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  Julia Set (c = -0.4+0.6i)

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  Julia Set (c = -0.19-0.67i)

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  Mandelbrot Set

JuliaSet

A program that generates Julia Set images.

The Algorithm

This algorithm generates images that represent the interesting behaviour of complex numbers.

The imaginary number (i)

The square root of -1 is defined as i. This number doesn't exist naturally, but it is used often for it's unique properties. When multiplied by a real constant, say b, the result is an imaginary number bi, the square of which will be negative the square of the constant, or -b^2, since i^2 = -1.

Complex Numbers

A complex number can be defined as the sum of a real number with an imaginary number a + bi. The sum of two complex numbers is simply the sums of the real components plus the sums of the imaginary components

(a + bi) + (c + di) = (a + b) + (c + d)i

The product of two complex numbers is more interesting. The components must be distributed with each other, but since i^2 = -1, the product of the two imaginary components is a negative real number, rather than positive. So the real of the product is the difference of the two real product terms.

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Consider two complex numbers represented as shown:

A = r*(cos(a) + sin(a)i)
B = s*(cos(b) + sin(b)i)

The product of these two numbers reveals a consequence of imaginary multiplication:

A*B = r*(cos(a) + sin(a)i) * s*(cos(b) + sin(b)i)
    = r*s*((cos(a)cos(b) - sin(a)sin(b)) + (cos(a)sin(b) + sin(a)cos(b))i)

    = r*s*(cos(a + b) + sin(a + b)i)

Complex numbers can be represented as vectors on an xy grid, where the x component of the vector represents the real component, and the y component of the vector represents the imaginary component. The product of two complex numbers, then, can be represented by the vector whose magnitude is the product of the magnitudes of the two complex vectors and whose angle is the sum of the angles of the two complex vectors. Naturally, then, the square of a complex number will be a complex vector with a magnitude equal to the square of the original magnitude, and an angle equal to twice the original angle.

A   = r*(cos(a) + sin(a)i)
A^2 = r^2*(cos(2a) + sin(2a)i)

Iteration

Consider a complex function f(z) = z^2 + c, where c is a constant. Take a complex number z and put it through the function, then take the result and put it back into the function. Do this repeatedly infinatum. This iterative process can be written as z[a+1] = z[a]^2 + c. Because of the nature of complex numbers, after infinite iterations, the modulus of these numbers will either reach infinity, converge to zero, or remain bounded in an endless loop, depending on the value of z[1]. For example, consider a constant of 0 for now, so the iterative function becomes z[a+1] = z[a]^2. Start at z[1] = 2 + 3i. The first few iterations result in z[2] = -5 + 12i, followed by z[3] = -119 - 120i, then z[4] = -239 + 28560i. This set seems to go to infinity under infinite iteration. Now consider z[1] = 0.5 + 0.75i. The next few iterations of this number would be z[2] = -0.31 + 0.75i, z[3] = -0.47 - 0.47i, and finally z[4] = -0.0036 + 0.43i. This set will eventually converge to zero. The Julia Set, loosly speaking, is the map of all complex numbers that either reach infinity or remain bounded under infinite iteration. Because c = 0, the Julia set is a simple circle with a radius of 1, since any number that is less than 1 decreases in value when squared, and any number greater than 1 increases in value when squared. Adding a non-zero constant c will complicate the set.

Generating the Image

Julia Sets can be approximated by using a computer. Each image is a complex grid ranging from -2 to 2, and from -2i to 2i (the width bounds are extended according to the image aspect ratio). Each pixel represents an approximate complex number in the grid. The computer goes through these pixels, and performs 256 iterations of the function z[a+1] = z[a]^2 + c on the complex number. If the modulus of the number exceeds 2, then the number is assumed to reach infinity and the pixel is colored white. If the number does not reach infinity (greater than 2) after 256 iterations, the pixel is colored black. This will create a pattern of black and white spots that represents the julia set generated from the function z^2 + c with the given complex constant c.

The shapes generated by these sets can have rotational symmetry and mirror symmetry, but they almost always have fractal symmetry. This means that the shape is invariant when zoomed in by a certain factor. It contains copies of itself as parts of itself. This results in a crystal-like shape which is complex, but symmetric and regular.

For further analysis, the algorithm can measure the number of iterations made before the function reaches infinity (if at all), and assign a color to the pixel using a color map, revealing even more patterns and symmetry in the Julia Sets. They make for great wallpapers!

Usage

Installation (clone and build)

NOTE: Project can only be built on bash-based platforms (so macOS, Linux, or cygwin).

Make sure that you have make and g++ installed. Both of these can be acquired using apt-get. Clone the git repository into your local machine using git clone and entering the remote repository. You can then build and install using make install. This will build the project and install the binary in /usr/bin.

Command line interface

The program uses CImg option parsing to retrieve command line arguments. To set options, type the name of the option, followed by the value of the option: $ juliaset -[name] [value] .... If the value is a boolean, you can just type the name of the option to set it to true: $ juliaset -[boolean-option]. Options do not have to be put in any order.

Here are a list of options:

Option	Description	Defaults
-cr	The real component of the complex constant	0.0
-ci	The imaginary component of the complex constant	0.0
-mbrot	Generates the mandelbrot set image (overrides -cr and -ci)	false
-imgx	The width of the image	1920
-imgy	The height of the image	1080
-zoom	The zoom scale of the image	1.0
-offx	The x offset of the image	0.0
-offy	The y offset of the image	0.0
-rot	The angle of rotation of the image	0.0
-save	The filename to save to	jimage.jpg
-cmap	The colormapping used	rainbow
-cmaps	Lists all of the colormaps and returns	false
-test	Generates a 400x300 test image (saved to the savename) for the set colormap	false
-help (or -h)	Prints the help message	false

Built With

• c++
• cimg
• html
• makefile
• xml