Benoît Mandelbrot

Born in 1924, died in 2010.
Interested in roughness and irregularities seen in nature:

• Traditional geometrical shapes (e.g. circle, line) are perfectly smooth
• Searched for simple geometrical equations that would exhibit roughness

Established fractal geometry

• Rough/fragmented geometric shapes
• Self-similarity at all scales: zoom in on a small area and you can find the original shape in it.

The Applet

Click on this image to start the applet. The source code can be downloaded here.

Mandelbrot Set: Definition

The Mandelbrot set is a set of complex numbers (c) that meet the following condition:

Z_n(c) ≠ ∞  as n →∞

The function Zn is defined as follows:

• Start by setting n=0
• Initial value:        Z0 = c
• Compute:            Z_n+1 = Z_n² + c  (keep repeating this, incrementing n)
• Test whether Z_n goes to infinity as n goes to infinity

The Mandelbrot Set is self-similar at all scales:

• Zoom in on areas to find small sections that look like the original shape; can zoom in further on those.

Practicalities

Computers aren’t good with infinity!

• It has been shown that if Z_n > 2, it will tend to infinity
• We can compute Z_n up to a maximum of some value such as n=100, and see if it exceeds 2 during that time
• As you zoom into smaller areas, need to increase n

Complex numbers have two parts (real, imaginary)

• Plot real part on X axis and imaginary part on Y axis
• We can just treat these as two separate numbers

Square of a complex number:

• If (a,b) is a complex number, then its square is (a²-b², 2ab)

A commonly plotted area is from (-2,-1.5) to (1,1.5).

Plotting the Mandelbrot Set

Loop over all points in a graphics area.

For each point:

• Rescale it to the point on the complex plane that we are plotting
• If it’s a member of the Mandelbrot set, give it a fixed colour
• If it’s not, colour it according the value n where it reached Z_n > 2
• To set a pixel to a colour in the applet, draw line 1 pixel long

Colouring:

• I use blue for the members of the set
• I give other points a hue in the range 0-0.5, which is along the colour spectrum red-orange-yellow-green.

Applet Features

JLabel :

• Static text items

JTextField:

• Editable items

JButton:

• Plot and Reset buttons

ActionListener:

• One for each button: implement actionPerformed

FocusListener:

• One for each JTextField: implement focusLost

• When focus is lost on a position textbox, it has been edited, so parse their values next time Plot is pressed
• When focus is lost on Iterations textbox, it has been edited, so parse its value

MouseInputAdapter:

• Attached to applet overall
• mouseMoved: update mouse position textbox
• mouseDragged: first, record start position, then record current position
• mouseReleased: parse positions and repaint.

Feedback & Comments

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