This lesson builds intuition about surfaces and three-dimensional spaces from a topological perspective. There is a lot of opportunity for imagination and discussion but less computation and problem solving than in a typical math circle session.
The lesson starts with an informal explanation of topological equivalence and then introduces students to the torus, the Klein bottle, and three-dimensional analogs of these surfaces, using gluing diagrams as a tool. Along the way, students play Tic-Tac-Toe on the torus and imagine life in a Klein bottle universe. The lesson ends with an introduction to the Euler number of surfaces.