During the past two years, the PI's NSF grant has made it possible to generate a new collection of computer graphics programs for the geometry of smooth and polyhedral submanifolds in euclidean 3- and 4-space. This ongoing work has been the result of collaborations with undergraduate and graduate students. It has been applied in the proposer's research on the geometry of piecewise circular curves and on highly symmetric combinatorial structures on the complex projective plane. Preliminary versions of this research program have been used to provide the illustrations for proposer's Scientific American Library book and several articles, as well as the images used for Mathematics Awareness Week. The P.I. and his assistants intend to continue the development of this system, and to apply it to extensions of the problems, as well as other problems in the geometry of submanifolds. The work of the P.I. has interest to the mathematical Sciences community at large, as well as to mathematics educators.
