A night in the MacLab with Mathematica and Matt's formula for the top half of a torus in 3-space led to the following spew. We have done nothing but upload the Mathematica 3.0 notebook files. We have plans --- airy as they may be --- to write up a tour through our brains, but as of now it will be up to you to explore and discover on your own. A mental note has been made to create comments in the notebook files in the future. :). At any rate:

Formula for Torus' top half:

z = Sqrt(- x^2 - y^2 + Sqrt(x^2 + y^2) + R

Without R, this will generate a torus of outer radius one and inside "hole" of radius zero (it's just pinched at the origin). Adding the R value will scale both the torus' outer radius and the radius of its hole equally but oppositely. Bigger R means bigger hole, but smaller torus, and vice versa. Values of R betweem 0 and 1/4 are acceptable. (Why 0 and 1/4? When you hit 1/4, which is actually treated as Sqrt(1/4) (notice the whole expression is under a Sqrt), you have created a hole with 1/2 radius, but shrunk the torus' outer radius to 1/2... a cylinder.) More elucidation forthcoming.

Donuts    [5k]     : donut excursion (grafx removed)
Donuts II [93k]    : excursion in 2d (grafx present)
DonutFun  [4k]     : manipulations   (grafx removed)
Tori I    [2,500k] : tori excursion  (grafx present)
Tori II   [4,381k] : manipulations   (grafx present)

21st April 1998 the SRO
brainsik & strthrwr