Math 402, Assignment 1, Spring 2008

Due Wed January 23 (at the beginning of class)

  1. Refer to the "carpenter's method for checking straightness" in Chapter 1.  Using a scissor and paper (construction paper or light cardboard may work best), try to construct a straight line. Do NOT use a ruler. Freehand your original line and then try to improve it   How did this method work in practice for you?

    Explain why this method works in theory.  Why will light shine through if the edges are not straight, and why will no light shine through if the edges are straight?  Consider carefully what properties of straightness or what definition of straight you are using.

    Why do you think carpenters use this method instead of a straight edge?
  1. Refer to the axioms for straight lines which were handed out in class (they are also available on the class website under "handouts". Consider a plane with a disk of radius 1 (this means a circle and its interior) removed.  Note that a line ends when it hits this disk; it doesn't ``jump over'' the hole. You can construct one by taking a piece of paper and cutting out a hole with radius 1 inch.  Determine whether each of the axioms is true or not if you use a plane with a disk removed instead of a plane.  Explain each of your answers.  Diagrams may be helpful.

  2. 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.

Try to build your model so that it is roughly as wide as it is tall.