The Mandelbrot Set
A strange and haunting mathematical construct with endless complexity
The image is a computer generated plot of one of the most remarkable and intriguing structures ever to emerge from the depths of mathematics - the “Mandelbrot Set” -named after the Polish born mathematician Benoit Mandelbrot (1924 - 2010).
The Mandelbrot Set is a self-similar fractal.1 Zooming in reveals copies of itself. Take a look ….. Hardest Mandelbrot zoom in 2014, 10^198 - 350 000 000 iterations (youtube.com).
One might imagine that some extremely complicated equation is required to generate the Mandelbrot set but that is not the case. It is basically generated from:
f(z) = z2 + c
f(z) is the symbol for a function of z and f(z) is equal to z squared plus a complex number c.
A recursive method is applied to that function.
Complex Numbers:
Any complex number2 can be written in the form a + b i where a and b are real numbers and i is √-1. A complex number in that form can be illustrated using what is known as an Argand3 diagram. This is based on a complex plane in which the horizontal axis is the “real axis” and the vertical axis is the “imaginary axis”
Either a or b could be zero and so every number is actually a complex number. For instance, in a + bi we can make the real part a = 4 and if b = 0 the outcome is simply the real number 4
Recursion:
Recursion - (mathsisfun.com) - is the application of a rule or formula to its own result, again and again.
To generate the Mandelbrot Set, a starting value (e.g. z = 0) is applied to the function f(z) = z2 + c. This reaches a result and then that result is applied to the same function. The process is repeated.
Either the value of f(z) gets continually larger and larger (i.e. it diverges “escapes” to infinity) or it does not.
The Mandelbrot Set is interested in the values of f(z) which do NOT escape to infinity.
Exploring f(z) = z2 + c
Example A:
Let c = -2. Then, f(z) = z2 - 2
Beginning with z=0 has the result that f(z) = -2.
Next, make z = -2 and f(z) = 2
Make z = 2 and then f(z) = 2.
From here, f(z) will always have value 2.
Example B:
Now make a small change so that c = -2.01 and f(z) = z2 - 2.01
Again, begin with z = 0 and apply the recursive method. The results obtained (to 3 decimal places) are f(z) = -2.01 and then 2.03, 2.111, 2.448, 3.981, 13.837, 189.448 etc. The results get ever larger. In this case f(z) diverges to infinity.
In fact, for any real number less than -2, f(z) will diverge to infinity.
Example C:
Suppose f(z) = z2 - 1. Begin again with z=0 and f(z) is then -1. Now make z = -1 and then f(z) = (-1)2 – 1 = 0.
z again becomes 0 and the outcome is an endless cycle with the value of f(z) alternating between 0 and -1. It is said that the orbit of f(z) is between 0 and -1.
Example D:
If c = 0 then f(z) will always be equal to zero (0) but what happens if c becomes positive?
When c is greater than zero but less than 0.25, the value of f(z) does not escape to infinity. But if c is greater than 0.25 then f(z) does diverge to infinity.
For example, suppose f(z) = z2 + 1 and begin with z=0. In that case f(z) = 1. After that, f(z) will take values of 2, 5, 26, 677, 458330 and so on as it escapes to infinity.
Example E:
c can be any complex number and the same recursive process can be used to find the value of f(z) and so determine whether it either remains bounded and is within the Mandelbrot Set or diverges to infinity and is outside the set. In some instances, that is discovered after only a few iterations but in other cases a considerable number of iterations can be required.
In this Argand diagram, i (the square root of -1) is shown at point 0, 1.
Make f(z) = z2 + i and then apply the same recursive method. It will be found that the value of f(z) eventually alternates between -i and -1+i and this does not diverge to infinity. The point in the complex plane representing i is therefore in the Mandelbrot Set.
See Unveiling the Mandelbrot set | plus.maths.org.
All points in the set are within radius 2:
Example A showed that when c = -2, the point is in the Mandelbrot set and could be plotted at -2, 0 on an Argand diagram. In Example B, f(z) diverged to infinity when c was -2.01.
All points within the set are found to be at or less than distance 2 from the origin.
Drawing the set:
Numerous points in the complex plan have to be tested to discover whether, for each point, f(z) diverges to infinity. If it does, the point is outside the Mandelbrot set. Computers are ideal for this type of repetitive computation and graphics are used to colour the various points. For instance, points found to be within the set may be coloured black and points outside coloured blue.
In practice, more colours are used based on the number of recursions required to ascertain whether a point is within or outside the set.
Video:
Numerous videos are available. I found the following helpful …..
The Mandelbrot Set - Dr. Holly Krieger - Numberphile (youtube.com)
The Complete Idiot's Guide to the Mandelbrot Set (youtube.com)
Mandelbrot and Fibonacci:
The Fibonacci sequence (previous Viewpoint) is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610… and so on forever. After the initial 0 and 1, each number in the sequence is the sum of the two numbers immediately preceding.
A close look at the Mandelbrot Set shows what might be described as antennae (or wispy lines) appearing at various points around the edges of the set
The number of “antennae” at the various points are in line with the Fibonacci sequence. This fascinating fact is well-explained in the Numberphile video on Youtube - Fibonacci Numbers hidden in the Mandelbrot Set - Numberphile (youtube.com).
Julia Sets:
Julia sets - named after Gaston Julia (1893 - 1978) - is a type of fractal that is created using complex numbers and iterative processes.
NotesJuliaMandelbrot.pdf (cornell.edu)
Julia Sets | Sarah Bricault (mit.edu)
A more formal definition of the Mandelbrot Set:
The Mandelbrot set is the set of values of c in the complex plane for which the orbit of the critical point z = 0 under iteration of the quadratic map
Zn+1 = zn2 + c
remains bounded. Thus, a complex number c is a member of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded for all n > 0
The Mandelbrot Set (malinc.se)
Other Materials:
A vast amount of material is available on this topic. The following is a selection …..
Argand Diagrams - (physicsandmathstutor.com)
How Mandelbrot's fractals changed the world - BBC News
Mandelbrot set - Wikipedia
What is the Mandelbrot set? | plus.maths.org
A quick explanation of the Mandelbrot set | by Alonso Del Arte | Medium
Mandelbrot Sets – The Chaos Hypertextbook
NOVA | Hunting the Hidden Dimension | The Most Famous Fractal | PBS
Online Mandelbrot Set Plotter - ScienceDemos.org.uk
Complex Analysis (complex-analysis.com)
The Mandelbrot at a glance (paulbourke.net)
Let’s draw the Mandelbrot set! – The Mindful Programmer (jonisalonen.com)
An almost one-liner to construct the Mandelbrot set with Mathematica | Let’s talk about science! (ekamperi.github.io)
The Mandelbrot Set (bu.edu) and Exploration #1 (bu.edu)
What real numbers are in the Mandelbrot set? - Mathematics Stack Exchange
Mandelbrot Set | The Math Chronicles
https://www.mathchronicles.org/mandelbrot-set
Calculator:
Calculate an Iteration (rechneronline.de)
Complex Number Calculator - Symbolab
Fractals - see What are Fractals? – Fractal Foundation and How Fractals Work | HowStuffWorks
Complex numbers have an interesting history - History of Complex Numbers | SpringerLink
Jean-Robert Argand 1768 - 1822