分片常负曲率空间中螺旋Tiling的保形构造及其埃舍尔艺术计算机辅助设计

埃舍尔数学艺术(EMA)具有良好的美学和数学价值，每年蕴含百亿美元的市场需求，EMA研究具有重大意义。然而，由于新的EMA模型开拓不易，且数学与艺术家缺乏深度协作，EMA的研究态势进展缓慢。鉴于常负曲率空间不存在相似对称性，本项目拟在分片常负曲率空间引入保形意义下的相似概念，探讨一类兼具负曲率和螺旋特征的新型几何对象——PCNCST。通过寻求一组恰当参数，遥控PCNCST的负曲率和螺旋特征，籍此阐明其存在条件和允许类型；针对PCNCST特有的螺旋循环群结构，采取对称群生成元的解析分解策略，建立收敛、稳定、有序的PCNCST保形构造方法；综合数学、艺术和计算机的优点，生成数学结构严谨、艺术形态美观的EMA图案。本项目拟探讨的科学问题处于数学、艺术和计算机的交叉领域，对丰富和拓展非欧几何，探索新型埃舍尔数学模型，挖掘保形理论的美学潜力，都可以提供新的思路，在理论、艺术和经济方面具有重要价值。

Escher mathematics Art (EMA) has good aesthetic and mathematical value, which contains ten billion dollars of market demand each year. EMA research is of great significance. However, due to the difficulty in developing the new EMA model and the lack of deep cooperation between mathematics and artists, the research of EMA has made slow progress. In view of the fact that there is no similarity symmetry in the constant negative curvature space, this project intends to introduce the concept of self-similarity in the sense of conformal mapping, and discuss a new kind of geometric object PCNCST with both negative curvature and spiral self-similarity symmetry in the piecewise constant negative curvature space. We expect to control the negative curvature and spiral characteristics of PCNCST by seeking a set of appropriate parameters, which aims to clarify the existence conditions and allowable types of PCNCST. In view of the special cycle group structure of PCNCST, the analytical decomposition strategy of symmetry group generator is adopted to establish converge, stable and ordered construction method. Based on the advantages of mathematics, art and computer, the digital EMA patterns with rigorous mathematical structure and beautiful artistic form are generated. The scientific problems to be discussed in this project are in the cross field of mathematics, art and computer. It can provide new ideas for enriching and expanding non Euclidean geometry, exploring new Escher mathematical model, and tapping the aesthetic potential of conformal theory, which is of great value in theory, art and economy.