Closed curves on a torus representing elements $(p,q) \in \mathbb{Z} \oplus \mathbb{Z}$ of the fundamental group
The fundamental group of the torus is Z⊕Z, with generators λ (the loop encircling the rotation axis) and μ (the meridional loop). This program depicts closed curves on a torus of revolution representing the free homotopy class pλ+qμ for integer pairs (p,q), illustrating how the algebraic structure of the fundamental group is realized geometrically as curves winding around the torus.
Controls: Click and drag to orbit, scroll to zoom.
