referred as ‘The fig tree’.
As I had mentioned in one of my previous articles Fractals are shapes that are rough and irregular. This roughness and irregularity can be easily calculated. How? By calculating their Fractal dimension. Felix Hausdorff and Abram Besicovitch found out that fractals had non-integer dimensions. They described that the fractals are curves that have dimension ‘in between’ the integer dimensions. These fractal dimensions are, thus, also referred to as the Hausdorff-Besicovitch dimension. But how to calculate these dimensions? There are two main methods that can be used to easily calculate the dimension.
One, by using the property of self-similarity that fractals posses. Let’s take shapes with known dimensions 1,2 and 3. For dimension one, let’s take a line of length 1 unit and scale it down to 1/4th it’s original length. So, it’s length now is 1/4 units. To obtain the original length, we have to multiply that 1/4th of the line four times. Let the factor, the line is scaled down by, be ‘s’, the number to which ‘s’ is multiplied to obtain the original length be ’n’ and the dimension be ‘D’. Thus, you would observe that in this case,
This formula is valid for any dimension. Suppose we try to prove this by using the area of a 2 dimensional shape. So, let us scale down each side of a square having unit length to 1/2 it’s original length so that it’s area is scaled down by. 1/4th. Thus to get back the original square, we need to multiply the scaled down square 4 times.
Thus, D = 2, which was the dimension required.
Similarly, it can be proved for a 3 dimensional shape.
Thus, the general equation found is,
Equation (2) is one of the formula that can be used to find fractal dimension of a shape. Now, suppose we take a Koch curve,
With the above given values of n and s, if we try to calculate its fractal dimension with equation (2), we approximately get 1.26 . This is the dimension of the fractal, Koch curve.
Two, by using a grid counting method.
In this method, you need to just draw grids on the fractal image, each box in it having a scale of 1 unit. Then again draw a grid on it, but each box having a scale of 1/2 this time. The again, with each box having a scale of 1/4. Count the number of boxes through which the fractal is passing. You can the calculate the dimension using the following formula,
where n( ) is the number of squares containing the image and 1/s is its grid scale. We can now calculate the Dimension of the Koch curve. Given below are three grids of scale in the ratio 1 : 1/2 : 1/4. By counting, the number of boxes of the first, second and third grid was found to be 18, 41 and 105 respectively.
Calculating dimension using the grid of scale 1 and 1/2,
Calculating dimension using the grid of scale 1 and 1/4,
Calculating dimension using the grid of scale 1/2 and 1/4,
By finding the average of these three values, it was found to be approximately 1.27. This is close to 1.26 which is the original dimension of the Koch curve.
Thus, these are two simple ways in which you can calculate the fractal dimension of a fractal image.
