These investigations are designed to support students' constructions of geometry through their own activity, beginning from the spatial (mental) action of reflection. Particular actions of sweeping out lines and swinging circles defined geometric construction for the Greeks became the basis for Euclid's axioms. Questioning those axioms (and thereby modifying the spatial actions of geometric construction) led to the development of non-Euclidean geometries, such as spherical and hyperbolic geometry. Expanding the group of spatial actions (as described by Klein) led to the development of projective geometry.
