John Baez has a nice article on Klein's quartic. Early on in our discussion, I tried making one out of paper, but I reached a point where it looked too crumpled to illuminate the topology. Greg Egan made a movie of the dual of the heptagonal tiling, but we still haven't seen a good rendering of the original. Since there still seems to be some interest, here are pictures of my attempt. We start with one quarter of the surface:

Topologically, it's a disk with two holes and an outside edge, but it's clear there's trilateral symmetry. Here's the skeleton of this piece:
This piece corresponds to the grey heptagons:

Now we join two of these pieces together to get half of the Klein quartic:

And here's its skeleton:

There are four heptagons on each piece that interlock with each other (where grey meets green).

Now we add two more pieces to get all the heptagons (though we haven't connected all the edges yet):

The corresponding skeleton--

--curls up into a tetrahedron:

It's possible to make the paper model curl up like that, but it gets so crumpled that it's not worth taking a picture of. It should, however, illuminate the structure of Klein's quartic enough that you can match up the heptagons in this hyperbolic tiling to the ones made of paper: