Hyperboloid of one sheet generator

Physical version of this manipulative https://www.geogebra.org/m/psvbff8u

You can twist the top to change the shape (in theory… the problem is the height changes so might work better with elastic)

You need to thread the model with some sort of string or fishing line (alternating up and down, then tie a neat knot joining the two ends when you're done)

Tinkercad version of the base here. The “lid” has a star hole with the diameter increased by 0.5mm (from 10mm to 10.5mm)
https://www.tinkercad.com/things/ah9LkHkcEFJ-hyperbolic-surface-generator