Relaxing topological surfaces in four dimensions
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ORIGINAL ARTICLE
Relaxing topological surfaces in four dimensions Hui Zhang1
· Huan Liu1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Visual mathematics · Relaxation · Optimization · Knot theory · Four dimensions
1 Introduction We often use figures to help communication about geometric objects [21], and our perception and understanding of geometric properties can be strongly facilitated by drawing and viewing these figures. However, it can be a far more challenging task to communicate about topology [15,31], which studies geometrical objects under the equivalence relation of homeomorphism, i.e., the geometric properties and spatial relations unaffected by the continuous change of shape or size of figures. Our paper is mainly concerned with the illustration of the topology of mathematical surfaces (2-dimensional objects) embedded in 4-space. The main properties of these surfaces to be studied and visualized in our work are related to their topology. In topology, a very small circle is the “same” as a huge one, and a small sheet of surface is the “same” as a big one, because we can stretch the small ones to make them exactly like the big ones. More generally, two sheets are going to be considered the “same” if it is possible to deform one into the other without cutting. Therefore, we think of surface as rubber sheet geometry—we can stretch or shrink them as much as we want, without tearing or losing the “thinness.”
B 1
Hui Zhang [email protected] University of Louisville, Louisville, USA
Generating illustrative drawings to communicate such topological properties are not trivial, and unique challenges for this class of math visualization problems include:
– Length and area are of no concern in topology problems When relating surfaces under the equivalence relation of homeomorph
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