Chaos and Crystallographic Symmetry

赖特9805507混沌和晶体对称性最近的研究已经导致了吸引子的图解，这些吸引子产生于函数的迭代，这些函数是可测量的混沌的，但仍然具有特定的非平凡对称性。这需要确定具有适当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于吸引子的可视化技术。这项研究旨在将这项工作推进到晶体空间群。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，以及在高分辨率下常规创建的固定图像和动画。这项任务具有挑战性和趣味性，因为230个晶体空间群的数学复杂性，以及以对称性明显的方式呈现三维吸引子的困难。预计这部作品将让观众以新的方式体验这些对称群的特征。这个项目的总体重点是使用一个高性能的工作站在三维空间中发现和可视化某些复杂的数学对象；这些特定的对象是具有晶体对称性的混沌吸引子。这些数学对象具有潜在的混沌结构，同时表现出高度的对称性。将要考虑的对称性是那些与晶体有关的对称性；由于这个原因，化学家们对这些对称性非常感兴趣。最近，数学家对类似的二维物体进行了大量的研究，这个项目旨在将我们对这类物体的理解扩展到三维。将使用具有海量内存的图形工作站来创建这些复杂物体的高质量图像。因为这些复杂的对象是数百万或数十亿个数据点的结果，创建有洞察力的、有意义的数据视觉表示对于理解它们是必不可少的。数学家和化学家等人应该会对这些物体的三维对称性的高级可视化技术和由此产生的洞察力感兴趣，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，因此这些学生不仅将获得使用这一高性能工作站的宝贵经验，而且他们也应该是第一批以本研究项目产生的新方式来看待这些对称群的人之一。

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.