Mandelbrot lemniscate

where f_n is defined by the quadratic recurrence relation f_n+1(z) = f_n²(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: Cassinian oval
  | f₂(z) | = | z²+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 |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: ((x-1)²+y²)((x+1)²+y²) =4².