Vi Hart — Beadwork
I don't have very much experience in beading, but it's not too hard to make a hyperbolic plane! The above version has seven beads around each hole, and three beads meet at every corner. Theoretically you could extend this model infinitely, but eventually the beads would be packed too densely to fit into our limited Euclidian space (or, in my case above, I ran out of beads).
Here's a drawing of the pattern, with the outer beads squished to fit it all onto a flat plane:
I started with the center ring and the seven rings surrounding it, to get an initial 'flower'.
I then added the next layer, being careful to always make sure I had seven beads in each loop. The diagram below shows specific instructions for how many new beads are added to get each new loop of seven around the border of the central flower.
Another layer could be added to the outside, or perhaps decorative fringe or dangly bits. It is also lovely left unadorned.
It looks very nice when using different colours:
Also, while this particular section of the hyperbolic plane is vaguely circular, once you get a feel for hyperbolicness you can extend in other direct ions, such as making a long strip of hyperbolic plane for a necklace. There are also many other ways of beading a hyperbolic plane, such as this dense packing where four pentagons meet at each vertex:
If you attempt to make one, send me an email! Also let me know any comments or suggestions for improving these instructions.
If you're interested in making the hyperbolic plane out of other things besides beads, you could also try making it out of balloons.