## SGU: Science Picture of The Week: Mandelbrot Set

There’s nothing quite like a beautiful scientific image. They are just so appealing on so many levels.

- For pure aesthetics, many are simply gorgeous.
- Under that layer, there’s often a specific scientific point or new discovery being showcased.
- Sometimes the image conveys complex information effortlessly putting any similar attempt using just words to shame.
- Often, the creation of the image is a fascinating topic in and of itself.

Every week I’m going to cull the web in search of a scientific image that I think fulfills as many of these as possible. Along with each image I will include a description of why I think the image is fascinating and the science behind what is being displayed.

The beautiful image above is one most people have probably seen. It’s the iconic Mandelbrot Set which has been called by Scientific American the most complex object in mathematics. This shape is an example of a fractal. The hallmark of this peculiar breed of shapes is that they are self-similar at all scales. That means you will see similar shapes regardless how close you get. These forms exist throughout nature from coastlines, mountain-shapes, capillary branching etc

This visualization was discovered by Benoit Mandelbrot in 1979. This claim is a bit controversial though so I will include Robert Brooks, J. Peter Matelski, and John H. Hubbard as co-discoverers. There is no controversy though that Benoit was instrumental in getting the Set that was named after him firmly in the consciousness of the world (and on its t-shirts).

Part of its wonder is that such complexity can come from a simple equation using complex numbers (z = z^{2} + c). More specifically:

The Mandelbrot set is defined to be that set of points c such that the iteration z = z

^{2}+ c does not escape to infinity, with z initialized to 0.

To create the image, you plot on a graph the various values of z. The image that forms appears to be a weird beetle shape; a series of distorted and connected circles. As you peer closer however, the outline comes to life as a shimmering infinitely intricate wonderland. Throughout your never-ending journey into this image though, you will come across again and again, variations on that same weird beetle.

Many like Dennis P. Sullivan of the City University of New York, believe the image itself had tremendous significance in the scientific community. Her refers to it as a “crucible” for learning about how chaotic or dynamic systems behave. He said: “It is really quite fundamental”

Not bad for a weird Beetle.

Image Credit: http://www.fractalposter.com/users/haeygen-1030.html

You really should include the fact that z and c are complex numbers.

Nice to see a mathematical subject on the SGU site. Maths has been a rather neglected area in the SGU up to now, depite some interesting new discoveries that should be worth a mention. For example, the discovery of a new Mersenne prime number back in 2013, and a recent controversial “proof” of the “abc” conjecture. The maths of these are very difficult, but the basic ideas of the problems are quite easy to state. There have also been advances in the twin-primes conjecture.