基于群论的柏拉图立体着色方案三维模型构造

提出用柏拉图立体的空间旋转群来完成柏拉图立体着色方案三维模型构造的思想,解决与对称性直接相关的着色方案三维模型的构造问题;提出一种柏拉图立体旋转群群元新的分类方法;提出群元抽象对称性、局部色数和饱和色数3个新概念;提出对抽象对象进行抽象着色方案的构造,然后再将抽象着色方案映射到具体的轮换上,最后映射到三维模型空间去的构造方法,设计了实现该方法的算法,并用Visual C++6.0和Direct 3D实现了算法及三维模型可视化.软件运行结果验证了所提出的方法及算法的正确性.

Construction of Three-Dimensional Model of Platonic Solid Coloring Mode Based on Group Theory

An idea of implementing three-dimensional model for coloring mode of Platonic solids in accordance with space rotational group of Platonic solids is raised, and the problem in constructing three-dimensional model directly related to symmetry is thus resolved. A new classification method of group elements in a rotational group of Platonic solids is given. Three new concepts such as abstracted symmetry of group elements, local number of colors and saturated number of colors are put forward. Constructing coloring mode for the abstracted object and then constructing a method that the abstracted coloring mode is mapped into a three-dimensional space are raised. An algorithm for implementing this method is developed, and the algorithm and the three-dimensional model are visualized by means of Visual C++6.0 and Direct 3D. The results validate both the method put forward in this paper and the correctness of the algorithm.