Exploring New Geometry by Touching, Seeing, and Feeling

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The understanding of geometric shapes in everyday life combines visual impressions, tactile sensation, and their unification in the process of seeing something and reaching out to touch it to determine its physical properties, which may convey new information. Interactive computer graphics systems can be combined with 3D-touch-responsive (haptic) interfaces to permit not only the simulation of our familiar 3D world, but also the extension of our available sensations to additional dimensions and unfamiliar geometric objects by using the abstract power of computer modeling.This research focuses on exploiting interactive graphics combined with haptics to develop new methods for investigating geometric structures whose understanding is beyond the normal capabilities of the unaided human. Results will be made available in the form of pedagogical animations as well as the software environments used to create them. Typical problems include the direct manipulation and study of deformable volumes resulting when a 3D knot is removed from the solid block of space it lives in, leaving the knot complement, and the exploration of the intricate collision-free deformations of surfaces in 4D space that are essential for studying the properties of functions of two complex variables.The anticipated outcome of pursuing such approaches will be to enhance human capabilities and intuition pertaining to specific challenging geometric visualization problems, to provide methods enabling the discovery of scientific questions related to complex geometric structures that might not otherwise have been asked, and ultimately to advance the entire concept of empowering human understanding using computer-based technology.