Introduction to Symmetry and Regularity
Symmetry is not only one of the most fundamental concepts in science and engineering, but it is also an ideal bridging idea crossing various branches of sciences and different fields of engineering. In the past, symmetry has been considered important for
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Introduction to Symmetry and Regularity
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Symmetric Structures Definition of Symmetry
Symmetry is not only one of the most fundamental concepts in science and engineering, but it is also an ideal bridging idea crossing various branches of sciences and different fields of engineering. In the past, symmetry has been considered important for its aesthetic appeal; however, this century has witnessed a great enhancement in its recognition as a basis of scientific and engineering principle. At the same time, the meaning and utility of symmetry have greatly expanded. It is not surprising that many valuable books are published in this field and regular annual conferences are devoted to symmetry in various fields of science and engineering. In the following, different definitions are provided for symmetry. Symmetry is the classical Greek word ΣΥM-MΕΤΡIA, the same measure, due proportion. Proportion means equal division, and ‘due’ implies that there is some higher moral criterion. In Greek culture due proportion in everything was the ideal. The word and usage have been taken over as a technical term into most European languages. Some other definitions and comments on symmetry are as follows: To say that an object or a situation is symmetrical in space–time coordinate x, y, z, t means that part of the object (etc.) has the same measure as another part. Measure implies identity to within the limits of the measuring device employed. ‘Symmetry’ is one more fundamental scientific concept, which alongside with the ‘harmony’ concept has a relation practically to all frames of the nature, science and art. The outstanding mathematician Hermann Weil highly evaluated the role of symmetry in modern science: Symmetry, as though is wide or narrow we did not perceive this word, there is the idea, with the help of which a man attempted to explain and to create the order, beauty and perfection.
A. Kaveh, Optimal Analysis of Structures by Concepts of Symmetry and Regularity, DOI 10.1007/978-3-7091-1565-7_1, © Springer-Verlag Wien 2013
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Introduction to Symmetry and Regularity
In a simple language, when we look in the mirror, we can see in it our reflection; this is the example of the ‘mirror’ symmetry. The mirror reflection is the example of the so-called ‘orthogonal’ transformation varying orientation. In the most general case, ‘symmetry’ in mathematics is perceived as such transformation of space (plane), at which each point of M passes in other point of M0 regarding to some plane (or straight line) a, when the line segment of MM0 is perpendicular to the plane (or to the straight line) a and is divided by it in halves. The plane (or the straight line) a is called as the plane (or axis) of symmetry. The plane of symmetry, symmetry axis and centre of symmetry are fundamental concepts of symmetry. The plane of symmetry is called such plane, which divides the figure into two mirror-equal parts arranged from each other as some subject and its mirror reflection. It is easy to establish that the cube has nine planes of symmet
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