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The Hyperbolic Toolbox - Introduction: Models for Non-Euclidean Geometry

Stephen Szydlik

In The Proof,  NOVA's program about the proof of  Fermat's last theorem, Andrew Wiles speaks of  his experience of doing mathematics:

"Perhaps I could best describe [it] in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it's all illuminated. You can see exactly where you were."

Though we as mathematicians with some research experience might recognize Wiles' description of mathematics, as teachers of the subject, it is easy to forget that our students have similar experiences with mathematics that is familiar to us.  In particular, college students taking a first course in non-Euclidean geometry are entering a "dark mansion" in many ways.  For most of them, the term "geometry" is synonymous with "Euclidean geometry";  a world in which Euclid's fifth postulate does not hold is counterintuitive, confusing, and perhaps even ridiculous to contemplate.


Euclid's Five Postulates

I. We can draw a [unique] line segment between any two points.
II. Any line segment can be continued indefinitely.
III. A circle can be described with any center and radius.
IV. Any two right angles are congruent.
V. (Playfair's form)  Given a line m and a point P not on m there exists a unique line n through P that does not intersect m.
Collectively, these five assumptions are referred to as "Euclid's postulates," or "Euclid's Axioms."  The fifth postulate is called "Euclid's parallel postulate," or the "parallelism axiom."  (See Baragar, 2001, p. 12)

The confusion of students is understandable, particularly when one considers the historical development of hyperbolic geometry.  From the time Euclid's Elements was published around 300 BC until the beginning of the 18th century, mathematicians attempted to prove Euclid's fifth postulate from his first four axioms.  The idea that a consistent geometry could be developed from his first four axioms and the negation of the so-called "parallel postulate" was not even considered as a possibility.  Although some mathematicians (e.g. Lambert, Saccheri) unknowingly discovered theorems in hyperbolic geometry in the 18th century, it was not until 1829 that Lobachevsky published the first paper on non-Euclidean geometry.  The pioneers of non-Euclidean geometry (including Gauss and Janos Bolyai, who are generally given credit with Lobachevsky for its discovery) had an axiomatic approach to the subject:  they assumed the negation of the parallel postulate, and subsequently established results that must follow as a logical consequence (e.g. the angle sum in a triangle in hyperbolic geometry is less than 180 degrees).  In some sense,  they were "stumbling around" in the dark mansion of hyperbolic geometry while they learned about its structure.

The work of these mathematical pioneers was particularly impressive, for they had no examples of a non-Euclidean geometry to help them visualize their results;  they were purely abstract.  Though there were no inherent contradictions in this new geometry, neither was there any evidence that this geometry possessed any relationship to reality.  Only later in the 19th century following the work of Beltrami, Klein, and Poincaré were there models of hyperbolic geometry available in which the theorems of the subject could be visualized.  One might say that the models provide a "lightswitch," illuminating that dark mansion of hyperbolic geometry.

As teachers of geometry, we are called to help our students find their way in the dark mansion.   Just as it did for the pioneers of the subject, hyperbolic geometry proves to be a challenge for our students who have no experience with a non-Euclidean geometry.  However, the NCTM's Principles and Standards for School Mathematics (p.18) tells us,

"Teaching mathematics well involves creating, enriching, maintaining, and adapting instruction to move toward mathematical goals, capture and sustain interest, and engage students in building mathematical understanding."

It is the objective of  this document to describe some tools and related activities that may help students develop their understanding of hyperbolic geometry.  These tools employ the Geometer's Sketchpad software, allowing the user to perform standard straightedge and compass constructions in three of the standard models of hyperbolic geometry:  the Beltrami-Klein disk, the Poincaré disk, and the Poincaré half-plane.  To be specific, the tools are "scripts" in Geometer's Sketchpad, or macros that automate ten "standard" straightedge and compass constructions in each of the models.  These constructions include, among others, constructing  hyperbolic lines and circles, dropping and raising perpendiculars, and constructing  midpoints.  A more complete description of the tools themselves can be found in the What's in the Toolbox? section (Page 6).

The purpose of the tools is to provide occasions for students to become actively engaged with hyperbolic geometry.  Though the tools may be used by the instructor to develop applications for classroom demonstrations, their real power is that they can be used as building blocks for students to investigate the subject interactively, through constructions in the models.  In this way, the tools can be used to foster an active classroom environment, giving students an opportunity to find the lightswitch themselves.

To get a fundamental sense of the nature of the Sketchpad tools, consider the problem of constructing a rectangle in hyperbolic space:


A First Example:  Constructing a Rectangle


The purpose of the rectangle construction exercise is to help students visualize and understand a fundamental concept in hyperbolic geometry.  This construction using the Poincaré half-plane makes a counterintuitive result transparent.  More importantly, the hyperbolic construction tools allow students to find the result themselves.  While the example above is necessarily shown as a demonstration, a much more pedagogically powerful exercise is to have the students use the tools themselves, in their model of choice.  In this way, rather than being passive receivers of knowledge, students become active participants in the learning process.  More examples of ways in the tools can be used in the classroom can be found in the Teaching Examples section (Page 8).

Stephen Szydlik, "The Hyperbolic Toolbox - Introduction: Models for Non-Euclidean Geometry," Convergence (January 2005)