Math 302, Assignment 1, Fall 2000

Due Wed August 30 (at the beginning of class)

  1. Using that the formula for a straight line is ax + by = c , prove that there exists a line through any two points.
  2. Refer to the axioms for straight lines which were handed out in class (they are also available at   http://www.math.uiuc.edu/~stolman/m302/handouts/axioms.html ).  Consider the subset of the plane consisting of all points (x,y) so that y is greater than 0. This is called the open upper half plane.   Determine whether each of the axioms is true or not on the open upper half plane   Explain each of your answers.  Diagrams may be helpful.
  3. Build an approximate model of the hyperbolic plane, useing the sheet of cutouts given to you and following the directions below. You should build this carefully, with scissors and tape. Please construct your model with care, as you will use it for many explorations of hyperbolic space.  Bring it to class in your shoebox.

The 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). No pieces should be allowed to overlap.

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 two peices 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.