The Shape Of Things

Geometry, the study of shapes, form the basis of much of the mathematical curriculum from elementary schools through universities. There is so much more than this, however, from tilings of the plane to minimal surfaces, from non-Euclidean geometries (like hyperbolic space) to non-orientable surfaces (like the Möbius strip or Klein bottle). Much of the study of these topics is accessible to a high school student willing to visualize.

 This is what's known as a minimal surface. These surfaces generalize the behaviour of soap film or bubbles. A bent wire dipped in a soap solution will form a soap film, and this film is the smallest (in terms of area) such film that will bound this wire. The figure at the right is not a soap film, but instead is a surface that has the same local shape as a film. It is a classic surface known as Enneper's surface.
 This video game universe has the same shape as a donut. The picture to the left is what a mathematician would call a torus. The game, by identifying opposite vertical sides, turns the universe into a cylinder. By then identifying the top and bottom sides, the game has wrapped this cylinder into the donut-shape shown.
 A Klein bottle has the same number of sides as this Möbius strip. To find out more, enrol in the SOAR Winter course!
Our approach is motivated by mathematics research. We will search for patterns in hands-on examples and discover (and prove) rules based on these patterns. The study of geometry allows us to ground our abstract results in terms of straightforward examples; that is, our mathematically rigorous results will come from experimenting with shapes via problem-solving, games, and computer programs.