Fractal Pages

Fractals are extensions of traditional Euclidean shapes, such as lines, squares, and circles, with two fundamental properties:
    1.   When you view fractals, you can magnify them an infinite number of times,
          and they contain similar structure at every magnification level.
    2.   You can generate fractals using finite and typically small sets of instructions and data.

Fractals grew out of the goal of mathematicians to completely describe the world using the standard geometrical expressions.   While at IBM as a mathematician in their research staff, Benoît B. Mandelbrot, PhD, proved and published the theory behind fractals; coining the term fractal in 1975, and later expanding and updating those ideas in 1982 within The Fractal Geometry of Nature.   The well-known Mandelbrot Set is named in his honor.   Another famous fractal researcher, with a set also named for him, is French mathematician Gaston Julia (whom Mandelbrot studied under).

The standard Mandelbrot fractal equation takes the form z(n+1)=z(n)²+c, where c is the complex number x+iy corresponding to any point on the (x,y) coordinate plane.   But where the x axis is an ordinary, vanilla real number, the y axis is an imaginary number, i.e. a real number times i, where i is the square root of -1.   Fractal equations are iterative, in that the result of one calculation of the fractal equation becomes the z input to the next calculation.   Over repeated evaluations of a fractal equation, values for each point in the (x,y) coordinate space either converge at single points, move toward the (0,0) origin point, or move toward infinity.   The diverse colors in fractal plots reflect the rate of this movement for each point.   Discussions of chaos theory frequently use fractals as examples, because slight variations in the fractal equation produce radically different results.

But first, a quick set of definitions:

FORMULAS are the mathematical statements (especially equations) of a fact, rule, principle, or other logical relation, used to perform the basic calculations of fractal images.   Many fractal generating applications are capable of using various formulae, whether hardcoded within the program or using a formula parser.

PARAMETERS are the constants in an equation that vary in other equations of the same general form.   They are values supplied to, and used by, a formula within the generating application.   These are established by the user of the software to adjust various settings, quantities, etc. to achieve varied results in the image itself.

IMAGES are the the pictorial representation as used in computer-aided design, the process by which a computer displays data pictorially.

MAPS, or GRADIENTS, are the color palettes used to cause variations in the way an IMAGE may be colored.   (Thousands of these are in the public domain already.)

The PARAMETERS and IMAGES made by an individual are their own property, and are considered to be copyrighted.   Any use by another person of said items should require the permission by the creator/author/originator.   Whereas FORMULAS are usually considered to be public domain, and virtually can not cause an IMAGE without supplying PARAMETERS.

After going to one of the above webpage links, select any of the various thumbnail images to see the fractal graphics.   After viewing each image, hit your browser's BACK button to return to that particular selection page.

(More of my images may be found within various links listed below.)

Due to the current limitations on available web-space, these images have been reduced in size and reformatted from their original creations.   Therefore, some of the crisp sharpness and details may be lost.   All of the current images are shown here in 1024x768 resolution as 256-Color GIF format files (varying in size from 225,104 bytes to 753,346 bytes).

If interested in one of the original images (for personal and non-profit use), then send an email to me and I will return the image back to you in the original size and format.   Or, under special circumstances, whatever format or size you may desire.   Posters can also be printed and shipped for a reasonable price, and are produced with a high-end commercial ink jet printer, using oil based, UV resistant inks, on heavy, high quality paper.   A typical size of 24-inches by 18-inches at 300-DPI would normally run around $55.00 in US dollars (which includes shipping within US).

The Fractint Discussion List, during the latter portion of 1997, held a contest to see all the various images that could be made from the same formula.   Forty-nine fractal artists entered three images each.   (My contest images were:  PaulLee1 / PaulLee2 / PaulLee3.)   The first round of voting selected ten favorites, then the second round selected the top three images from the first round.   You may see all of the Contest images and the Contest information from here.

The Fractal-Art list held a contest at Summer's end of 1998 to see all the various images that could be made by any participant that wished to submit images.   One hundred one fractal artists entered up to three images each, for a total of 498 images.   (My contest images were:   "Turbine Engine" / "Fertilized Cell Division".)   I was lucky enough to get third-place on one of my images in the category of "Best Mechanical Device".   You may see all of the Contest images and the Contest information from here.

And if you are really curious about the images I have been creating recently, then try viewing my other fractal graphics at the following web site:   http://www.Nahee.com/Fractals/