I have been doing some outreach work on conveying notions of hyperbolic space to people with limited math backgrounds. One idea I like to use is to talk about the hyperbolic plane as being like the opposite of the sphere. For instance, if I start with a circle (which is flat), then I shrink the boundary down to a point, it will bulge out into a sphere. So do the opposite: start with a circle and expand the boundary out to something infinite. This can be used to give some intuition about the classical models for the hyperbolic plane (upper half-plane, Poincaré disk, Klein disk, hyperboloid). In each of those models, one can think of it as having a boundary that is a circle of infinite radius, in different ways.
An audience member asked me an interesting question: why don't we use the hyperbolic paraboloid (familiar from calc 3) as a model for the hyperbolic plane? It captures this notion of a boundary needing to bulge out in different directions as one moves away from the center, like a Pringles potato chip or like choral. I'm wondering if anyone has ever made that idea precise, or if there is some reason why that can't be done. Is there a formula for a metric on the hyperbolic paraboloid that makes it a nice model for the hyperbolic plane? Is there some useful geometric interpretation for this? Are there interesting applications?