Reiter 9805507 Chaos and Crystallographic Symmetry Recent investigations have led to illustrations of attractors arising from the iteration of functions that are measurably chaotic but nonetheless have specified nontrivial symmetry. This requires determination of classes of functions with suitable equivariance properties along with searches for parameters yielding nontrivial behavior and visualization techniques suited to the attractor. This investigation is designed to advance that work to the crystallographic space groups. The classes of equivariant functions need to be identified, nontrivial parameters located and the attractors visualized. It is expected that viewing parameters can be changed in real time at low resolution and fixed images and animations routinely created at high resolution. This task is challenging and interesting because of the mathematical complexity of the 230 crystallographic space groups and the difficulties in rendering attractors in three dimensions in a manner in which the symmetries are apparent. It is expected that this work will allow viewers to experience the features of these symmetry groups in new ways. The general focus of this project is to find and visualize certain complex mathematical objects in three dimensions using a high performance workstation; the particular objects are chaotic attractors with crystallographic symmetry. These mathematical objects have an underlying chaotic structure while at the same time exhibiting a high degree of symmetry. The symmetries which will be considered are those associated with crystals; these symmetries have been of great interest to chemists for that reason. Similar two dimensional objects have been the subject of considerable recent study by mathematicians and this project is designed to extend our understanding of objects of this type to three dimensions. A graphics workstation with massive memory will be used to create high quality images of these complex objects. Sin ce these complex objects are the result of literally millions or billions of data points, creating insightful, meaningful visual representations of the data is essential to understanding them. The high level visualization techniques and the resulting insights into the three dimensional symmetries of these objects should be of interest to mathematicians and others, like chemists, who can benefit from understanding these symmetry groups. The project is designed to fully involve several undergraduate students in the research and hence these students will not only gain valuable experience with using this high performance workstation but they should also be among the first to view these symmetry groups in the new ways resulting from this research project.
