Chaos and Crystallographic Symmetry

混沌与晶体对称性

• 批准号: 9805507
• 负责人: Clifford Reiter
• 金额: $ 14万
• 依托单位: Lafayette College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9805507&HistoricalAwards=false
• 关键词: Chaos; Crystallographic; Symmetry

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.

Reiter 9805507 混沌与晶体学对称性 最近的调查导致吸引子的插图所产生的迭代功能，可测量的混乱，但仍然指定非平凡的对称性。 这就需要确定具有合适的等方差特性的函数类沿着搜索参数，从而产生适合吸引子的非平凡行为和可视化技术。 本研究旨在将这项工作推进到晶体学空间群。 等变函数的类需要被识别，非平凡参数定位和吸引子可视化。 期望可以在低分辨率下真实的时间内改变观看参数，并且以高分辨率常规地创建固定图像和动画。 这项任务是具有挑战性和有趣的，因为230晶体空间群的数学复杂性和在三维中呈现吸引子的方式，其中的对称性是显而易见的困难。 预计这项工作将使观众以新的方式体验这些对称群的特征。 该项目的总体重点是使用高性能工作站在三维空间中找到并可视化某些复杂的数学对象;特定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。 我们要讨论的对称性是与晶体有关的对称性，因此化学家对这些对称性很感兴趣。 类似的二维物体一直是数学家最近研究的主题，这个项目旨在将我们对这种类型物体的理解扩展到三维。 一个具有巨大内存的图形工作站将被用来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，因此创建有洞察力的，有意义的数据视觉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见解，这些对象的三维对称性应该感兴趣的数学家和其他人，如化学家，谁可以受益于了解这些对称群。 该项目的目的是充分参与研究的几个本科生，因此这些学生将不仅获得宝贵的经验，使用这个高性能的工作站，但他们也应该是第一个看到这些对称群在新的方式从这个研究项目。