Classic Computer Magazine Archive COMPUTE! ISSUE 137 / JANUARY 1992 / PAGE 88

Fractal compaction. (Column)
by Steven Anzovin

As the incredible Shrinking Man discovers at the end of the classic science fiction film, there are always new levels of wonder in the universe, no matter how small you get. Legions of computer users are discovering this truth as they play with the curious mathematical entities called fractals. Fractals, you see, are pretty much the same at every scale, from the cosmic to the miniscule.

A coastline provides a good example of fractal geometry. From space, the coast of California has a certain rough irregularity. A mile above the land, the coast has a similar roughness. Get down on the beach on your hands and knees, and the irregular boundary between sand and surf looks remarkably like the coast seen from space. That self-similarity is an essential property of fractals.

A number of software programs make it possible to explore fractal geometry on your home computer. It isn't necessary to master the mathematical mystery of IFS attractors, metri spaces, and affine transformations in the Euclidean plane to have fun with fractals (though you'll understand the theory better if you got past analytic geometry in school).

Most programs let you take a colorful image, like the well-known Mandelbrot set (kind of a mathematical black hole with an infinitely complex boundary), and simply zoom in and out at will. At any level, you can find whirling vortices, flaming dragons, Amazonian river systems, and complex Escher-like tillings. Their rhythmic, psychedelic quality is fascinating.

Two absorbing fractal exploration programs are The Beauty of Fractals Lab for the Macintosh, based on the book The Beauty of Fractals by Peitgen and Richter (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010; 212-460-1500; software $49.00, book $39.00). The image that accompanies this column is from The Beauty of Fractals. For PCs and Macs there's Desktop Fractal Design System by Michael F. Barnsley (Academic Press, 465 South Lincoln Drive, Troy, Missouri 63379; 800-321-5068; $49.95), companinon software to Barnsley's book Fractals Everywhere also by Academic Press ($44.50). It's considered by fractal mavens to be one of the few classics in the field.

Beauty creates beautiful abstract graphics. It even has a cool 3-D option but requires a color Mac with a math coprocessor. Desktop has fewer options but runs on any AT-class machine with 640K and EGA or VGA. It's more of a teaching tool for using fractals to model real objects like ferns, clouds, and even human faces.

Fractals are good for more than creating calculation-intensive eye candy on your PC, however. Another essential property is that they can be described with relatively small amounts of information--as little as a single mathematical formula.

Thus any computer image that can be described with fractal geometry can be stored in a very small amount of space. Barnsley's company, Iterated Systems (5550A Peachtree Parkway, Suite 650, Norcross, Georgia 30092; 404-840-0633), has a fractal-transform compression process called P.OEM that compresses a 768K 24-bit color image down to 10K with little loss of detail.

The weirdest, most fractal-like thing about P.OEM images is that their resolution is practically unlimited. You can even view them with more detail than in the original image. The transform process adds all the extra detail! Iterated Systems and Jones and Bartlett Publishers have copublished a 1.44MB floppy disk called Floppy Book (Jones and Bartlett Publishers, 20 Park-Plaza, Boston, Massachusetts 02116; 800-832-0034; $24.95) that contains 100 "pages" of full-screen 24-bit images and text. That's packing about 77MB of data onto an ordinary 3 1/2-inch floppy.

P.OEM PC floppy books can also contain compressed video (two minutes per disk), digitized sound, and ASCII text. They might well supplant CD-ROMs as a digital publishing medium for single books rather than entire encyclopedias or databases. The floppy book is faster (you can load the P.OEM file to your hard disk for access speeds no CD-ROM player can match), it's cheaper to duplicate, and every computer has a floppy disk drive.

You don't need special hardware or software to read a P.OEM floppy book, but you need special hardware to make one. The compression development kit costs up to $13,000, but you can have Iterated Systems or a service bureau compress your files for a low piecework rate, starting at $25 per picture.