Relaxing topological surfaces in four dimensions

In this paper, we show the use of visualization and topological relaxation methods to analyze and understand the underlying structure of mathematical surfaces embedded in 4D. When projected from 4D to 3D space, mathematical surfaces often twist, turn, and fold back on themselves, leaving their underlying structures behind their 3D figures. Our approach combines computer graphics, relaxation algorithm, and simulation to facilitate the modeling and depiction of 4D surfaces, and their deformation toward the simplified representations. For our principal test case of surfaces in 4D, this for the first time permits us to visualize a set of well-known topological phenomena beyond 3D that otherwise could only exist in the mathematician’s mind. Understanding a fairly long mathematical deformation sequence can be aided by visual analysis and comparison over the identified “key moments” where only critical changes occur in the sequence. Our interface is designed to summarize the deformation sequence with a significantly reduced number of visual frames. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical surfaces and their deformation in topological space.