![[Cubic surface]](../images/cubic.jpg "A cubic surface with an Eckardt point")
Here on the left (click to enlarge) you can
see—to contradict an earlier
statement I made that they are difficult to picture—a
(nonsingular) cubic surface, one of the beasts that I've spent a good
part of my thesis studing (the arithmetic of). Specifically, this is
the surface with (affine) equation
y³-3x²y+z³-3z=0.
The white rods (fifteen of them, if you count well) are not part of
the surface itself—or rather, they are, but they've been
emphasized for clarity: they represent the straight lines lying on the
surface. There are always twenty-seven straight lines on a
nonsingular cubic surface, but all might not be “real” in
the sense that some are actually pairs of complex conjugate lines; and
this particular surface has fifteen real lines and six pairs of
complex conjugate lines. Sometimes three lines on the surface
(necessarily in the same plane) meet in one point: then that point is
called an Eckardt point; this surface happens to have six
Eckardt points (all real), three of which have been shown on the
picture as small bright pink spheres (the other three are at infinity
so you cannot see them); one of them (namely (0,0,0)) is at the center
of the image. I've already mentioned
Eckardt points on this 'blog; they have many remarkable properties,
but they make the arithmetic of the surface rather harder to study
when they exist. The sort of question one might ask is this: given
that the surface has one point with rational coordinates (namely
(0,0,0)), and since its equation has rational coefficients, is it true
that there are points with rational coordinates arbitrarily close to
any real point? (And the answer, for this surface, is
yes.)
The image was made with the Persistance of Vision
(“POV-ray”) raytracer. I don't deserve much credit
since POV-ray has a primitive (quite appropriately called
cubic!) which draws a cubic surface. My work as a
mathematician in composing this image was limited to finding the
equation of a nice cubic surface having some Eckardt points and then
computing the equation of all the lines on it (a horrendous task in
general, but relatively easy for this one surface since it has a very
simple equation in which variables are separated). Note incidentally
that the colors are not on the surface but come from three colored
light sources.
I've also made a little animation of the rotation of the cubic surface (984kbyte AVI file) from the images computed by POV-ray. (Don't ask me how to read it or what codec it uses, I don't know anything about this stuff: I just fiddled around with MPlayer/MEncoder, randomly tweaking the command line options until it produced something that seemed like it was an animation.)
Would you believe it? Cubic surfaces have their own Web site!