Mandelbrot Set

The Sareoso diagram Ogi Almendra Orotasun looks like the Mandelbrot set, and in fact the name of the diagram Ogi Almendra is a bit of a word play, since it means Almond Bread, as does Mandel Brot.

The Mandelbrot Set is defined by very simple maths, but is endlessly complex. If you look at one part of it in detail, you see different shapes, similar but different, and the detail keeps on going forever. Who knows what we might find as we zoom in!

Here’s an example of some detail you can see (there are many other videos which look at different parts of the set, all looking different). It reminds me of a voyage through space!

 

What is the Mandelbrot Set?

Mathematics is used to determine if a point is in the Mandelbrot set or not. There is a particular simple calculation that is applied over and over again to the point. Points that are in the Mandlebrot set don’t move very far from the centre as this process goes on. Points that are not in the Mandlebrot set keep on moving further away as the calculation is repeated. (See the notes at the end for some more detail on the calculation).

In the Ogi Almendra Orotasun diagram, the dark blue area is the main part of the Mandlebrot set. The light blue surround and the other colours are points which are outside the Mandlebrot set – the colours show how quickly the points move away from the centre.

diag02sm

Ogi Almendra Orotasun

All the detail and complexity of the set is in the border.


Some Geometry

In the diagram, the two largest parts of the set are the heart-shaped cardioid area at the top, and below it the smaller circular area. It turns out that these have some interesting relationships. A cardioid can be drawn as the locus of a point on a circle as it rolls around another circle.

It turns out that the two circles that make the Mandelbrot cardioid are the same size as the circle below it, as illustrated below on a modified version of the diagram:

ogi-almendra

 


Notes: The Mandelbrot Set Calculation 

The calculation used to figure out if a point is in the Mandelbrot Set is simple:

  1. Start with the point you want to test as the value.
  2. Multiply the value by itself, and then add in the point again.
  3. Repeat step 2 with the new value.

In a one-dimensional example, I would test the point at 1.5 by working out:

Calculation 1: 1.5 x 1.5 + 1.5 = 3.75
Calculation 2: 3.75 x 3.75 + 1.5 = 15.5625
Calculation 3: 15.5625 x 15.5625 + 1.5 = 243.6914…

You can see that the value is rapidly getting bigger and bigger, so this point is not in the Mandelbrot set.

Testing the point at 0.1 gives:

Calculation 1: 0.1 x 0.1 + 0.1 = 0.11
Calculation 2: 0.11 x 0.11 + 0.1 = 0.1121
Calculation 3: 0.1121 x 0.1121 + 0.1 = 0.11256641

If I keep going for 40 more calculations I get 0.11270…, so you can see that the value is staying small, meaning it is in the Mandelbrot set.

In fact the borderline is at one quarter (0.25), which the dip of the cardioid in the picture!

The actual calculation uses two dimensions of course, so it is a little bit more complicated than the one-dimensional example, but only a bit – mathematicians use a scheme called complex numbers to work out the calculation.