The Mandelbrot Set is a mathematical curiosity with infinite complexity. The pages of this section of this website are to demonstrate the complexity and beauty of the Mandelbrot Set
 
The Mandelbrot set is derived from iterations of a rather simple equation for numbers in a complex number plane. The iterated equation is: "z_n+1 = z_n² + c", where at the start z₀ = 0 and “c” is a complex number (having two-dimensions) representing points of the number plain (in this case the display area of the screen), and is expressed in the following form: “0.7075+0.3088i”. The first part of the number “0.7075” is known as the real component (x-axis) and the second part “0.3088i” is known as the imaginary component (y-axis). And the “i” represents the square root of -1 (√-1). To clarify things, the √-1 is a number that is at right angles to the number line, thus creating the number plane.
 
To square the complex number “z” we must multiply all the components of the complex number. For example: (0.7075+0.3088i) * (0.7075+0.3088i). Let’s replace “z” with the “x” and “y” component.
 
z_next = z² + c   becomes   z_next = (x + y) * (x + y) + cx + cy
 
We expand this to   z_next = x² + 2xy + y² + cx + cy
 
Now we must now separate the real from the imaginary:
 
x_next = x² - y² + cx
y_next = 2xy + cy
 
Note that i² = -1 therefore y² becomes a real number, but negative.
And where x is muntiplied by i, it becomes an imaginary number.
 
I feel that it is important to say that this complex numbers methodology is used extensively in engineering particularly in electrical engineering with AC circuits, telecommunications, mechanical & structural engineering, fluid dynamics and aerospace, etc. This is a very real system and not just something to draw pretty patterns.

For this image to be generated, each and every pixel on the display area represents a single point on the number plane and this calculation is done multiple times until either the resulting value of “z” tends toward infinity or results in “z” stays small when we reach the computation limit. Numbers for “c” that remain small are part of the Mandelbrot Set; numbers that tend toward infinity are not part of the Mandelbrot Set.
 
We know that the number will tend toward infinity if the number crosses a circle that has a radius of 2 from the 0+0i point. The number of iterations that it takes to cross that circle will determine the colour of that single pixel. As you can imagine, if we have a display area, say 1200x900, there are a lot of calculators to be done.

Normally the colour of numbers in the set (the numbers that stay small) are coloured black. However, in some colour maps in these pages (and in other sites or programs), numbers in the set are coloured white. Within these pages you are also given the option to change that colour to anything you choose.

Experiment, explore options, do different things and see what happens, gain understanding, be creative, share it with your friends, but most of all have fun.

Don’t shy away from getting understanding, it will equip you for living life. God designed you with an enquiring mind, and He intended us to use that mind to gain understanding and perhaps even understand Him. When He laid out the foundations of the universe, He set out the laws of physics and mathematics. He intended us to explore and understand. Jesus is King.

Images generated from this site for private purposes are free to use as you feel fit without cost. Feel free to share the images with your friends and family, make that coffee mug for your grandmother, a print for a sibling, a t-shirt for your nephew, etc. I only ask that you share this site with friends and family so that others can also have fun exploring the world of mathematics.
 
If you intend to use the product for commercial purposes, such as printing on a t-shirt, mugs etc to sell as a product, or for a business logo, or any other commercial purpose, then I ask that you get in contact with me. A lot of work has gone into making this program and it is only right that money is to be made from this, then the one who has written the software should get a small portion of the proceeds.

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