August 5, 2005

Shodor intern Ellie Grano taught today's class. She began with a discussion on the basics of geometry. First, she explained how Euclidean geometry was based on two-dimensional planes, and how all the concepts within it apply to two-dimensional figures. Next, she introduced Euclid's five postulates of two-dimensional geometry.

She then introduced them to non-Euclidean geometry, geometry in non-two-dimensional space. They began with spherical space by dividing up into teams and working with plastic spheres they could draw on. Ellie asked them to try and apply the five Euclidean postulates to the spheres. They quickly worked out the first four, but found that it was impossible for two straight lines on a sphere to not intersect less than twice. In spherical space, she explained, parallel lines are defined as lines that do not intersect more than twice.

After snack Ellie asked the students how they would draw a triangle on a sphere. She explained that you would have to draw three line segments that form three angles, just as you would in Euclidean space. Using a computer program, however, they found that the angles of the triangles they constructed did not add up to 180 degrees.