||last updated: 2004-05-21
where fn is defined by the quadratic recurrence relation fn+1(z) = fn2(z)
+ z with initial f0(z) = 0.
These curves are called equipotential
curves by Peitgen and Saupe (1988).
In general, the curve is a polynomial in x and y of degree 2n.
The first three Mandelbrot lemniscates are:
- n = 1: circle
because for the unity circle holds that | f1(z) | = | z | = 1
- n = 2:
| f2(z) | = | z2+z | = 1 leads to a Cassinian oval
- n = 3: pear curve
Although the resembles more a bulb, the curve has been given
the name of the pear curve.
When n goes to infinity, the curve is the Mandelbrot
1) We can write |z2+z | = |z| |z+1| =1 as (x2+y2)((x+1)2+y2)
=1. Converting this into the Cartesian form of the Cassinian oval leads to: