【作者】 闫河；

【导师】 王秀峰；

【作者基本信息】 陕西科技大学 ， 材料学， 2013， 博士

【摘要】 目前，对称物体或者图形对称性的认识主要是传统的、平移条件下的对称规律和对称要素的作用。自1984年科学家发现准晶体后，逐渐打破了基于具有“平移重复”的传统对称性理论的界限，但仍试图通过种种对称操作建立与“平移”条件相应的对称要素。研究发现，自然界中更广泛存在的材料结构、生物和自然体系等具有另一类对称性，即非平移对称性。这类对称性与平移条件下的对称性有本质的不同，是对称性的更复杂、更普遍的一类规律，有待进一步去认识。例如，材料的团簇结构、玻璃相结构；叶序、渐开线形贝壳等动植物的生物对称性；云团、湍流以及电磁波，等等。可以认为，这些材料结构、生物和自然体系中存在着具有共性的一类规律，即非平移条件下的对称性。研究这类对称性并建立相应的对称要素具有十分重要的意义。本文研究非平移条件下材料和体系的对称性，分析这类对称结构的对称性特点。研究发现，非平移结构都具有对称收敛性；总结了各种收敛性结构的对称要素；建立了达到对称结构重复的对称操作方法和相应的对称要素，提出了描述解决这类问题的新方法。本文在以下几方面开展了研究工作：最先关注对称要素的是晶体学，晶体学中传统的对称要素也是一些几何要素，它是可以让图形或者一些结构发生规律变化的依据，但是传统的对称要素不能够准确描述准晶体的对称性，需要寻找具有更广泛意义的对称要素来对其描述。准晶体是一类没有平移对称性质的物体，但是其具有收敛对称性，对准晶体对称性完整的研究需要建立在非平移对称的基础之上。分析了超材料中左手材料、光子晶体和超磁性材料在传统对称要素下的对称性，发现超材料中很多特性不能够用平移对称来解释，必须借助非平移对称才能进行更深层次的研究。建立了非平移条件下四种新的对称要素，包括直螺旋轴、曲螺旋轴、曲对称面和收敛心，并建立了其相应的数学模型。直螺旋轴类似于螺钉的螺纹，或者是球面的渐开线，通过建立螺纹数学模型得出了螺纹测量的新算法。曲螺旋轴指变直径螺旋状曲线，典型结构是牵牛花，建立了变螺距圆锥的数学模型，得出曲螺旋轴切线和法向量计算方法。曲对称面是传统对称要素经过一定的组合之后产生的新的对称要素，其能够描述一些传统对称要素无法描述的对称性，建立了曲对称面的数学模型，利用该模型能够更好的对具有曲对称面的物体或者图形来分析。在螺旋扩大物体或者图形中总会有一个或者几个收敛点，该点就是收敛心。给出了收敛心典型模型向日葵花盘拥有两个收敛心，鹦鹉螺的壳有一个收敛心，通过对典型模型的分析建立了收敛心的数学模型。证明了晶体中对称定律的正确性，验证在晶体中只存在一次、二次、三次、四次和六次对称轴，不可能存在其他的对称轴。对非平移条件下对称性进行扩展分析，包括在材料、控制、生物、机电等一些领域的应用。讨论了动物外形及其DNA结构在非平移条件下的对称性，对植物叶序、花瓣分布规律的分析得出其与斐波那契数列的关系，并研究了宇宙和混沌世界所具有的对称性。论文在以下几方面取得了创造性成果：分析了传统对称性的运用范围和对称要素特点，论述了这些对称性描述原理和对称要素的局限性，总结了用传统对称性原理方法描述的各类结构和现象。提出了描述非平移对称结构的新原理，即收敛性对称结构的描述方法，并归纳了各类具有非平移对称性结构的特点，建立了非平移对称性的描述体系，用更具有普遍性的曲线、曲面等来描述这类现象。定义了直螺旋轴、曲螺旋面、曲对称面、收敛心等全新的对称要素，并结合实际，推导出相应的对称群。运用非平移对称性理论，描述了材料结构、生物和自然体系。事实证明，提出的新理论和方法能够准确、全面的描述自然界中更多存在的非平移对称现象。这一理论和方法将对材料结构、认识和利用自然等具有重要的影响。更多还原

