I learned about the ovals of Cassini recently, which are curves defired by the product of their distances to two points, and as with any geometric shape defined in terms of distance, I became curious how they looked in taxicab geometry. Here are several curves, drawn for different constants:
This is the first time I've seen taxicab geometry produce a curve. Taxicab circles, ellipses, parabolas, and hyperbolas are all comprised of only straight lines.
Hidden in the above curves is the taxicab equivalent of the lemniscate of Bernoulli:
I like how this shape resembles the infinity symbol we're used to but also combines elements of taxicab geometry (the half circles on the ends) and traditional Euclidean geometry (the curves).
If you know of any other interesting shapes to draw in taxicab geometry, email me.