This question is naive. My association with Möbius strip comes from not being able to smoothly extract positive solutions of the diophantine equation
I got a parametrization (must be classical):
$\quad x\ :=\ \alpha^2 + 2\alpha\cdot\beta - \beta^2$
$\quad y\ :=\ \alpha^2 + \beta^2$
- $\quad z\ :=\ \beta^2 + 2\alpha\cdot\beta - \alpha^2$
(see that $z^2-y^2=y^2-x^2$), and I considered also the other three related parametrizations, obtained by replacing one or the both $x\ z$ by $-x\ \ -\!z$ respectively. I still don't seem to parametrize the positive solutions alone. It feels that the positive solutions flow seamlessly into the negative solutions (or mixed solutions). I'd appreciate some expert comments about this situation to educate me, please.