【Abstract】 Currently， our understanding of symmetry objects and graphic symmetry isthe function of symmetry rules and symmetry elements under traditional andtranslational condition. Since the crystal was found on1984， the limit oftraditional symmetry theory based on the translation repetition was graduallybroken， but the scientists are still trying to build the corresponding symmetryelements with translation tradition by kinds of symmetry operation.The studies found， the widely spread material structure， biology andnatural system in the nature possessing another different symmetry， which isnon-translational symmetry. The establishing of the non-translational symmetryis fundamentally different from the symmetry under the transitional condition；being a kind more complicated and normal rule of symmetry which waiting forus to explore further.In this article， we studied the symmetry of the material under thenon-translational condition to find the characteristics of these symmetricalstructure， the astringency of structure； summarized the symmetry elements ofdifferent kinds of astringency structure； established the repeated operationalmethods and corresponding symmetry elements to reach the symmetry elements，promoting the new way to solve these problems. The first symmetry element wefocused on is the crystallography. In the crystallography， the traditionalsymmetry elements are some geometric elements which provide foundationallowing the graph and some objects to have regular variation. But the traditionalsymmetry elements could not describe the symmetry of the crystal， so we needto find a symmetry element with more extensively meaning to describe it.Quasicrystal is a kind of object without translational symmetry， while it hasastringency symmetry. The integrated study of Quasicrystal must be based on thenon-translational symmetry. After analyzing the symmetry of left-handedmaterials， photonic crystal and super magnetic materials under the traditionalsymmetry elements， we found that the some characteristics of the super materials could not be explained by the translational symmetry but thenon-translational symmetry， which would help us to do the deeper research.Some new symmetry elements under non-translational condition was given，including vertical screw axis， twist screw axis， twist symmetry plane andastringency center， and the corresponding mathematic model was built as well.Vertical screw axis is analogous to the thread of the screw， or the involute ofthe sphere surface. And we get the new algorithm of measuring the screw threadby building the screw thread mathematic model. The twist screw axis is thevarying diameter spiral curve whose typic structure is morning glory. And webuild the mathematic model of varying pitch conical to get the computingmethod of tangent line and normal vector of twist screw axis. Twist symmetryplane is a new symmetry element by combining some of the traditionalsymmetry elements which could describe the symmetry that the traditionalsymmetry elements could not. Build the mathematic model of twist symmetryplane would help us analyze the objects or graphs with twist symmetry plane.There will always be one or some astringency points in the spiral dilating objectsor graphs， and the point is the astringency center. And the astringency centertypical model-sunflower disc having two astringency centers， and there are oneastringency center in the shell of pearly nautilus. Its mathematical model wasbuilt by analyzing the typical model which prove the correctness of law ofsymmetry in crystal and verify that there only exist monogyre， axis of binarysymmetry， axis of trigonal symmetry， four-fold axis of symmetry and axis ofhexagonal symmetry in crystal， and other axis of symmetry is impossible toexist.Have a extended analysis on the symmetry under the non-translationalcondition， including its application in material， controlling， biology andelectromechanical field. We also discuss the symmetry of animals’ shapes and itsDNA structure under the non-translational condition； as well we analyze therule of plant phyllotaxis and petal distribution and get its relation with Fibonacciseries. At the same time， we study the symmetry of the universe and the ChaosWorld.The creative achievements in this article through analysis are:Analyze the application scope of traditional symmetry and the characteristics of the symmetry elements， and state the limits of these symmetry descriptiontheory and symmetry elements. At last we conclude different kinds of structureand phenomenon described by traditional symmetry theory.Propose the new theory to describe the non-translational symmetrystructure， which is the description of astringency symmetry structure. And wealso conclude the characteristics of all kinds of non-translational symmetrystructure and build the descriptive system for non-translational symmetry， usingmore general factors like curve， or surface to describe this phenomenon.Define the whole new elements like vertical screw axis， twist helicoids，twist symmetry plane， astringency center， etc. and elicit correspondingsymmetric group combining practice.Use the non-translational symmetry theory to describe the material structure，biology and natural system.Facts proved that the newly proposed theory and method could describe theexisting non-translational symmetry phenomenon in the nature accurately andcomprehensively. This theory and method would bring a significant influence tothe material structure， learn and make use of the nature.更多还原