While the most efficient method for communicating math concepts is the use of real world objects or virtual manipulatives, creating illustrations of mathematical phenomena beyond three dimensions has been a particularly challenging task and many mathematical phenomena have thus only existed in the mathematician's mind. This research seeks to investigate the question of whether computer graphics techniques can help expert mathematicians and general public to visualize and communicate the higher-dimensional mathematical objects and their deformations. The ultimate goal of this research is to establish a mechanism by which an expert human viewer can manipulate a higher-dimensional geometric object that they can only see in part, i.e., via a slice or projection into two or three dimensions. Theoretical contributions of this research will impact and improve methods in mathematical visualization, particularly graph visualization, computer aided design, and large-scale spatial visualization, while the project deliverables will have direct and transformative impact on the ability of mathematicians to study higher-dimensional objects, and to communicate what they have learned in person, in presentations, and in archival works. The success of this project will ultimately translate into more rapid advancement in areas of pure and applied mathematics where higher dimensional geometry plays an important role, and provide mathematical visualization tool sets to facilitate college and K-12 students in their geometry courses. The project will make research outcomes including open source software freely available, and will disseminate mathematical sciences to the general public by rendering and presenting pedagogical animations at the University of Louisville Planetarium. The project also includes integrated educational and outreach activities for K-12, undergraduate, and graduate students.
The research will explore an interactive visualization paradigm that makes use of energy-driven self-deformable object models embedded in higher dimensions, supplemented by reduced-dimensional analogies for expert human viewers to guide the deformations towards their final goals. The investigators will begin by assigning a deformation energy to the higher-dimensional object, so that the aspects of the configuration that are unseen and unfamiliar can be controlled in a principled and well-posed manner by constraints or manipulations on the aspects of the configuration that are seen and familiar in our dimensions. Often times mathematical simulations are concerned with heavily vectorized operations performed over and over in a large number of iterations. The project will exploit hardware-enabled parallelism to accelerate mathematical simulations, and to extract key moments where successive terms differ by one critical change to represent and analyze various mathematical evolutions. By combining guided relaxation method and accelerated computation, this research can potentially make a novel contribution to building intuition about classes of geometric and topological problems that otherwise would be nearly impossible to communicate, perceptualize, and disseminate; and can potentially further the entire concept of the assistance and empowerment of human understanding by computer methods, specifically via the power of visual and computational spatial visualization tools. All outcomes of this project, including technical reports, research articles, links to educational and outreach activities, open source software, and pedagogical animations will be accessible from the project's web site (www.cecsresearch.org/vcl/nsf1651581/).