Mathematical Presentation In Escher's Painting

The thesis takes the mathematical presentation in Escher's plane and space works as the main starting point to analyze the important role that mathematics plays in Escher's works,and interprets the evolution of space in the three-dimensional plane as well as explores the possibility of its application and expression in contemporary art from the perspective of mathematics.On the basis of drawing lessons from the relevant research results of predecessors,this thesis classifies Escher's works and makes a comprehensive and detailed study.The first chapter is the citation.The second chapter analyzes the two major reasons for Escher's way of mathematical thinking,and points out Escher's basic mathematical form in his works.This chapter focuses on the characteristics of mathematical forms presented in Escher's works,and classifies them into categories.The third chapter is the main research part of the paper,which takes the mathematical form characteristics of Escher's plane structure works as the research object.The research content is listed as "the principle of symmetry in Escher mosaic pattern" and "the infinity and cycle of Poincare model in Escher's circular limit",and the "mathematical structure" embodied in Escher's planar works is discussed separately to explain the existence of mathematical content in Escher's planar painting.The key point in this chapter is the second section,which is not only related to the performance of a surface on the plane,but also the transformation of two-dimensional plane to three-dimensional space,which is a transitional research for the following chapter 4.The fourth chapter takes the mathematical form characteristics of Escher's spatial structure works as the main research object.The "mathematical structure" in Escher's spatial structure works is roughly classified into two categories: the innovation of curvilinear perspective and Plato's three-dimensional.Both of them are the products of Escher's exploration of spatial paintings,which broke the traditional three-dimensional space and laid a foundation for the existence of illusory space works.The fifth chapter is the key research part of this paper,which takes the mathematical form characteristics of Escher's illusion space works as the mainresearch object.This chapter mainly studies Escher's works in the two mathematical forms of Mobius ring and Riemannian surface in topology,and explores these forms of scientific paradoxes used by Escher in his works of spatial structure.The sixth chapter is the end of this paper,which focuses on the influence of the "mathematical structure" in Escher's art on all aspects of new art and the breakthrough of thinking consciousness.In our time,the new art is booming,the writer hopes to explore the possibility of its application and expression in contemporary art by studying the new "mathematical structure" in painting.Under the influence of modern art,"the way of mathematical structure" for artists can be regarded as a conceptual tool,and a new way to observe objects,space and structure.The emergence of the new structure in art is not only an innovation of the original structure,but also gets rid of the sense of restraint of the painting space.As a catalyst for the development of new art,it breaks the stereotyped spatial thinking to a great extent.The last part is the summary and the prospect of this thesis.