Mandelbrot Set

The Mandelbrot set is probably the most famous mathematical fractal for its beautiful and 'trippy' details. This design is a representation of the first published visualisation of the Mandelbrot set from 1978, since it is more approachable to represent in embroidery!  (Scroll down to see the original image).

How does this design represent maths? It is a visualisation of the properties of a specific equation that acts on complex numbers. To understand the Mandelbrot set, you can think of a complex number a bit like a coordinate on a standard (x,y) plane. That is, we have two numbers that correspond to a point in our 2D space. We call the combination of these two numbers z, and there are special rules for how complex numbers are added and multiplied. 

The Mandelbrot set is governed by the equation zn+1=(zn)2+z0. Here’s how it works: you start with a point called z0, then square it and add the result back to z0 to get z1. You keep repeating this process, using the new result each time. If the answers do not grow forever (for example, if there is a repeating cycle of answers or the size of the answers shrinks), the original point is in the Mandelbrot set. All the values inside the outline (usually coloured black) are in the Mandelbrot set. If the answers keep growing forever, then z0 is not in the Mandelbrot set.  The colourful areas outside the set are these points. They are coloured by how quickly the answers grow.