Mandelbrot Set- The bottomless wonders of fractal geometry

In our school days whenever a math problem would start creating “the problem” in our minds making us scratch our head, there was this common expression showing the worthlessness of learning all this math if we cannot implement it in real life. One would say – “Where do I use these sines, cosines, tangents while doing my groceries?” or “What is the use of solving this calculus problem if I see no use of it? – I never needed calculus to do my routine works?” And these questions have one important thing that detaches most of us from the mathematics and the rigor it demands.

That thing is the great difference between of regularity, consistency, idealistic nature of the mathematical laws, formulae, theorems and the irregularity, inconsistency, chaos, non-ideal nature of the things around us, the reality we live in. We have reduced this difference by using tolerance, probability, acceptable limits of accuracy, modelling.

Still mathematics always seems to be out of reach and only the intellectual venture for almost all of us. The reason is (again) the feeling of incoherence between math and the real world. Today, I will dive into one such idea and the person behind it named Benoit Mandelbrot who bridged this gap in a big way, revealing the secret mathematical ways in which the real-life objects and the nature works.  

What we learnt in Schools – The Euclidean geometry

Point, lines, triangle, square, circles, parallelograms – these are the shapes that we have always crunched our fingers in our school day. The are based on the Euclid’s geometry where the shapes are largely regular sizes. Hence, the mathematics is very structure and there are logical, structured ways, theorems, formulae to work with these figures, hence we can say that there are always some ways available to figure out the unknown information.

But you rarely see perfect geometric objects in real life. Real life objects are not perfectly shaped the way we learn them in an ideal nature hence we are expected to have the tolerances in our calculations. Now, here are some questions – “Can you describe the size of the cloud floating in the sky in a mathematical way?”, “How would one describe the branching of a tree in a mathematical equation?”, “How would one describe the path traversed by the lightning strike in the sky?” (Again, what good is math if I couldn’t implement it into the real world?)      

Hidden Dimensions

We all have awareness of 3 dimensions, it is the way we experience the real world. We also know that there is fourth dimension in nature but out senses are limited to 3 dimensions. There was a problem that few were working on called the ‘Space Filling Curves’.

How would one draw a line so that it will touch every on the plane?

Giuseppe Peano was the first person to discover a curve which would cover every point on the plane with a small repeating unit pattern- curve. This basic unit on repetition and angle of rotation would cover all the plane.

After that, many others like David Hilbert, Henri Lebesgue, Wacław Franciszek Sierpiński demonstrated different types of space filling curves where you can see a simple repeating unit when iterated enough times can fill the complete area in a given plane.

Theoretically, a curve or a line has single dimension further clarifying, it has no thickness. Then, how can we explain the idea of curves which when iterated enough in a set direction yield a higher dimensional geometry. This is where the fun really began.

We know how addition of one dimension enhances the geometric description of every object. Thus, we are aware of 1D, 2D and 3D objects. From a line to square to cube that is how it works. But, as we have seen in the last idea of space filling curve, how would we fit the idea of 1D curve of zero thickness curve filling the 2D plane? How would one describe the events happening between 1D world and 2D world?

Hausdorff dimension

From the premise of above problem, where we were stuck was our understanding of the nature of the dimensions- we were considering the dimensions to be a whole number. What is happening between ‘space filling curve and the space they are occupying’ is between 1D and 2D – indicates that there are some dimensions which lie between each whole number dimension. This led to the non-whole numerical concept of dimension. This means that the dimensions could be fractions.    

This created whole new world of dimensions which can be described as hidden dimensions between two whole number dimensions. For example, when we look at a tennis ball, it is spherical in shape but when we zoom significantly enough, we find that the boundaries of this ball are not perfectly continuous rather the boundaries of the ball are made up of small polymeric fur-like hair. Even when we would zoom into the perfectly round rubber ball its roughness is revealed at significant observation level.

Cantor Set – Smaller Hidden Infinities

Henry John Stephen Smith, an Irish mathematician described idea of splitting a line into three parts, then removing the middle part. Then, after one has removed the middle portion, he will split the remaining tow parts into three and again remove the middle part from it. This will be iterated until the sum of the line remained will be zero. The final remaining parts coordinates are the part of the Cantor set.

One can easily find out that even after we keep on splitting the line into three equal portions and deleting the middle portion, there is no way we will stop on this. Every new part generated will be split into the same style it was split earlier indicating iterative process. The iterations will go on to infinity thus for every new iteration some new coordinates will be added indication that the Cantor set has infinite elements but the length of the line will eventually become zero as we are infinitely removing 1/3^rd part from the set continuously. Sounds weird already!

What I explained above are mathematical ideas purely generated from the human mind, thought experiments and their logical explanations. These ideas remained ideas because nobody could find a way to find their implementation in real life world. There was no coherence of these mind-boggling ideas to the real-life objects around us.     

Benoit Mandelbrot and The Rise of Fractal Geometry

There were many events in the life of Benoit Mandelbrot which converged to the development of Fractal Geometry. The world ‘fractal’ which has now become common in pop culture and sci-fi was actually coined by Benoit Mandelbrot himself.

Mandelbrot was interested in a problem related to the length of the coast of Britain. He found this idea in a research paper by an English mathematician, meteorologist Lewis Fry Richardson. According to Richardson if we were to measure the length of the total coast line of Briton the results will very based on the basic scale used for the measurement. As we go on taking practically smaller and smaller scale the length of the coastline will approach closer and closer to the realest of real length of coastline.

