functional objects; these systems were generally developed after the lambda cube paper was published. The idea of the cube is due to the mathematician Henk Barendregt (1991). The framework of pure type systems generalizes the lambda cube in the sense that all corners of the cube, as well as many other systems can be represented as instances of this general framework. This framework predates the lambda cube by a couple of years. In his 1991 paper, Barendregt also defines the corners of the cube in this framework. See also In his Habilitation à diriger les recherches, Olivier Ridoux gives a cut-out template of the lambda cube and also a dual representation of the cube as an octahedron, where the 8 vertices are replaced by faces, as well as a dual representation as a dodecahedron, where the 12 edges are replaced by faces. Homotopy type theory Notes Further reading External links Barendregt's Lambda Cube in the context of pure type systems by Roger Bishop Jones Lambda calculus Type theory