## The Hyperbolic Toolbox: Non-Euclidean Constructions in Geometer's Sketchpad

2.  Selling the Tools: A Pedagogical Argument

At the University of Wisconsin Oshkosh, the mathematics department teaches two half-semester courses in geometry, one in Euclidean geometry and one in hyperbolic geometry.  Together, these courses make up a four-credit sequence, designed for prospective secondary teachers who are in their junior year of college.  The sequence has both content and pedagogical goals, including the development of

• an enriched understanding of Euclidean geometry,
• a deeper conceptualization of the nature of proof and axiom systems,
• exposure to a practice of teaching consistent with the NCTM's vision of mathematics education.
It is with the above goals in mind that the hyperbolic toolbox has been completed.  The models considered -- the Beltrami-Klein disk, the Poincaré disk, and the Poincaré half-plane -- provide a context for visualizing the theorems of hyperbolic geometry.  By giving students examples of universes where Euclid's parallel postulate fails, they give students a chance to experience hyperbolic geometry in a concrete way:  to see how it is possible for two lines through a point to be parallel to a third line, to appreciate a world where rectangles do not exist, and to understand how it is possible for the angle sum in a triangle to fall short of 180 degrees.  The Teaching Examples page illustrates several ways in which the tools accomplish this.

Most of the students taking the geometry sequence at the University of Wisconsin Oshkosh have previously completed a course on the nature of proof.  For many, however, the geometry course provides the first opportunity to apply their background in a specific subject.  The geometry course emphasizes proper justification, proof, and the nature of axiom systems.  A benefit of courses in Euclidean and hyperbolic geometry is that students can clearly see the role of axioms, for the only difference between the two geometries is the parallel postulate.  Students learn to be careful about what statements are theorems in Euclidean geometry, which are theorems in hyperbolic geometry, and which are theorems in both.  Constructions in the models aid in making these distinctions in two ways.  For example, consider the statement "Two lines perpendicular to a third line are parallel to each other."  After exploring that statement in the hyperbolic models, a student should get a sense that it is a theorem in hyperbolic geometry as well as Euclidean geometry.  The student then knows that proving the statement will not require a parallel postulate.  This is indeed a powerful connection for a student to make.  Second, a compelling aspect of straightedge and compass constructions is the clear necessity of justification.  When a student finishes a construction, only rarely does an instructor need to ask, "How do you know you are correct?"  Students naturally ask that question themselves.

By examining hyperbolic geometry we obtain a deeper understanding of Euclidean geometry.  The study allows for a contrast; we are able to see what makes Euclidean geometry "Euclidean" by negating a single axiom.  Working within the hyperbolic models provides us with another quite different way of enriching our understanding of the Euclidean world.  Though each of the models is "hyperbolic", the underlying space for each is the Euclidean plane.  For example, the Klein model consists of a fixed circle and its interior, and "lines" in the model are merely chords of the circle. It is a purely Euclidean object.  When viewed in this manner, the models can be used as a means of better understanding Euclidean geometry.  One useful construction in the Klein model, for example, is the pole of a hyperbolic line:  it is the intersection of tangents drawn to the Klein disk at the endpoints of the chord defining the line.  Constructing the pole of a Klein line thus involves standard Euclidean straightedge and compass constructions (specifically, raising a perpendicular to a line from a point on the line).  In this way, the models offer an opportunity to understand Euclidean geometry through the window of hyperbolic geometry.  This argument is fundamental to the nature of the geometries, for the relative consistency of hyperbolic geometry is demonstrated by the existence of models of hyperbolic geometry in Euclidean geometry.

It should be noted that the fourth goal is an implicit one.  The course is content-driven, and discussion of teaching methodology is not an explicit part of that content.  However, the NCTM's Professional Standards for Teaching  tells us, "Learning is an active, dynamic, and continuous process that is both an individual and a social experience." (see Professional Development Standard 3.)  We would like our future teachers to support such a learning environment, and so providing them with a model of such a classroom is vital.  The examples provided on the Teaching Examples page offer possible ways in which the tools can be used to stimulate an active learning environment.

