Mandelbrot lemniscate


last updated: 20040521 
where f_{n} is defined by the quadratic recurrence relation f_{n+1}(z) = f_{n}^{2}(z)
+ z with initial f_{0}(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 2^{n}.
The first three Mandelbrot lemniscates are:
 n = 1: circle
because for the unity circle holds that  f_{1}(z)  =  z  = 1
 n = 2:
Cassinian oval
 f_{2}(z)  =  z^{2}+z  = 1 leads to a Cassinian oval
(a=4) ^{1)}.
 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
set.
notes
1) We can write z^{2}+z  = z z+1 =1 as (x^{2}+y^{2})((x+1)^{2}+y^{2})
=1. Converting this into the Cartesian form of the Cassinian oval leads to:
((x1)^{2}+y^{2})((x+1)^{2}+y^{2}) =4^{2}.
