N-dimensional geometry using APL2

Abstract

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 and fundamental domains in n-dimensional space and their display and manipulation. APL2 [API] 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 and an example solution including drawings of a fundamental domain in 4-space and portions of the APL2 code for finding n-dimensional convex hull boundaries and picking fundamental domains. The overall approach has been to generalize to n dimensions the techniques of [HVl] and [HV2], which attacked the same problem in three dimensions.