Math 302, Fall 1999

Instructions for building approximate models of the hyperbolic plane

Friday in class we passed out two sheets of cutouts for building approximations to the hyperbolic plane. You should build both of these carefully, with scissors and tape, before class on Monday. Please construct your models with care, as you will use them for many explorations of hyperbolic space.
The first, "rainbow" handout is a sheet with many annular arcs to cut out. This is used to build the "annular hyperbolic plane" as described in Chapter 5 of Henderson's book. To build this model, cut out each piece, and note that it is one-sixth of an annulus, with an inner rim, an outer rim, and two short radial ends.

Next, tape these together. The rules are as follows. The inner rim of each piece gets taped to (most of) the outer rim of another one. A short radial end of one piece can be taped to such an end of another piece, oriented so they form a larger part of one annulus (and NOT in an "S" shape).

Note that because, in the flat plane out of which we cut these, the inner and outer rims don't quite have the same curvature, so they don't quite want to be taped together. It is important to tape carefully, and avoid getting any wrinkles near the seam.

Also note that you should never form loops in your hyperbolic plane: avoid taping a short end of one piece to a short end of another piece already in your model. The annular hyperbolic plane continues infinitely in all directions: left and right within the annular strips, and radially inward and outward.

The second, "mickey mouse" handout has a number of hexagons and heptagons, printed in groups of three. Cut out each of these "mice", and note that between the hexagonal "ears", you can find a concave part of its boundary where we see three edges, of first one hexagon, then the heptagon, then the other hexagon.

The rule for taping these pieces together is simple: ensure that each vertex ends up surrounded exactly by two hexagons and one heptagon. (Also, again, avoid making any loops in this space: always use new pieces to extend the part you've already built.)

In practice, you can get pretty far by doing the following: Look for some place on your model where on the boundary you find a hexagon with three free edges. Take a new "mouse" piece, and tape the concave part of its boundary (between the "ears") to these three free edges.

This model seems easier to build than the annular hyperbolic plane, because we are matching straight edges against other straight edges. One piece of tape will do for each pair of matched edges. You do need to be careful at each vertex, since we're producing a cone point of greater than 360 degrees there.