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Author Casselman, Bill
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract We shall be concerned with geometrical figures with a high degree of symmetry, in both 2D and 3D. In 3D the most symmetrical figures are the five Platonic solids, which we shall see how to construct in the last section. There are many ways to do this, and many described in the literature, but the most satisfactory method is one which extends to a wide variety of regular figures of all kinds. This depends on understanding its symmetry transformations, which make up what is called a Coxeter group, generated by reflections of a particular kind. 1. Mathematical symmetry In common English usage, the term symmetry seems to have meant at first the property of being balanced or well-proportioned. This original meaning continues in a slightly more technical sense in the phrases bilateral symmetry and mirror symmetry applied to a figure which looks the same as its image in a mirror. For example, the triangle on the left has mirror symmetry while the one on the right does not. In effect, the reason we say the figure on the left has mirror symmetry is that we can slice it with a line to divide it into two halves which are congruent to one another, but with orientation reversed, as if reflected in a mirror. The line is called an axis of symmetry of the triangle. Geometrical symmetry and the fine structure of regular polyhedra 2 In mathematics, we distinguish between degrees of symmetry. For example, the equilateral triangle shown below has more symmetry than the triangles above.
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study