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An Interesting Equation - And What Is It Actually Telling Us?

Updated: Oct 8, 2023

Author: Arpan Dey

Do you have an Internet connection? I bet you have, since that is how you can read this. Then, assuming you haven't, do type the following equation in an online graphing calculator (I used Desmos when I first learned of it).

The following screenshots are different portions of the graph of the above equation.

Are you surprised? You should be! How can such a simple equation have such a complicated graph? Well, the equation may not seem that simple to all of you, but it is relatively simple.

You may use this to impress your friend. But that sort of thing aside, what else can you infer from this? Well, simple inputs can have complex outputs and vice-versa. This is not supported by classical science, but is precisely a postulate of chaos theory.

Not so sure? Look no further than the Mandelbrot set. In case you don't know about it, it is actually generated by reiterating a simple equation on the complex number plane (Do a bit of studying online if all that sounds like jargon to you). The Mandelbrot set is, perhaps, the most complex image ever. It looks innocent from far. But zoom in - and pattern after pattern emerges. It never ends - it goes on infinitely. You don't know what is down there, or even if anything is there at all... Watch the following YouTube video for an idea!

[Orson Wang. "Deepest Mandelbrot Set Zoom Animation ever - a New Record! 10^275 (2.1E275 or 2^915)". 2010.]

Mathematics is a very fundamental subject. It is the purest form of human thought, and can be referred to as "applied logic" (just like physics can be called "applied mathematics"). It is true that we humans have developed the language of mathematics. "2" may mean what we refer to as by "3", in some parallel universe. But the underlying philosophy, as some believe, must be the same. Some have argued that the truths of mathematics are objective. They are true independently of any human activities, beliefs or capacities. This philosophy is known as mathematical realism.

It is, and may forever be, beyond the reach of our limited minds to comprehend this idea at a very deep level. But that does not take away the awe of learning and thinking about it...

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