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Listed below are some of the non-proprietary methods used by the artist to create digital art. Links are provided if you want to learn more. Digital Images A digital image is a two dimensional array of colored dots, called pixels. These pixels are lined up in rows and columns but since there are so many of them, you can't see them individually. Typically there are several megapixels in an image. The color of each pixel is set by the device that is creating the image. For example, when you snap a picture with a digital camera, the color of the pixel that is in the middle of the array of pixels is determined by the color of the light that goes through the center of the lens. With PhotoShop, or similar software, you can manipulate the color of the pixels directly. Each digital image that you see on this website was created by a mathematical algorithm developed by the artist for the express purpose of making that picture. Here is an example, suppose that we have a 256×256 array of pixels that range in color from red, at the bottom left to blue at the top right - as shown below. In mathematical terms, the color of the pixel in row i and column j, is defined by the intensities: Red = i, Green = 0 and Blue = j. The resulting image would be:
The color of a pixel in a fractal image is determined by the behavior of a sequence of numbers that is based on the location of that pixel. Consider the pixel in row r and column c of a 256×256 array of pixels. Let x = (c - 128)/64 and y = (r - 128)/64. We look at the complex number z = x+iy. Notice that the center of the array corresponds to z = 0. Now form a sequence of numbers using the square function, so that if we start the sequence with z, the sequence would be
z, z2, z3, z4, etc.
For example, a pixel in row 128, column 256 (half way down on the right side) would give us z = 2 and the sequence would be
2, 4, 16, 256, 65536, etc.
The "behavior" of this sequence is described by saying that it goes to infinity, so we we will color the pixel a shade of gray. For a pixel in 128, column 96 we get z = -.5 and the sequence is
-.5, .25, .0625, .00390625, etc.
This sequence converges to 0 so we will color the pixel a shade of red. The shades depend on how fast the sequences "tend" to their final destinations. Some sequences neither go to infinity nor converge. Pixels, whose sequences have this property, or whose sequences are on the borderline between convergence and non-convergence, are colored yellow. The resulting fractal looks like this:
For a fuller account of fractals, click here.
Iterated Function Systems This is another way to generate fractals by repeatedly applying a set of functions to a point. A classic book on the subject is "Fractals Everywhere" by Michael Barnsley. Another reference is Iterated Function Systems.
This is a method for converting a three dimensional scene to a digital image. We start with a set of three dimensional objects, a light source and a location and direction from which to view the scene. A rather complete description of the process is given in ray tracing especially the section "Overview of Ray Tracing".
3D Images - Lenticular Lens Method Most of us have seen images that, when viewed with 3D glasses, have the appearance of being three dimensional. The lenticular lens method also involves making multiple images of the subject from different angles. These images are then "interlaced" together so that when a special plastic sheet, called a "lenticular" lens, is placed over the interlaced image, a three dimensional effect is produced. By interlacing images in a slightly different manner, this method can also be used to change the picture when viewed from different angles. Using ray tracing techniques to produce images, we have made some of our images appear three dimensional. A good, but technical reference, is stereoscopy.com.
Further Reading One of the best technical references on the subject is "Computer Graphics" by Foley, vanDam, Feiner and Hughes.
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