A First Example: Constructing a Rectangle in Hyperbolic Geometry
Formal and precise definitions are vital to a rigorous approach to geometry. This necessity is not always apparent to students, in large part because of preconceptions about geometric objects. One example that can help students to "focus on the formal" is the problem of constructing a rectangle.
The standard definition of a rectangle is "a quadrilateral with four right angles." Students' experiences in Euclidean geometry, however, lead them to believe (and rightly so) that this definition is somewhat more than they need. In Euclidean geometry, a rectangle can be equivalently defined as "a quadrilateral with at least three right angles," or "a parallelogram having at least one right angle." These equivalences fall apart in hyperbolic geometry and this provides an opportunity to highlight for students the importance of relying on formal definitions.
In studying Euclid's parallel postulate, one fundamental theorem
establishes the logical equivalence of the postulate and the
existence of rectangles. An immediate consequence, of course, is that in
hyperbolic geometry, rectangles do not exist. Although students may seem
to understand
and can even reproduce the proof, the counterintuitive nature of the result
makes the theorem a difficult one to visualize. An extremely useful
activity for the students is to have them propose different methods of constructing
rectangles in Euclidean geometry, then seeing how those methods fail in
hyperbolic geometry. For example, here is one way to construct a
rectangle:
- Choose arbitrary points A and B in the plane and draw segment AB.
- Raise a perpendicular m to AB at A.
- Raise a perpendicular n to AB at B.
- Choose an arbitrary point C on line n.
- Drop a perpendicular from point C to line m. Let D be the foot of this perpendicular.
- Then quadrilateral ABCD is a rectangle.
Where does this construction go wrong in hyperbolic geometry? From an abstract point of view, a good student will be able to recognize that following the construction, angle BCD is not necessarily a right angle. Nevertheless, without a model it is difficult to visualize a quadrilateral constructed in this manner that does not have four right angles. Below is the attempted construction of a rectangle in the Poincaré half-plane model of hyperbolic geometry. Click on "Rectangle Construction Steps," then click on each step in turn to see the construction:
Given that the Poincaré half-plane is conformal (i.e. it represents angles faithfully), the difficulty with the construction becomes clear: angle BCD is certainly not a right angle. This highlights the importance for students of relying on formal definitions rather than on their experiences in Euclidean geometry when studying hyperbolic geometry. In hyperbolic geometry, it is possible to have a quadrilateral with exactly three right angles!