Geometry as a Laboratory Course

“Geometry is the art of careful reasoning from

badly drawn figures.”

   —Henri Poincarè, Dernières Pensées, 1913

How long does the professor have to wait for the students to show up for class? That very question became an issue in the Spring 2006 semester of MAT 405, Foundations of Geometry. Something very strange was going on; something different from the traditional format for a geometry class.

Hilbert’s approach was to use the concepts of formal logic. He started with undefined terms (Euclid had provided written definitions) and added postulates to specify the properties of the undefined terms, demonstrating that there are actually only five axioms for Euclidean geometry.

Despite Hilbert’s re-stating of Euclid, students studying Euclidean geometry continued until very recently to construct figures using an unmarked straight edge and a compass, the tools of ancient Greece. Poincarè’s observation certainly was true about figures drawn using these tools. No matter how carefully drafted, there are bound to be flaws.

During the past twenty years, dynamic geometrical software has become available. The Math Lab at Salem State College has been using an application called Geometers’ Sketchpad (GSP) since version 3 was published in 1995. While GSP uses the same fundamental building blocks—points, lines, and circles—the tools are a little more sophisticated and produce figures with fewer flaws.

In addition to allowing for more accurate figures, GSP is dynamic in the sense that various parts of the figure can be selected and moved around. That means that if one wants to study how a property of a triangle changes for acute, right, and obtuse triangles, one only has to distort the figure and watch the computer screen.

How has this changed the way geometric concepts are studied? Here’s an example of an extension of Pythagorean’s theorem that states, “the area of the square on the hypotenuse of a right triangle equals the sum of the areas on the other two sides of the triangle.”  The students construct a general triangle ABC—any triangle at all. Then they construct equilateral triangles on each of the sides of the original triangle. The question now becomes an investigation: First, calculate the areas of each of the equilateral triangles, then drag the vertices of triangle ABC and explain what happens when triangle ABC is acute, right, or obtuse. Make a conjecture. Prove the conjecture. And the final touch is: what happens if semicircles are constructed on the sides of triangle ABC instead of equilateral triangles? What about rhombuses?

Computers aren’t the only newer tools that can be used. Modern geometry courses are not limited to the Euclidean plane. We have spheres! Lenart Spheres to be specific. These are clear plastic with a diameter of about 8 inches and can be marked upon with washable marking pens. They have a “spherical” compass for constructing circles and a great circle attachment to construct great circles (a circle on the surface of a sphere that has the same circumference as the sphere, dividing the sphere into two equal hemispheres). Using the Lenart spheres, we can compare Euclid’s fifth postulate on the plane and on the sphere. It turns out that Euclid’s fifth postulate is true on the sphere. However, Playfair’s Parallel Postulate, “Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and never intersects the first line, no matter how far they are extended,” is false on the sphere.

Paper models have not gone completely by the wayside. Cones of various sizes are still constructed using paper and tape. Students can construct a triangle that has the cone point as an interior point then draw a few auxiliary lines to help in the calculation of the sum of the measures of the angles of the triangle, which turn out to depend on the size of the cone. There is a formula in terms of the cone angle. Using the counterpart of a straight line on the plane, students can discover that on some cones a straight line can intersect itself and on other cones it is not possible to construct a single straight segment joining any two arbitrarily picked points. Cones provide examples where all right angles are not equal and that can be demonstrated using paper folding and Euclid’s definition of what it means to be a right angle: “When a straight line intersects another straight line such that the adjacent angles are equal to one another, then the equal angles are called right angles and the lines are called perpendicular straight lines.”

Finally, there is the hyperbolic plane. While the Euclidean plane is flat (no curvature), and spheres have positive curvature (the curvature of a sphere is the reciprocal of the radius), hyperbolic planes have negative curvature. GSP has a Poincarè model of a hyperbolic plane built into it. Using the model, students can discover that both Euclid’s Parallel Postulate and Playfair’s Parallel Postulate are false on the hyperbolic model (and hence on all hyperbolic planes).

Given all these tools available to Math students through the Math Lab, it turned out that many of them didn’t make it to Foundations of Geometry on time last Spring because they were in the Math lab experiencing geometry for themselves. So, I guess the answer to the initial question is that it would be best for the professor to join the students in the Math Lab. It’s an exciting place to be.

Mary Platt, Mathematics Department

Beacon, a regular column of ASpect, features noteworthy items in the School of Arts and Sciences.