Quadrics via Semigroups

This is the story of the rediscovery of classical three-dimensional geometry, especially the geometry of quadric surfaces, while studying the semigroup $M_2(\mathbb{R})$ of linear endomorphisms of a real plane. One of the surfaces that appears prominently in this context is the hyperboloid of one sheet, referred to as {\em spaghetti bundle} in \cite{Samu:88}. In this story the spaghetti presents itself as the set of idempotents in $M_2(\mathbb{R})$, the cone emerges as the set of nilpotent elements and the hyperbolic paraboloid as the set of semigroup-theoretic inverses of a singular element.