## Example 6.5:  An Absolute Standard of Length

In Euclidean geometry, there an absolute standard of angle measure.  Not only do we all have the same conception of what a "degree" is, but the definition of a "degree" is standard.  It is 1/90 of a right angle.  The limitations of the straightedge and compass notwithstanding, one can construct an angle of any measure to any required degree of accuracy, without a protractor.  One does not require the use of a measuring tool (i.e. a protractor) to construct an angle.  Said differently, if you constructed a right angle, then called a friend and asked them to do the same (with straightedge and compass), you will end up with congruent angles, although neither of you have used a protractor.

On the other hand, there is no absolute standard of length in Euclidean geometry.  If one wants to construct a segment of any particular length, one requires a ruler of some type.  If you constructed a segment and somewhat arbitrarily defined it as your "unit length," then called a friend, you would be unable to describe to your friend how to construct a segment congruent to yours with straightedge and compass without making some reference to a ruler.

Big Idea:  In hyperbolic geometry, it is possible to define an absolute standard of length, as well as an absolute standard of angle measure.

This fact is a direct result of

Big Idea:  In hyperbolic geometry, if two triangles are similar, then they are congruent.

These two ideas are additional examples of the strange and wonderful properties of hyperbolic geometry, as well as the counterintuitive notions with which students may struggle.  Having the students illustrate these concepts in the models can prove tremendously illuminating.

Exercise:

1. In Geometer's Sketchpad, open one of your hyperbolic models and construct an equilateral triangle.  Measure the lengths of the sides (using the appropriate tool) to ensure that your triangle truly is equilateral.
2. Now measure the angles of your triangle using the appropriate tool.  Again, make sure that the result agrees with what you would expect.
3. What is the angle sum in your triangle?  What happens to the angle sum as you change the length of a side of the triangle?  Be specific.
4. What angle measure corresponds to a side length of 1 unit?  Compare your answer with someone using the same model on a different computer.  How does this illustrate the "absolute standard of length" in hyperbolic geometry?
5. Construct a second equilateral triangle in the same model and measure its side lengths and angles.  Find a triangle similar to your first triangle, but not congruent.

Discussion:

This exercise can enhance student understanding in several ways.  Note, for example, that the first question requires the student to recall the construction of an equilateral triangle.  The standard construction is, in fact, Euclid's first proposition and does not require the use of a parallel postulate; it is as valid in hyperbolic geometry as in Euclidean geometry.  The construction in a non-Euclidean geometry helps students to recognize this.

More explicitly, the exercise either reinforces (or introduces, depending upon how the instructor uses it) the notions of angle sum, absolute unit of length, and congruence vs. similarity in hyperbolic geometry.

The interactive Sketchpad diagram below can be used to answer the questions in the Poincaré half-plane model.  In particular, one could define a unit of length in that model to be the length of the side of an equilateral triangle whose angles measure approximately 52.7 degrees.