The Mandelbrot Set

Computer generated graphics have improved enormously over the last forty years.

The first computer generated graphic of the Mandelbrot set \(1978\) :

OldTrout, Going Postal

A contemporary zooming animation :

I particularly like these new images and animations because they can help people see the beauty of mathematics and perhaps to understand it a little better and more deeply.  The numbers may be daunting but the beauty is there for all to see.

The coloured pixels in the animation above are not part of the Mandelbrot set which is black. They are the complex numbers which belong to the escape set \(E_c\) and are colour coded according to how many iterations it took to escape to infinity.  The complex numbers which belong to the keep set \(K_c\) are the Mandelbrot set.

Details from the zoom animation :

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

OldTrout, Going Postal

The Mandelbrot set :

OldTrout, Going Postal

A complex number \(z\) is of the form :

\(x + iy\),

where \(x, y \in \Bbb R\) and \(i\) is a symbol with the property that \(i^2 = -1\).

The real part of \(z\) is \(x\),  \(x = \) Re \(z\) and the imaginary part is \(y\),  \(y = \) Im \(z\).

Hence, \(z = x + iy\).

The set of all complex numbers is denoted by \(\Bbb C\).

Complex numbers can be thought of as based vectors and we are mainly interested in their distance from the origin.  This absolute value is denoted as \(|z|\).  They are often studied within a disc or an annulus.

The Mandelbrot set is a set of complex numbers contained within the disc of radius \(2\) centred at the origin \(0\) :

After various numbers of iterations :

OldTrout, Going Postal

A mathematical viewpoint :

OldTrout, Going Postal

The numbers are generated by iterating a complex quadratic function to obtain a sequence of terms.  Either these sequences are attracted to a fixed point and stay in the keep set or they escape to infinity and belong to the escape set.

The function for the Mandelbrot set is :

\(P_c(z) = z^2 + c\)

We take \(z = 0\) and \(c\) as any complex number in the disc.

The sequence of terms generated is thus :

\(c, \, c^2 +c, \, (c^2 + c)^2 +c, \, ((c^2 +c)^2 + c)^2 + c, …\)

When \(c = 0\), the sequence is :

\(0, 0, 0, 0, …\)

This sequence is constant with a super-attracting periodic cycle of \(p = 1\).  All the points in the cardiod are attracted to the point \(0\) and orbit around it.

When \(c = -1\), the sequence is :

\(-1, 0, -1, 0, …\)

This sequence is a super-attracting periodic cycle of \(p =2\).  This is the main disc.

When \(c = -1.755\), the sequence is :

\(-1.755, 1.325, 0, -1.755\)

This sequence is a super-attracting periodic cycle of \(p = 3\). The West Midget is \(p = 3\) as are the discs directly above and below the cardiod.

When \(c = -1.311\), the sequence is :

\(-1.311, 0.408, -1.145, 0, -1.311\)

This sequence is a super-attracting periodic cycle of \(p = 4\).  An example is the disc to the left of the main disc.

The periodic cycles :

In more detail :

Bifurcations occur :

It maps to this :

The cobweb is the iterations of a quadratic function.

What happened to the escape set?

When \( c = 1\), the sequence is :

\(1, 2, 5, 26, …\)

Off to infinity along the real axis at the third iteration.

When \(c = i\), the sequence is :

\(i, -1 + i, -i, 1 + i, 3i, …\)

It orbits around \(0\) at first and escapes to infinity at the fifth iteration.

The Buddhabrot :


 

© OldTrout \(2018\)