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?
Best Answer
I think the hyperbolic paraboloid is a great model to build intuition about what negative Gaussian curvature is, and how it would interact with things like angle sum. You might also look at images for “crocheted hyperbolic plane”, or create one yourself, to build some even stronger intuition how adding more matter than what would be needed for a planar setup will automatically lead to curvature, and how that curvature might have different embeddings which do not affect the intrinsic metrics.
As comments pointed out, the curvature is not constant, so at that point it's a poor model for the hyperbolic plane. Other models like the tractricoid or Amsler's surface have constant negative curvature, so they can represent a portion of the hyperbolic plane with accurate metrics.
But the key problem with all these isometric 3d models is that they at best only represent a portion of the plane, not all of it. They are ill suited to studies of the global structure of the hyperbolic plane. So at some point you have to give up hoping for an isometric embedding, and get used to dealing with somewhat non-intuitive metrics.
As for the first paragraph, about building intuition by speaking about a circle infated to infinite radius, I'm not sure I'd be happy with that. I can see many ways how this could be misread. But that's my personal impression; if your students feel that it helped them, that's good enough in my opinion.