A toolkit to describe and interactively display three-manifolds embedded in four-space

Abstract

A data structure and toolkit are presented here that allow for the description and manipulation of mathematical models of three-manifolds and their interactive display from multiple viewpoints via the OpenGL 3D graphics package. The data structure and vector math package can be extended to support an arbitrary number of Euclidean spatial dimensions. A model in 4-space is described by its bounding pure simplicial 3-complex. By intersecting a 3-flat with this 3-manifold, the algorithm will extract the requested closed pure simplicial 2-complex surface enclosing the desired 3D slice. The user can interactively rotate, pan, zoom, and shade arbitrary 3D solid or wire-frame views of the revealed 3D object created by intersection, thus exploring both expected and unexpected symmetries or asymmetries in the world of 3-manifolds in 4-space.