In the classes I teach I often use GeoGebra for demonstrations; I assign GeoGebra constructions as homework, and our senior math majors use it in their oral presentations. GeoGebra, if you don’t know, is a Java-based dynamic geometry program freely available on the web. GeoGebraTube has thousands of applets developed by members for use in the classroom.

Now, GeoGebra is used mainly by high school teachers, and most of the applets available online are developed with that in mind. But the emphasis on algebra and coordinates by state curricula means that the program’s possibilities are vastly curtailed in common practice.

In trying to explore the software’s capabilities, I’ve set myself the task of constructing some of the fractals described in Mandelbrot’s *Fractal Geometry of Nature*. Images of my first attempts are shown below. At this point I’m not going to go into what makes a fractal a fractal, contenting myself merely with the following quatrain, which comes from Jonathan Swift:

*So, Nat’ralists observe, a Flea,*

Hath smaller Fleas that on him prey,

And these have smaller Fleas to bit ’em,

And so proceed ad infinitum.

Click on an image to view the slideshow and read the captions. Interactive versions can be found by going to my GeoGebra profile page. I won’t expand on my cleverness in the succinct programming that produced these images, but some of them are extremely clever.

An Apollonian net. Given three mutually tangent circles, it is possible to construct two other circles tangent to all three. One of the five circles bounds the other four. This algorithm is a special case of one discovered by the Greek mathematician Apollonius of Perga. The process is repeated for each triple of tangent circles, and so on ad infinitum. The figure that results is called an Apollonian net; the figure bounded in the circular “triangle” of three tangent circles is called an Apollonian gasket.

Given a chain of circles tangent in pairs (a Poincaré chain), the entire chain can be reflected into the interior of each link, a transformation known as inversion (cf. Coxeter’s Introduction to Geometry). Repetition of this process ad infinitum produces a fractal Jordan curve separating the plane into two disjoint connected sets, the exterior and the interior.

Another Poincaré chain.

A Koch snowflake. At each step we trisect each side and erect an equilateral triangle upon the middle segment, and so on ad infinitum. The boundary of the resulting snowflake is a fractal curve.

A fractal tree. At each branch we reproduce two copies of the preceding “stick,” but rotated by two angles that remain constant and scaled by two constant factors. Changing the angles and factors produces a wide variety of figures resembling various species of trees and plants, brochi, blood vessels, and so forth.

This is merely a variant of the Koch snowflake in which the relative lengths of the new segments are allowed to be different from one another, but with ratios that remain constant at each step. A circle is included to represent Euclidean geometry, the antithesis, one might say, of fractal geometry.

This shows the first five steps in the generation of a Peano curve that fills the Koch snowflake. The generator is a seven-segment polygonal curve; at each step, each segment is replaced by a scaled copy of the original. The resulting figure consists of a single open, simply connected polygonal curve. In principle the entire curve can be traced on paper without lifting one’s pencil or crossing a line, beginning at the lower left-hand point of the star and ending at the lower right-hand point. This curve was designed by Benoit Mandelbrot and described in his Fractal Geometry of Nature.

The polygon resulting from “filling in” the polygonal curve in the preceding. As Mandelbrot says: “This advanced teragon, shown as boundary between two fantastically intertwined domains serves better than any number of words to explain what plane-filling means.”

A self-inverse fractal patchwork. We begin with four tangent circles with centers at the vertices of a rhombus, and four lines, each of which is perpendicular to a diagonal at a vertex. (This collection is shown faintly in the background.) We then construct six circles, each of which is orthogonal to a triple of circles (the blue and green circles) or a pair of circles and a line (the red, yellow, orange, and purple circles). This figure is then repeatedly reflected across the four original circles and four lines, producing a cloth whose pattern is invariant under the group generated by the eight transformations.