Experimental mathematics using APL and graphics

Abstract

Mathematics is ordinarily done by examining computations to extract likely conjectures. Only later are the conjectures proved, and, in traditional practice, the computations and motivations suppressed. We describe a computational investigation of the geometry of fundamental domains and units in totally real algebraic number fields. Our approach requires the finding of convex hulls in three dimensional space and their display and manipulation. APL and graphics are used for this purpose and appear to be essential for any practical pursuit of this approach. This paper presents a definition of the problem, an example solution including drawings and portions of the APL code, and a sample of the results so far obtained. In addition, lessons learned from our own mistakes and fumblings are drawn to help and encourage other experimental mathematics explorations.