**Create fun and interesting computer generated wallpapers for your iPhone!**Fractalate is an iPhone application that creates fractal images for the Mandelbrot and Julia fractal sets. These images are easily saved to the iPhone photos folder so they can later be used for wallpaper images for your phone.

Fractals are constructed from mathematical formulas. Because of this you can theoretically zoom into any point of the fractal to infinity and it continues to display interesting and new patterns.

The refresh action creates a brand new palette of 512 colors and redraws the current fractal view. This new palette is made through a series of algorithms that provide a brand new and completely unique set of colors.

Fractalate uses double precision floating point values and allows zooming in to the images that you can see approximately 30,000,000,000,000 aka 30 trillion times. If you took the average screen size of 14" and resolution 1280x1024 you'd have a pixel that was about 5 million square miles, about twice the size of Arcturus in the following video!!

#### Sizes of our planets and stars

### Interesting math

The fractals generated by fractalate use 512 iterations for each pixel. That equates to a worse case of doing over 65 million iterations to draw one screen. (Whew... that's why it takes a while to draw the screen sometimes).

#### Mandelbrot

The mandelbrot fractal is best described by wikipedia:

http://en.wikipedia.org/wiki/Mandelbrot_set

Essentially for each point that we draw we work out the number of iterations it takes to either escape or we give up trying after a maximum number of iterations (in my case 512). The number of iterations that it takes to escape is assigned a palette color.

You can find the actual formula all over the web for calculating this, so at the risk of repeating it yet again here it is:

z

_{n+1} = z

_{n}^{2} + c

z & c are complex numbers that can be easily represented on a graph so that z = z(r) + z(i) is represented by z(r) as the x axis and z(i) as the y axis.

The point we are calculating is represented as c in the equation above. The color that this point receives is calculated by the number of iterations of the above equation until

|z_{n}| > 2

#### Julia

The Julia fractal is computed quite similarly to the Mandelbrot fractal. The main difference is that Julia uses a

*seed *value. This seed can be any value but many of them will yield almost null Julia sets. Fractalate is initialized with an initial value of (0.11, 0.65) which creates an interesting Julia view. There are many other values that create very different Julia fractals. You'll have to explore and find your own!

#### Palette

Generating an interesting mandelbrot palette is also worth attention. In fractalate it is generated by taking a unique number of distinct colors and the spreading the palette between these colors. Each color is determined to be sufficiently unique if it has enough color separation from the previous color.

So for example if the two colors where RGB values and the first was (00,00,01) and the second distinct color was (00,05, 05) then I generate x colors in between these

- (00,01,02)
- (00,02,03)
- (00,03,03)
- (00,04,04)
- (00,05,05)

This gives a resulting pleasant visual representation of the fractal as it has several distinct colors but several immediately adjacent colors are quite similar.

### Support

Support, comments, or kudos for fractalate;

Email me at: fractalate@gmail.com address.

You can download fractalate directly from the itunes store.