(I didn't realize phpBB imposed a length limit on the subject line... took a while to get one that fit. :?)
Over the past few days without an internet connection, I've been intensely interested in enumerating all possible types of quadric manifolds in 4D (i.e., those representable as a 2nd degree polynomial in 4 variables). As a result of a little independent research, I've come up with a scheme for enumerating all possible geometrically-distinct quadrics in n dimensions, and a formula that states how many non-trivial, non-degenerate distinct quadrics there are in n dimensions.
I'll state, without proof for now, the result: in n dimensions, the number of geometrically-distinct, non-trivial quadrics is:
n<sup>2</sup> + n - 1
where n >= 1.
(Trivial in this case means it either has no solutions, or contains only one point.)
Now, this count includes degenerate quadrics, which are products of 2 hyperplanes, corresponding with 2nd-degree polynomials that can be factored into a product of lines. The count without degenerate quadrics is:
0 (for n=1)
n<sup>2</sup> + n - 3 (for n > 1)
Plugging in n=2 and n=3 gives the familiar number of conic sections and quadric surfaces. Plugging in n=4 gives the number of quadric manifolds in 4D, which happens to be 17, broken down as follows:
1) The hyper-ellipsoid
2) Five hyperbolic manifolds (including the ellipsoidal cone)
3) Two parabolic manifolds
4) Nine non-degenerate cylinders of lower-dimensional quadrics. (I'm using "cylinder" here in the calculus sense; in wendy's terms it's the prism product.)
Among the cylinders are the spherinder (sphere-prism in wendy's terms), the cubinder (circle-square prism), and the conical cylinder (cone prism). The duocylinder is not representable by a single 2nd-degree polynomial in 4 variables.
One of the two parabolic manifolds I call the Supersaddle or Supercone (in analogy with how conic sections derive from the 3D cone): its intersections with different hyperplanes yield all 4 types of paraboloids and hyperboloids in 3D. The equation of the Supersaddle is:
x<sup>2</sup> + y<sup>2</sup> - z<sup>2</sup> - w = 0
Sadly, I could not find any 4D quadric that can give rise to all 9 3D quadrics. The Supersaddle is the closest candidate, but there is no ellipsoid in it. Interestingly enough, the 4D ellipsoidal cone can only yield hyperboloids of 2 sheets, 3D cones, and ellipsoidal paraboloids on intersection with hyperplanes. There's no way to get a hyperboloid of 1 sheet out of it, nor the 3D saddle.
Now, in 5 dimensions, there are:
- 1 hyper-ellipsoid (as always in every dimension)
- 6 hyperbolic quadrics
- 3 parabolic quadrics
- 17 non-degenerate cylinders (according to the number of quadrics in 4D)
giving a total of 27 quadrics.
If people are interested, I'll explain how I arrived at these numbers. :-)