We just finished talking about quadric surfaces in the Calculus III class that I’m teaching, so the timing was perfect to make my own.
A hyperboloid of one sheet has the amazing property that it is a ruled surface. That is, it is the the union of straight lines. Actually, it is doubly ruled—there are two straight lines through every point on the surface. The fact that the hyperboloid is doubly ruled allows us to make it out of skewers.
Here are photos of the final product. Left on its own the rubber bands pull the sculpture into a solid column. Twist the top and bottom (being careful of the pointy ends) and it opens into a hyperboloid.
(I am a little concerned about the lifespan of this object since rubber bands degrade over time. I’m thinking of putting a drop of wood glue in each joint and taking the rubber bands off.)
A few years ago I made another physical model of a hyperboloid. I used a jig saw to cut two circles out of wood, drilled holes in them, and threaded string between them. Then, when the two circles are twisted opposite directions the strings stay straight and form a hyperboloid.