The Greatest Love Story

As lines, so loves oblique may well

Themselves in every angle greet;

But ours so truly parallel,

Though infinite, can never meet.

Thus Andrew Marvell finds in geometry a powerful image for classical doomed love: the perfectly matched pair, originating and pointing in the same direction, identical in all but location, incapable of ever getting closer and at the same time locked together with no possibility of moving to the next town. The greatest of the Greek geometers, Apollonius of Perga, gives us a similarly striking image of hopeless separation, which turns out to be simultaneously an image of infinite closeness. It is buried in Book 2 of his Conics (tr.Taliaferro, Green Lion Press, 2000), and to understand it we have to go back a few steps and perform the strange, interesting act of cutting a cone with planes.

A cone can be cut into five shapes. If you pass a plane through the very tip of it (the vertex), the section you will make is a triangle; and if this triangle bisects the circular base of the cone it is called an “axial triangle.” If you cut the cone with a plane more or less horizontally, you will get a circle or ellipse. If you cut it with a plane of which the diameter is parallel to one side of the axial triangle, you get a parabola. And if the diameter of the  section is not parallel to a side of the axial triangle, you get a hyperbola. Apollonius proves that these are in fact what the figures are, and then goes on to characterize them. Triangles, circles, and ellipses are all closed figures, but parabolas and hyperbolas open up  indefinitely and indeed go on forever.

In Book 2, Apollonius shows that it is possible to construct a pair of lines that enclose a hyperbola in such a way as to be asymptotes (“not capable of meeting”). In the diagram below, DF and DE are the asymptotes.

The extraordinary thing about these lines is that not only do they never meet the hyperbola, but they will eternally continue to get closer. He proves this in proposition 14: The asymptotes and the section, if produced indefinitely, draw nearer to each other; and they reach a distance less than any given distance.

In other words, as you go way down the hyperbola, no matter how close the asymptote and the section might be, it is always possible for them to get closer — and yet they will never touch. The hyperbola will keep expanding indefinitely, and yet will always curve inwards sufficiently so that it will never meet the straight lines enclosing it.

And there we have it: a love between two things as fundamentally different as a curve and a straight line, always tending in the same direction, always seeming to know where the other is, always getting closer, but destined never to converge.  It is an image that lies between separation and consummation — an affair that can never be fulfilled but also never finished. They can never get enough of each other.

At the end of Plato’s Symposium, Socrates convinces Aristophanes and Agathon that “the genius of comedy was the same with that of tragedy, and that the true artist in tragedy was an artist in comedy also.” In his own uncanny, abstract way, Apollonius is an artist who can suggest the tragic and the comic at the same time.


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