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(2*u*)sin(*theta*/2) )

*z* = *a* ( cos(*u*)sin(*theta*/2) + sin(2*u*)cos(*theta*/2) )

*theta* = *theta*

Where:

0 <*u* < 2p

0 <*theta* < 2p

and "*a*" is a constant that gives the aspect ratio.

Where:

0 <

0 <

and "

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