Interactive Mathematical Visualization

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This is the first year of a two year continuing grant. One long-term goal of this research is to develop and exploit techniques to make it possible for scientists to study and solve problems in mathematics and physics that could not be undertaken without automated assistance. Another long term goal is to deduce general principles about the nature of human capabilities and limitations in mental engagement with abstract data spaces. intelligent constraint-based interaction (rule-based) methods will be combined with semi-automated optimization techniques develop a domain-independent interactive system of constraints. This system will be designed to allow manipulation of objects that are transforms of non-observables which are conceptually difficult mathematical properties. Simulations that transform non-observables into manipulatable observables are useful to exploit human learning processes to master abstract problems. The abstract concepts range from issues of relative size and speed estimation, unfamiliar dimensions, and chaotic processes in physics and interactions in many-body problems. Domain-dependent constraints on representations are used to form a model of a desired class of spatial configurations. This leads to a rational automation methodology. Direct, context-free manipulation of objects, rather than manipulation of objects, rather than manipulation of corresponding surrogate "control" objects, will also be examined. This effort seek to lay a foundation for understanding human capacity for mental engagement with visual representations of abstract concepts. This will be useful as a basis for subsequent experiments on psychophysical theories of visualization.//