Hyperbolic geometry differs from Euclidean geometry in many fundamental ways. One of the significant differences is that many objects which can be constructed in Euclidean geometry cannot be constructed in hyperbolic geometry. Perhaps the most notable is the rectangle, and the example discussed in the introduction to this document provides some insights regarding the difficulty of constructing this figure. However, it can prove just as rewarding for students to consider other geometric objects, deciding whether they, too, exist or fail to exist in hyperbolic geometry. Moreover, attempting these constructions can help students gain important insights into the necessity of making careful definitions. Constructing (or attempting to construct) rectangles, Saccheri quadrilaterals and Lambert quadrilaterals in hyperbolic geometry, for example, helps to highlight distinctions between these objects. This leads to a note of warning: before assigning the exercise, either make sure that you have given the students precise definitions for the objects in question, or be prepared for the ensuing discussion!
In hyperbolic geometry, decide which of the following objects could exist. For those that do exist, describe how you would carry out a construction (justify your steps). For those that do not, explain where a construction that would work in Euclidean geometry fails here:
- Saccheri quadrilateral
- Lambert quadrilateral
- equilateral triangle
The exercise provides a balance between formal justification and exploration. In order to show that the object truly exists, one needs to demonstrate the construction in the abstract, providing formal justifications. On the other hand, attempting constructions in the models can provide insight on those objects that cannot be constructed. It can also lead to a nice discussion on the difference between, "It cannot be done," and, "I am unable to do it."
Some of the objects may seem trivial to us, but provide surprising challenge to students. Certainly, if rectangles do not exist, then squares do not either, but some students need to try to construct a square before they realize this. On the other hand, since circles are somewhat strange-looking in the models (particularly the Beltrami-Klein disk), students may find their existence somewhat hard to justify.
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