Conservation principles and action schemes in the synthesis of geometric concepts

Abstract

In this paper a theory for the synthesis of geometric concepts is presented. The theory is focused on a constructive process that synthesizes a function in the geometric domain representing a geometric concept. Geometric theorems are instances of this kind of concepts. The theory involves four main conceptual components: conservation principles, action schemes, descriptions of geometric abstractions and reinterpretations of diagrams emerging during the generative process. A notion of diagrammatic derivation in which the external representation and its interpretation are synthesized in tandem is also introduced in this paper. The theory is exemplified with a diagrammatic proof of the Theorem of Pythagoras. The theory also illustrates how the arithmetic interpretation of this theorem is produced in tandem with its diagrammatic derivation under an appropriate representational mapping. A second case study in which an arithmetic theorem is synthesized from an underlying geometric concept is also included. An interactive prototype program in which the inference load is shared between the system and the human user is also presented. The paper is concluded with a reflection on the expressive power of diagrams, their effectiveness in representation and inference, and the relation between synthetic and analytic knowledge in the realization of theorems and their proofs.