Using Diagrams to Understand Geometry

This paper describes ARCHIMEDES-STUDENT, a computer program that constructs and modifies its own representations of diagrams from instructions supplied by a human who is demonstrating a theorem of geometry. The program's representation permits it to make inferences from its constructions and to find a justification for the conclusion of the theorem. It is argued that the sort of perceptual reasoning displayed by this program represents one important aspect of understanding because it relates the abstract mathematical theorem to knowledge of spatial relations. For humans this approach grounds abstraction in experience and thus provides a more compelling demonstration than a formal proof. Because ARCHIMEDES-STUDENT is a well-defined computer program, it provides a precise suggestion of how this aspect of understanding can be achieved.