New directions in four-dimensional mathematical visualization

• Main Author: Yan, Xiaoqi
• Other Authors: Fu Chi-Wing
• Format: Theses and Dissertations
• Language: English
• Published: 2015
• Subjects: DRNTU::Engineering::Computer science and engineering::Computing methodologies::Computer graphics
• Online Access: https://hdl.handle.net/10356/65695
• Institution: Nanyang Technological University

Four-dimension visualization is an interesting topic as we live in a 3D space, we do not have the chance to directly “see,” manipulate or “touch” 4D objects. Four-dimensional visualization is an important topic since mathematicians have long been wondering how to visualize beautiful geometries such as knotted spheres, quaternions, and Calabi-Yau space cross-sections, while physicists have been wanting to visualize the hypersphere, which is Einstein’s universe. However, only computer graphics can be used to create 4D visualization tools which have the power to accurately represent objects for which physical models are difficult or impossible to build, and which have the ability to allow the user to interact with simulated worlds. Without 4D visualization tools we cannot even plot a complex-valued function with only one independent variable. Graphics pioneers did great work on 4D visualization, but it is not enough: the major problems to be addressed in 4D visualization (how to present these models to the eyes and how to enhance the user’s intuitive experience of the abstract geometric world that we are trying to understand) have still not been completely solved. In more detail, projecting a 4D object onto 3D space is actually a dimension reduction. Certain geometrical information, such as symmetry and curvature, is unavoidably lost after the projection because we rely on the 3D projection view to explore the geometry that actually exists in four dimensions. Moreover, projecting the 4D surface onto a 3D space may lead to false occlusion. Although a 4D surface does not have any intersections in 4D space, projecting it onto a 3D space may cause intersecting (occlusion) lines. In addition, the user interaction is not intuitive or efficient, and the potential of improving accessibility of 4D visualization system for public users is virtually unexplored. This project aims to explore new research directions that address the above questions.