As noted in the Background section, others have produced Geometer's Sketchpad construction tools for the Poincaré disk and (in part) for the Poincaré half-plane models.  Completing the tools for the half-plane and contributing tools for the Beltrami-Klein disk rounds out a package that allows students to analyze hyperbolic geometry in a concrete way from several different perspectives.  As the NCTM notes in the Principles and Standards for School Mathematics,  "It is important for teachers to highlight ways in which different representations of the same objects can convey different information and to emphasize the importance of selecting representations suited to the particular mathematical tasks at hand." (p. 362)  The different models do indeed offer different representations of hyperbolic geometry.  The Poincaré models are conformal, and their faithful representations of angle measure make them attractive for this reason.  On the other hand, the hyperbolic axiom is most immediate to visualize in the Klein disk, and constructing the common perpendicular between two divergently parallel lines in the Klein model is an accessible problem.  The three models together provide a powerful means of considering hyperbolic geometry from different perspectives.

### The Role of Technology

"Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning."
- The Technology Principle, the Principles and Standards for School Mathematics (p. 24)
Technology in the form of a dynamic geometry package such as Geometer's Sketchpad offers a powerful means of studying geometry.  As the value of this technology has become recognized, instructors have integrated it into their geometry courses (see, for example, the text by Baragar).

Dynamic geometry software offers access to hyperbolic geometry in at least three fundamental ways. First, Geometer's Sketchpad allows the user to perform basic constructions in the hyperbolic models. Indeed, though not conceptually difficult, many of these constructions are quite challenging from a practical standpoint. Consider, for example, the problem of constructing a hyperbolic line in the Poincaré disk model. That is, given two points A and B interior to the disk, construct with straightedge and compass, the arc of a circle orthogonal to the boundary of the disk that passes through A and B. Theoretically, this is not a particularly difficult construction. If the center of the Poincaré disk is labeled O, then the steps are as follows: Constructing a Poincare line by hand using a straightedge and compass offers a challenge in careful record-keeping. (Click for an enlarged view).

1. Construct ray OA.
2. Construct the line through A perpendicular to ray OA. Label the points where this line intersects the boundary of the disk as C and D.
3. Construct the lines that are tangent to the boundary of the disk at C and D. Assuming A and B are not on a diameter of the circle, let A' be the point of intersection of these tangents.
4. Construct the Euclidean circle through A, B, and A'. The arc of this circle
5. is interior to the disk is the desired Poincaré line.

This construction is not conceptually difficult. However, it requires quite a competent student to carry out the steps by hand using a straightedge, compass, and pencil without getting lost in the details of the construction. The capability of the Geometer's Sketchpad software to hide and show parts of a construction on demand helps to eliminate this difficulty.

Second, Geometer's Sketchpad allows users to use basic constructions to build more complex structures.  Though understanding how to construct a line in the Poincaré disk is a worthwhile and important skill,  one does not want to go through each of the above steps when a hyperbolic line is called for.  If one wants to discuss dropping a perpendicular from a point to a line, for example, one does not want to spend too much time on the construction of the given line!  As with other technologies, Sketchpad extends the range of problems accessible to students.

Third, through its dynamic nature, Geometer's Sketchpad presents an opportunity for students to explore, experiment, and make conjectures.  With its "click-and-drag" capabilities, students are able to change a defining object in a sketch and watch the resulting effect on the rest of the sketch.  For example, a typical activity in the Poincaré models is to construct a triangle and observe that its angle sum is less than 180 degrees.  However, with dynamic geometry software, one can change the defining vertices of the triangle and observe how the angle sum depends on the triangle, specifically that the larger the triangle, the smaller its angle sum.  This investigation can provide a context for an introduction to Gauss's theorem on the proportionality of a hyperbolic triangle's area to its defect.  See the Teaching Examples section for more examples.