Chaos Theory
Chaos is everywhere around us. Fractals are born out of chaos. They are the patterns and orders formed in chaos. Fractal geometry is a part of a wider topic called the chaos theory. So, what is ‘chaos’ and what is ‘chaos theory’ all about?
Fractals is born out of a bigger theory called ‘Chaos theory’. So, what is this chaos theory? Let’s play a small game to find out…
[I first saw Chaos Theory explained by this game on the YouTube channel Numberphile and it fascinated me a lot. So, I’ll try my best to reiterate and explain it based on my understanding.]
Let’s call this game, the chaos game. So, what do you need to play it? Just a blank sheet of paper, something to write with and a rolling die.
Now, let’s begin,
We’ll start with marking three dots on the paper, almost like the three vertices of a triangle. Name the dots A,B and C. Draw another point as the starting point. Now roll the die, if 1 or 2 comes up, then, mark a dot between the starting point and A. But, if 3 or 4 comes up, then, mark a dot between the starting point and B, or otherwise, mark a dot between the starting point and C, that is, if 5 or 6 comes up.
Say 2 comes up, so you mark a dot between the starting point and A.
Then you roll the die again. Suppose 2 comes up again. Then you mark a point between the previous dot that you had put and A. Then you repeat the process.
After some trials, let’s say you get 4, you mark a point between the previous point that you had marked and B.
You get the idea, right?
But doing this for sometime, you would only notice some random and chaotic dots on the sheet. But, what if these random dots lead us to something unexpected? To see some results, we can ask a computer to do this. There are many websites where this is possible. I tried one of them. (To try yourself, click here). In that, I set the number of points to be 3 and clicked start and set the speed to ‘fast’. The starting point, or referred here as the trace point, started moving, marking points. After waiting for about 5 mins, this was what I had noticed:
It is indeed the Sierpinski Triangle! Random dots and chaos lead us to such an order and symmetrical pattern. This is a part of Chaos theory. Chaos theory is usually defined as,
“The branch of mathematics focusing on the behaviour of dynamical systems that are highly sensitive to initial conditions.”
So, the initial conditions in this case were the three dots from where we began and the rule by which we were marking the dots. If instead of 3, you begin with 5 dots, then you would get a different pattern. In chaos theory, apparent randomness leads to various patterns, fractals, symmetry, etc. The Barnsley fern can also be created through the chaos game.
(To try yourself, click here).
Surprisingly, the concept of Chaos theory wasn’t first discovered by a mathematician, but, by a meteorologist named Edward Norton Lorenz. He was a mathematician dressed in the coat of a meteorologist. It all started by a concept that became very famous and was used in books and movies. This is the Butterfly effect.
The Butterfly effect basically says that the flapping of the wings of a butterfly may cause, several weeks later, a tornado on the opposite side of the world. Seems impossible, doesn’t it? Specifically, it is the sensitivity of dynamic systems to initial conditions. The Butterfly effect was a concept discovered only due to approximations.
Weathermen are known for making approximations about the weather and being hilariously inaccurate about it, but, it is not entirely their fault. It’s due to the Butterfly effect.
Edward Lorenz worked in the weather forecasting department. He was one of the forecasters there. He had initially found some values related to the forecasting that had about 6 decimal places. Feeding this to his computer gave a particular graph. Then, he fed the same values again. Only this time, the values were rounded off and approximated to 3 decimal places. Initially the graph was the same. But after sometime, the second graph had so much difference and variation compared to the first that the forecast changed completely. Just a difference of about 0.001 in the values caused so much change in the graph. This is the Butterfly effect. This was where the entire idea of chaos theory was born.
Fractals is a part of this theory of chaos. The chaos theory was also summarised by Edward Lorenz as:
“Chaos: When the present determines the future, but, the approximate present does not approximately determine the future.”
