David Madore's WebLog: Behold the cubic surface!

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Behold the cubic surface!

[Cubic surface]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!

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