Cardioid and Nephroid – and so on

Created with the iPad app Fractals

The image above shows you eight fractals, all Mandelbrot sets ranging from z^2+c to z^9+c. As you can see the number of petals in each set is the exponent of the imaginaire component minus one. So the z^9+c fractal has 9-1=8 petals.
The one with one petal below (z^2+c) is called a cardioid, because it resembles a heart.

Cardioid

You can see a cardioid every day when you look into your coffee cup and watch the reflection
Credit: ThatsMaths blog

If you construct it as a graph the function looks something like

    \[r(\theta)=c(cos(\theta)-1)\]

(with some parameter c along the way) as you can see below (with c=5).

This one below is called a nephroid, because it resembles a kidney.

nephroid

Why do these fractals end up as circle-eque? It’s because they are created in the complex plane. As you might know multiplication (and thus exponentiation) can be seen as rotating in the complex plane. The most basic explanation for this can be illustrated by the sequence 1,1i,1ii,1,iii,1 thus 1,i,i^2,1^3,1 or 1,i,-1,-i,1 as seen below.

Going from 1 to i and so on ends up being a counterclockwise turn through the complex plane.

And as a last illustration you can see the effect when you create a light source in a circle. There are a few leaks I couldn’t fix 🙂 but I hope the idea sticks.

Created with Ligthwave Studio on the iPad