Relaxing topological surfaces in four dimensions

Abstract

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.

A Flip‐book of Knot Diagrams for Visualizing Surfaces in 4‐Space

“…Traditional techniques for visualizing surfaces in 4D typically involve creating pictures of 4D entities intersecting in a 3D projection and associating the fourth dimension (i.e., the w “eye coordinate”) with visual cues such as 4D depth color, texture density, etc (see e.g., [HZ05,HIM99,ZH07]). Other representative efforts include a variety of ways to render 4D objects (see e.g., Banks' interactive manipulation and display of surface in 4D [Ban92], Chu's use of 4D light sources to render 4D surfaces [CFHH09], Noll's method of rotating hyper‐objects in four‐dimensional space [Nol67], and Zhang's cloth‐like modeling and rendering of 4D surfaces [ZL20]). Figure 3 shows some of the typical 4D visualization techniques.…”

“…We have been able to utilize our slicing tool to render flip-books of diagrams from an array of known surfaces in 4-space. In Figure 10, we show a series of movies rendered for a Klein bottle being relaxed from the traditional bottle shape into a pinched torus shape [ZL20]. The Klein bottle is a closed non-orientable surface that has no inside or outside, first described by Felix Klein [Wei03].…”

“…The Klein bottle is a closed non-orientable surface that has no inside or outside, first described by Felix Klein [Wei03]. Our tool starts with the standard shape of Klein bottle, and renders movies across difference phases while the Klein bottle was being relaxed under an energy model [ZL20]. The Klein bottle reaches its minimum energy state and appears to be a pinched torus in our dimensions (see the flip-book rendered in Figure 10(d)).…”