Mandelbrot Sets
The Mandelbrot set is probably one of the most popular fractals of all, possibly even the most
popular object of contemporary mathematics. Since Mandelbrot made his experiment in 1979, it has
been considered one of the most beautiful objects ever seen. The complexity of the Mandelbrot
set is in an altogether different class compared to that of Julia sets.
Consider a family of functions parameterised by a variable. Although any family of functions can
be studied I have chosen quadratic polynomials again for convenience. f(x) = x2 - z, where z
is a complex number. As the complex part z varies, the Julia set will vary on the complex plane.
Some Julia sets will be connected, and this causes a partition in the z-parameter plane. The
values of z for which the Julia set is connected is called the Mandelbrot set in the parameter
plane. The boundary between the Mandelbrot set and its complement is often called the Mandelbrot
separator curve. The Mandelbrot set is the black shape in the picture (below). x is an element
of [-1 to 2] and y is an element of [-1.5 and 1.5]. There are some surprising details in this
image, and it's well worth exploring.
With the simple algorithm f(x) = x2 - z, it is possible to separate the points of the complex
plane into two categories:
Points inside the Mandelbrot set. (The black cardioid discussed below)
Points outside the Mandelbrot set.
The bulk of the Mandelbrot set is the black cardioid. (A cardioid is a heart-shaped figure). It's
studded with circles all around its boundary. There are an infinite number of circles on the
cardioid, and each of these circles again has an infinite number of circles on their edges, and
on and on to infinity.
A Julia set with the parameter z, which is part of the circle on top of the
cardioid, under close examination, reveals strands of dark blue above the
original circle.
At the end of each of these strands it is possible to find further cardioid shapes, which also
have the same pattern of circle. Below is the main cardioid shape of the Mandelbrot Set:
with x in [-1.0,2.0];
y in [-1.5,1.5].
Next: What is the difference between a Julia Set and A Mandelbrot Set?
