Reading the Wolfram MathWorld™ article on the Möbius strip I came across an equation that was news to me.
After the familiar parametric form
it is said that “In this parametrization, the Möbius strip is therefore a cubic surface with equation…”
That’s a neat cubic equation for sure. But… wouldn’t the sign of the left hand side of (2) (call it ) distinguish the “positive” and “negative” sides of the surface, contrary to its non-orientability?
I went to SageMathCloud™ to check and got this plot of (2):
That’s… not exactly the Möbius strip as I know it. But it’s true that (1) implies (2): the cubic surface contains the Möbius strip. Here is the plot of (1) superimposed on the plot of (2).
The self-intersection arises where the “positive side” suddenly transitions to negative .