Ruled surfaces
Summary:
Craft: Using skewers and rubber bands, build wodden models of the hyperbolic paraboloid and the hyperboloid.
Math: Create ruled surfaces, surfaces made by tracing out lines. Analyze their gaussian curvature. Make doubly ruled surfaces, and observe how they flex.
Summary
We have seen several way to construct surfaces with negative curvature, based around excess angles or excess length. Now we will go backwards, starting from a way of constructing surfaces, and finding that they naturally manifest negative curvature. We will examine Ruled surfaces, surfaces traced out by a line moving through three dimensional space. The quintessential examples are the hyperbolic paraboloid (seen in [[Hyperbolic origami]]), and the hyerpboloid (like the large cooling towers in nuclear powerplant). Ruled surfaces are particularly easy to construct, so they are often used in architecture.
Activity
Here is a model of the hyperboloid.
Here is a top down image of a hyperboloid.
