Infinite Limits

In the intricate world of fluid dynamics, where the movement of liquids and gases forms the backdrop of countless natural phenomena, the unsolved Clay Institute Millenium Prize Problem of the Navier-Stokes equations stands as a beacon of understanding. These mathematical expressions encapsulate the intricate forces, pressures, and velocities that orchestrate the motion of fluids. George Gabriel Stokes Image Credit: Clay Mathematics Institute Fluid dynamics is a captivating branch of physics and mathematics that delves into the behaviour of fluids in motion. Whether it's the ocean currents, the streams, or the turbulence in a boiling pot, understanding the principles governing fluid motion requires strong mathematical concepts and physics insights. The Navier-Stokes equations, named in honour of the French engineer Claude-Louis Navier and the Irish mathematician George Gabriel Stokes, are the fundamental equations that explain fluid dynamics. In their general form, they can be expre...

The Mandelbrot Set is a set of complex numbers commonly used in art, due to its mathematical beauty. It was first defined by Robert W. Brooks   and Peter Matelksi in 1978, and the first visualisation was made by Benoit Mandelbrot in 1980.   Credit: American Scientist A lot of websites, videos, applets, and so on do a repeated zoom into the boundary of the Mandelbrot Set. But in mathematical terms, we define the Mandelbrot Set as the set of complex numbers which (under iteration) stays bounded within two. Here’s how it’s mathematically represented (don’t skip this, it’s easy math): f c (z) = z 2 + c, f c (z)<= 2 Let’s take an example here to understand this.   If c=-2,   f -2 (0) = 0 2 + (-2) = -2 (You start with z=0 as default) f -2 (-2) = (-2) 2 + (-2) = 4+ (-2) = 2 (You take the result of the previous function and input it as z in the next iteration) f- 2 (2) = (2) 2 + (-2) = 4+ (-2) = 2 We know that the result of the previous function (2), upon another itera...