Ariadne is a touch-based program for the learning of homotopies of paths, without the use of formalism, by building mental models. Using Ariadne, the user can construct points, paths by dragging points and homotopies by dragging paths as well as compute winding numbers of paths, all on a variety of surfaces, through touch gestures. Ariadne provides surfaces in two and three dimensions and an optional number of punctures.
This environment enables the user to tackle questions regarding the equivalence of points by paths or paths by homotopies, because it allows only mathematically valid operations, i. e., paths and homotopies cannot pass through punctures on the surface.
Ariadne is designed to let students of all ages and prior states of knowledge approach problems ranging from the construction of a path connecting points to the classification of all paths up to homotopy on a punctured plane, effectively calculating the fundamental group of a sphere.

Drawing paths:

Constructing manifolds: