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          Jason Doucette / Xona Games
           
          hometown – Yarmouth, NS, Canada
          residence – Seattle, WA, USA
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          Jason Doucette's Wallpapers

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          MANDELBROT SET
          WALLPAPERS / DESKTOPS / BACKGROUNDS


          What is the Mandelbrot set?

          The Mandelbrot Set is a fractal named after its discoverer, Benoit Mandelbrot. Mandelbrot coined the term "fractal" in 1975 from the Latin fractus or "to break". Fractals are things that are self-similar at various scales. Magnification of a fractal will reveal small details similar to larger characteristics. In the Mandelbrot Set, these small details do not replicate the larger whole exactly, and thus it is said to have only quasi self-similarity. The Mandelbrot Set is a fractal created by a very simple mathematical formula:

          Zn+1 = Zn2 + C

          It is an iterative process using complex numbers. Given a number C in the complex plane, iterate the above formula to infinity. If Z converges to a point, it is in the Mandelbrot set. If it diverges (i.e. goes off into infinity), it is not in the Mandelbrot set. This implies there are only two possibilities for each point in the Mandelbrot set. Each point is either:

          1. in the Mandelbrot set
          2. not in the Mandelbrot set

          So, if we are drawing a picture of the Mandelbrot set, where do the colors come from? Shouldn't the image be black and white?

          Yes, it should be. In fact, this is what it would look like:

          Mandelbrot Set

          But, that's boring. We can create astonishing pictures by coloring all points that are outside the Mandelbrot set according to their proximity to the Mandelbrot set. In other words, we color points outside of the Mandelbrot set according to how close to the Mandelbrot set they are. Besides, if we just colored it black and white, a lot of beautiful images would be all white, since often only a minute, indistinguishable percentage of the image's area is within the set (any of the images below that have an absence of black areas are examples of this). If we color the areas outside of the set, we are able to distinguish which portions are closer to the set than others, and it creates breathtaking patterns. Here is what the same image looks like with colors:

          Mandelbrot Set


          How can you tell how close to the Mandelbrot set a point is?

          We can find this out by looking how quickly each point is divergent. If it takes a lot of iterations to notice a point is divergent, it means the point is almost in the Mandelbrot set, but not quite. We color this a different color than points that are really far away from being inside the Mandelbrot set - those that are divergent right away. The result is a smooth palette of colors that signifies how close to the set each point is.

          Why are the colors different in each picture?

          The colors chosen for 'painting' these pictures are arbitrary - they were just made up, and have no real significance other than two similar colors represent places that are the same distance away from the Mandelbrot set. By creating a smooth flowing palette of colors, we can visualize portions of the image that are all relatively the same distance away from the Mandelbrot Set.

          Why does every picture look different?

          Perhaps the most amazing thing about the Mandelbrot set is that it has infinite detail. You can zoom in on any point with detail forever and get more detail. The images below are places I personally zoomed in on. Some of the images are zoomed in sections of others - if you look close enough, you will notice.

          Why are there two links for each picture?

          The first link ([Normal]) is the normal way Mandelbrot set images are created - without any anti-aliasing. You will see lots of graphic artifacts in these. This is not how the image would look in real life, if it existed.

          The second link ([Better]) is my method of creating Mandelbrot images, which removes aliasing. I do this by supersampling. I sample each pixel thousands of times to ensure that it is the right color. These images are the way you would see the Mandelbrot set with your own eyes, if this were possible.

          By comparing the two, you will notice a significant amount of greater detail and clarity with my version.

          The image files are huge! Why?
          I can see some fuzziness when looking really close. Why?

          I use the web standard .JPG image format to store these images. Unfortunately, the .JPG format is lossy, meaning that data is lost during the compression of the data. While this is okay with most real life images, you can notice unclear portions of this images if you look close enough (when dealing with images of infinite detail, one tends to look very closely at the detail of a single snapshot to find more detail). I have avoided data loss as much as possible by saving the images in a high quality setting, but these images still do not match the perfect image data of the originals.

          The original (perfect data) images stored in .TGA image format range from over 550 KB to almost 2,000 KB in size. In comparison, these .JPGs only range from 100 KB to almost 500 KB.

          Mandelbrot Set
          This is a view of the entire fractal.
          See below for closer looks of this fractal.

          640 x 480
          800 x 600
          1024 x 768
          1152 x 864
          1280 x 960
          [1600 x 1200]

          Note: Wallpaper was created at 1600x1200
          and was scaled to fit the other resolutions.
          Mandelbrot Set
          Mandelbrot Set
          Normal = no anti-aliasing. Better = anti-aliased.
          All images are 1024x768 resolution.

          Mandelbrot Set
          [Normal] [Better]
          Mandelbrot Set
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          Mandelbrot Set
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          Mandelbrot Set
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          MORE WALLPAPERS / DESKTOPS

          There are more wallpapers / desktops available here:
          http://www.matthewdoucette.com/wallpapers/
          This is my brother's site (Matthew Doucette).

           
          869,505 visitors since Monday, March 4th, 2002
          2,041,752 total page views since May 13th, 1999
          Jason Allen Doucette / Xona Games
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