GeoGebra Art

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.


The Quadrivium

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as “even,” “odd,” “perfect,” and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like “double,” “greater,” “smaller,” “half,” “one and one-half times,” “one and one-third times,” and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

And once more, inasmuch as part of “size” is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of “size,” geometry the part that abides and is at rest, astronomy that which moves and revolves.

…In Plato’s Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life; arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him, says: “You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld.”

–Nichomachus of Gerasa, Introduction to Arithmetic

liberal arts

The Eccentric Circle

When you were in high school, you undoubtedly encountered conics at some point. The most familiar example is the parabola, introduced in algebra courses as the graph of the equation y = x2 or, more generally, y = ax2 + bx c. You may have learned how to find the x– and y– intercepts, the coordinates of the vertex, and so on. These are all useful skills–although I personally have never used them outside of a classroom–but they tell you very little about the parabola as a geometrical object. The other conic sections–the ellipse and the hyperbola–don’t lend themselves to a description of the form y = f(x), hence are usually not studied at all. This is a terrible misfortune, as they are part of the treasure that comes down to us from the ancient geometers, and figure largely in physical sciences such as optics, acoustics, and celestial mechanics.

The most famous way to describe conics is as slices of cones, hence the name. In plane geometry, however, it is more convenient to think of them in a different way. We start with a line l (the directrix) and a point F not on the line (the focus). We select an eccentricity ε > 0. The conic is the locus of all points P such that the ratio of PF to the perpendicular distance from P to l is ε. If ε < 1, the conic is an ellipse; if ε = 1, a parabola; and if ε > 1, a hyperbola.


Here we see the eccentricity vary from 0 to ∞. The focus and directrix are in red. The parabola is the “boundary” between the family of hyperbolas and the family of ellipses, and occurs when the eccentricity is 1.

You may have observed that the ellipse becomes more and more “circular” as we slide the eccentricity toward 0. When the eccentricity reaches 0, then the circle collapses to the point F itself. But if we “blow up” the curve as the eccentricity shrinks to 0, then the limiting curve is indeed a circle. So we may view the circle as part of the same family of curves.

Many of the important properties of circles generalize to more universal properties shared by the other conics. For instance, from any point in the exterior of a conic there are precisely two tangents; or again, at each point on a conic there is exactly one tangent. It is possible to prove these properties by analytical methods, but this lacks geometrical finesse, involving arguments that would have been alien to the Greeks. The most excellent way to prove many of the basic theorems about conics is to use the eccentric circle.

eccentricConsider the conic with focus F, directrix l, and eccentricity ε. The interior of the conic is defined as the set of points P for which PF is less than ε times the distance to l. Let O be any point not in the interior and not on the directrix, and let s be its perpendicular distance from l. Construct the circle centered at O with radius εs.

Now let E be any point on the circle. Let R be the point where the line EF intersects l. Let P be the point where the line through F parallel to the radius intersects OR. An argument from similarity shows that P is on the conic.

Now imagine letting E traverse the circle. Then the conic is swept out by the point P as shown.


Here we have an ellipse. If the conic is a parabola, then the directrix is tangent to the eccentric circle; if the conic is a hyperbola, then the directrix intersects the eccentric circles at two points, and the two branches of the hyperbola correspond to the two parts of the circle.


The eccentric circle is what allows us to prove theorems concerning tangent lines. We already know that from any point in the exterior of a circle there are two tangents to the circle. Given an eccentric circle centered at O, the two tangents from F to the circle produce two tangents to the conic as shown.


Similarly, to show that there is exactly one tangent at a point of the conic, construct the eccentric circle at O. This circle passes through F, and there is exactly one tangent to the circle at F; this tangent produces the tangent at O.