Thread by @PulpLibrarian

Happy #FibonacciDay everybody!

Well I say everybody, but not everyone likes maths. But what about cool 80s maths like fractals? The Mandlebrot Set can be created in a few lines of computer code. But what are fractals?

Today in pulp I look at the Devil’s Polymer…

Benoît Mandelbrot was born in Poland and educated at the Paris Polytechnique. He was also fascinated by Johannes Kepler, the 17th century ‘father of planetary motion,’ and followed a more eclectic scientific journey that eventually took him to MIT and then IBM.

Like Kepler, Mandlebrot was interested in mathematical patterns, particularly the irregular mathematical shapes found in nature. From studying Zipf’s Power Laws to visualising telephone interference patterns his meandering studies led to his key insight: the fractal.

Fractals are recursive, and infinitely self-similar mathematical sets that we see regularly in nature. A cauliflower is a fractal: it’s made of florets, each of which looks like a smaller cauliflower. Strip down a single floret and the individual stems look like – a cauliflower!

So how far can you go before a natural fractal shape breaks down, and the cauliflower becomes something else? In 1967 Mandelbrot asked how long the coastline of Britain was. It’s not a straightforward question as Britain is a fractal.

From above the coast of Britain has smooth curves, but the closer you get the more irregular it becomes, and the more the rough patterns seem to repeat themselves as you reach the microscopic level. By 1980 Mandelbrot had the computing power to visualise such recursive patterns.

Complex numbers are based on the number i, defined as the square root of -1. Now this square root doesn’t actually exist, but 16th Century Italian mathematicians discovered that if you pretended it existed you could solve some quite complex equations. Which is always fun!

Better still you can create a two-dimensional space – the complex plane – that has real numbers along the x axis and imaginary numbers (computed using i) on the y axis. Any point on the plane represents a Complex number, i.e. it has a real and an imaginary component to it.

Why use a complex plane? Well it turns out there are a lot of things in the real world that are easier to understand if you use Complex rather than real numbers. From alternating currents to turbulance they help us understand irregular things.

You can use Complex numbers in a number of formulae; if you feed the answer back into the formula and repeat many thousands of times you’ll get a range of answers (including infinite ones – ignore them!) that you can map on to the complex plane.

Now look at the pattern…

Gaston Julia did this in 1915, and his Julia Sets show the complex fractal structure that can occur if you use a recursive formula. However Julia was limited in visualising this by the technology of the time. Mandlebrot however was at IBM!

Mandelbrot used IBM’s computers to work on the simplest of Complex equations: Z goes to Z squared plus C (where Z is a Complex number and C a Complex constant). He ran this several thousand times, feeding the result back into the equation. Then he mapped it on the complex plane.

The result, first visualised on 1 March 1980, was a bug-shaped image. But look closer at the edges: the patterns seem to be repeated, but more complex, and seem never-ending. The more cycles of the equation you run, the more of this detail you reveal.

A fractal has a definite geometry; there are parameters and rules in play. But the detail is seemingly infinite. Points very close to each other can show wildly different behaviour around the edges of the set. It is the mathematical expression of chaos.

In the Mandlebrot set there is always a path from one point of it to another but each section of the path is infinitely long and displays different detail at different depths. However the whole set is finite, and fits inside a circle of radius 2. It wasn’t geometry as we knew it!

In his 1982 book The Fractal Geometry of Nature, Mandelbrot suggested that these recursive patterns occur across the natural world; fractal geometry was the key to understanding turbulent systems and could be applied to everything from astrophysics to the stock market.

But it was the imagery, rather than the mathematics that caught the public attention. Soon fractal images were on posters, t-shirts and screensavers. It was the cool, crazy cutting-edge of mathematics.

At least for a while…

So what do we use fractals for? In life sciences it’s a useful way of mapping the growth and activity of dynamic systems. Fractal antennae help keep the mobile phone network going. It’s had a big impact on computer graphics: the Death Star in Return Of The Jedi is a fractal CGI.

Art criticism has looked at fractals in the work of Jackson Pollock, whilst literary criticism is now studying fractals in language and literature. What is the most fractal novel written so far? Finnegans Wake. Now it makes sense!

More maths another time… #SaturdayMotivation