Chaos and Crystallographic Symmetry

混沌与晶体对称最近的研究导致了从可测量的混沌函数的迭代中产生的吸引子的插图，但仍然具有指定的非平凡对称性。这需要确定具有合适的等方差性质的函数类，以及搜索产生非平凡行为的参数和适合吸引子的可视化技术。本研究旨在将这一工作推进到晶体学空间群。等变函数的类需要被识别，非平凡参数的定位和吸引子的可视化。预计观看参数可以在低分辨率下实时改变，固定图像和动画通常在高分辨率下创建。这项任务既具有挑战性又有趣，因为230个晶体空间群的数学复杂性，以及在三维空间中以明显的对称性呈现吸引子的困难。希望这个作品能让观众以新的方式体验到这些对称群的特征。该项目的总体重点是使用高性能工作站在三维中发现和可视化某些复杂的数学对象；特定的物体是具有晶体对称的混沌吸引子。这些数学对象具有潜在的混沌结构，同时又表现出高度的对称性。将考虑的对称性是与晶体有关的对称性；正是因为这个原因，这些对称性引起了化学家们的极大兴趣。类似的二维物体一直是数学家最近大量研究的主题，而这个项目旨在将我们对这种类型的物体的理解扩展到三维。拥有海量内存的图形工作站将用于创建这些复杂对象的高质量图像。由于这些复杂的对象是数百万或数十亿个数据点的结果，因此创建有洞察力、有意义的数据可视化表示对于理解它们至关重要。高层次的可视化技术和由此产生的对这些物体的三维对称性的见解应该会引起数学家和其他人的兴趣，比如化学家，他们可以从理解这些对称群中受益。该项目旨在让几名本科生充分参与研究，因此这些学生不仅可以获得使用该高性能工作站的宝贵经验，而且还应该是第一批以该研究项目产生的新方式观察这些对称群的学生。

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.