Calculus to Fractals
In the previous article I spoke about the Euler’s constant. This constant is an important part in calculus. Well, what is calculus? The twin tools of calculus are integration and differentiation. I know for a student in school terms like ‘calculus’, ‘integration’, ‘differentiation’ sounds like a big deal. But don’t worry! These words are very simple to understand. Differentiation in simple terms is finding the slope of a curve, that is, rate of change of a variable and integration is finding the area under the curve, that is, summing the rate of change at each instant of time. Integration is the reverse of differentiation and vice versa. That is all.
Newton and Leibniz were the ones who discovered this world of calculus and it became extremely successful. They discovered that by combining these changes defines the evolution of the system. Newton’s three laws of motion and the electromagnetic equations of James Clerk Maxwell followed from this discovery. The physical sciences were transformed. It was assumed that all phenomena could be understood in terms of these new techniques. Pierre Simon Laplace claimed that given the position of every particle in the universe and its rate of change, we could predict the entire future of the universe forever in its every detail.
The methods of the calculus apply wherever a curve is smooth. It was thought that any curve with bends or kinks could be split into separate smooth curves that would then succumb to calculus. That any curve could have only isolated corner points was not even questioned.
But, there is no rose without thorns. And this thorn was Karl Weierstrass. Always in a mood to find flaws in other’s arguments, he found a curve that was nowhere smooth and was filled with bends, corner points and kinks no matter how much you magnify the curve. Mathematicians were stumped looking at such an unusual curve that was nowhere differentiable. So, they just avoided it and called it “pathological”.
There were also many strange shapes that were discovered that did not follow any rules of the Euclidean geometrical shapes. The shapes such as a cone, cube, sphere, etc. are smooth shapes. These strange shapes were not smooth they had a very rough surface. Thus, the classical geometry rules could not be applied to this. Slowly, there were so many curves and shapes discovered that were irregular and rough, that it formed a completely new geometry of it’s own.
Benoit Mandelbrot, coined a name for these figures — fractals. This was derived from a Latin word ‘fracta’ which means “to break” or “fragmented”. And this new geometry discovered was called “Fractal Geometry”. Benoit Mandelbrot came to be called as the Father of Fractal Geometry. He said that this was the part of mathematics that was very close to nature and consisted of shapes that were really present in nature like the clouds, the coasts of lands, leafs, tree branches, etc. Nature obviously does not contain perfect squares, circles, or rectangles. Leafs are not triangles and mountains are not cones. Nothing in the real world is smooth. Even YOU are a fractal. This is one of the biggest reasons why fractal geometry has proved to be a more natural study of the world.
Not all were satisfied with this discovery. They called all the geometrical figures, shapes and curves in it as ‘monsters’. Monsters that were destroying pure mathematics. But today, this “Gallery of monsters” has been converted into a “Museum of Science”.
