The Symmetry Analysis Under Non-translational Symmetry Condition And Establish Symmetry Elements

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.