There is one tipping point to this idea that when one goes on further decreasing the length of the scale used for the measurement of the coastline, the final value will go on increasing- this is also known as the coastline paradox.

Mandelbrot found a good connection between the idea of fractional dimensions and the coastline paradox. He developed the idea of roughness of the objects. He devised this theory of roughness into a computer program.

Mandelbrot while working in IBM developed a computer program and a makeshift computer graphics generator to check whether his equation fits to the empirical equation given by Richardson. Which lead to the confirmation of the theory of roughness and the fractal geometry created by Mandelbrot.

The Mandelbrot Set

To understand the formation of Mandelbrot set one needs to understand the idea of iteration, convergence/divergence and the complex numbers. An iteration is repeated step which is connected to the output from the previous step. Iterations are wildly used in approximations.

Convergence in iteration is the condition where the output will reach to a fixed value after some continuous iterations. Further iterations will not change the output. The output becomes iteration independent.

Divergence in iteration is the condition where the output continuously goes on changing as the iterations go on increasing and there is no iteration independent output.  

Complex numbers are these numbers which are developed to solve the square root of negative numbers which were identified during the quest to find the roots of cubic equations. For, solving such problems where one needs to find the square roots of negative numbers, imaginary number called ‘i’ was created.

Though it is called as imaginary number, it is as real as the real numbers and has real life significance and applications. Every complex number is made up of imaginary and real part. The real part is represented on X-axis and imaginary part is represented on the Y-axis.

Mandelbrot set is the set of complex numbers ‘c’ for which the function given below, does not diverge to infinity, the sequence remains bounded to a fixed value.

For example,

For  z=0 and c=1,

and so on…

As we see from the outputs of the iterations the sequence is diverging, meaning it will not settle on a constant output. The sequence grows as the iterations go on, hence  does not belong to Mandelbrot set.

Similarly,

For z=0 and c=(-1)

and so on…

Here, we can clearly see that the output from the iterations is stable and just after few iterations it only fluctuates between values of 0 and -1. This means that the iteration output remains bounded to 0 and -1. Hence, c=(-1)  belongs to Mandelbrot Set.

This algorithmic work of iterations and checking for each possible complex number can be fed to a computer program. The graphical representation can be shown as follows.

The output image itself is mesmerizing. If you could point out that looks like created out of itself. The significant shapes are seen repeated everywhere you can observe. But what are real life applications of this outputs?

Mandelbrot Set in real life

The real fun begins when we identify the number of iterations it too for every number in Mandelbrot set. For example, if we assign a color spectrum from low contrast to high contrast and associate the increase in contrast to increased iteration to converge, we get following image.

Here, you can see the contrast bands around the Mandelbrot set.

Further if we get a complete color spectrum and correspond the number of iterations to the spectrum what we get is phenomenal. The colors and the pattern you will find in the Mandelbrot set are the patterns that are found in nature. There are many YouTube videos dedicated to zooming into specific regions of Mandelbrot set. In every area you will find different shapes and sizes, yet you will discover that they are made up of same basic shapes and that is the beauty of fractal geometry.

The equation given by Mandelbrot is also simply indicated as follows:

This simple equation has yielded son many and infinite number of patterns that when you see the Mandelbrot set zooms you can confirm that nature thinks in the ways of mathematics.

The mesmerizing patterns from the Mandelbrot set zoom:

The development of computers and image processing created an explosion in the world of Mandelbrot set. Few lines of code with such a simple equation can yield such a variety of patterns and these patterns are spitting image of the shapes that are available in nature. Only single video of Mandelbrot set zoom is sufficient to believe the coherent nature of nature and the mathematics behind it.

Mandelbrot’s ideas and his fractal geometry are popularly used in the many fields. The nature of branching of veins, branches in lungs, branches in liver, kidneys have been computer modeled and simulated using Mandelbrot’s Fractal geometry. This helps the doctors to identify the critical branches to be controlled and operated during the procedure. The pattern created by the lightening strike can be modeled mathematically by using the fractal geometry. The global satellite imaging has been largely improved by the use of measure of roughness and fractal geometry. Event the random motion of the atoms- Brownian Motion can be explained by using fractal geometry. The irregular but more natural, organic and realistic images are created using the fractal geometry. Even the string theory which believes in the requirement of 11 dimensions is trying to find the answers possibly in fractal dimensions.

The Buddhabrot

On further expansion of the visualizations of the iteration sin the Mandelbrot set, one interested rendering emerges out which is famously known as Buddhabrot. Because, the image looks similar to the Buddha sitting in his meditative state and the particular curls of meditative Buddha’s hair.

When we track the number of iterations it takes for the given complex number to diverge to infinity and also include that spectrum of iterations, it yields the Buddhabrot.

The most important thing about Benoit Mandelbrot’s contribution to the mathematics is the genius of his mind which bridged the gap of completely abstract and so called ‘pure mathematics’ to the irregular, organic, physical and rough nature of the mathematics of real-world objects.

The extent to which Albert Einstein disrupted the world of Physics with his equation  is the same extent of disruption created by Benoit Mandelbrot to the world of mathematics through fractal geometry.  

(P.S. – Interestingly both the equations are of square nature, actually the complete energy-mass equivalence equation of Einstein contains additive term to the square, represented as:

here, p is the momentum of object under consideration)

So, nature does think in the language of the mathematics.