WHAT ARE FRACTALS AND FRACTAL GEOMETRY?
The concept of fractal geometry is the basis of Vistapro's capacity to generate imaginary scenes. Many computer graphics enthusiasts have become interested in fractal graphics through programs such as Vistapro, and public domain Mandelbrot and scenery programs. The popularity of fractal graphics using personal computers traces back to the appearance of stunning images of the Mandelbrot Set (a type of fractal object) on the cover of Scientific American in August of 1985. That widespread exposure of these strangely beautiful abstract objects led many amateur and professional programmers to the original source book on fractals: The Fractal Geometry of Nature (by Benoit Mandelbrot).
While fractals and fractal geometry have become hot buzzwords in the computer graphics field, it is not obvious what they are. The following description is simplified, and interested students and readers should read Mandelbrot's book on the subject.
We owe the word fractal to Mr. Mandelbrot, a mathematician and Fellow at IBM's Watson research organization in New York. Fractal refers to objects with fractional dimensions, that is, objects which don't really fit into the ordinary world of things like lines (1-dimensional), surfaces (2-dimensional) and solids (3-dimensional). Fractals are objects which fit in- between these normal-dimensional objects.
Mandelbrot took an interest in a long neglected area of mathematics which originated at the turn of this century. Some devotees of geometry at that time began to study lines which didn't behave like ordinary lines.
If you read Mandelbrot's book you'll become familiar with some of the mathematical history of things like Peano curves, Hilbert curves, and Koch snowflakes. What makes these objects so strange, and what led Mandelbrot to look deeper, are two properties: these lines tend to fill up a 2-dimensional surface (they act as if they are something in between lines and planes), and their appearance seems to be identical no matter how much they are magnified. Magnified small portions of these fractal lines tend to look like the whole unmagnified line. Odd indeed!
Mathematicians at the turn of the century tended to call such objects pathological and didn't have a good way of integrating them into the rest of mathematics, especially geometry. Geometry was mostly dominated by the study of well-behaved, smooth, simple forms like lines, planes, and solids. Mandelbrot made a systematic study of these weird fractional dimension geometric forms and helped bring them into the mathematical fold.
Mandelbrot also showed how these objects are models of many things found in the natural world, like surface textures of mountains, coastlines of islands, and branching designs of plants, trees, blood vessels, and lung tubes (bronchi).
If you want to envision a mental picture of how Vistapro exploits fractal geometry to generate natural looking land surface textures, take the following little mental journey into the process of crumpling a sheet of paper:
1. Imagine a flat triangular sheet of paper.
2. Divide the sheet into a small number of sub-triangles.
3. Randomly select some of the intersection points and raise or lower them (by a large amount) above the original plane of the flat sheet.
4. Now divide the sub-triangles into smaller sub-triangles.
5. Randomly raise and lower some of the newly created corner points like you did in step 3, but by a smaller amount than in step 3.
6. Keep repeating steps 4 and 5 making smaller and smaller sub- triangles and raising and lowering corner points randomly by smaller and smaller amounts at each step.
7. Stop when you've reached a point where each smaller division into sub-triangles can't make any more difference in appearance on a limited resolution display like a computer monitor.
8. Now color all the little sub-triangles by a method which makes the highest corner points white (snow on the mountain tops), lower ones brown and green (mountain sides with trees) and the lowest ones blue (a lake at the bottom of the mountain valley).
If we did Steps 4 and 5 using some regular (non-random) technique, in the end the highly crumpled surface would be a lot like the first fractals explored by Mandelbrot; they would look similar at any degree of magnification they were viewed. The introduction of randomness to the process makes them look similarly random at different degrees of magnification.
