Klein Bagel showing Lemniscate cross-section.
The Klein Bagel
   What is the Klein Bagel? The Klein Bagel is a term I coined for a specific immersion of the Klein Bottle manifold into three dimensions. It can be parameterized by taking any figure 8 curve such as a Lemniscate or 2-1 Lissajous Figure and rotating it around the Z-axis to form a Torus, but give the figure 8 a 180 degree twist as you rotate it around the full circle. I discovered this parameterization myself in 1994, but if you know of anybody who invented it earlier let me know so I can give them credit! The 2-1 Lissajous version is particularly easy to parameterize (see below), but I think the Lemniscate version shown above is more aesthetically pleasing. Given cylindrical coordinates you have:

R = 1 + a ( cos(u)cos(theta/2) - sin(2u)sin(theta/2) )
z = a ( cos(u)sin(theta/2) + sin(2u)cos(theta/2) )
theta = theta

Where:

0 < u < 2
p
0 < theta < 2p
and "a" is a constant that gives the aspect ratio.
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