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Chapter 6 - Chapter 6 Symmetry Topics 1 Symmetry and Groups...

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Chapter 6 Symmetry Topics : 1. Symmetry and Groups 2. The Cyclic and Dihedral Groups 3. Finite Symmetry Groups a8 a8 a8 a72 a72 a72 a65 a65 a1 a1 a109 a0 a0 a64 a64 a8 a8 a8 a72 a72 a72 a65 a65 a65 a65 a1 a1 a1 a1 Copyright c circlecopyrt Claudiu C. Remsing, 2006. All rights reserved. 83
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84 M2.1 - Transformation Geometry 6.1 Symmetry and Groups There is an abundant supply of objects (bodies, organisms, structures, etc.) with symmetry in nature. Figures with symmetry appear throughout the visual arts. There are also many scientific applications of symmetry (for in- stance the classification of crystals and quasicrystals in chemistry). Theoreti- cal physics makes heavy use of symmetry. But what is symmetry ? When we say that a geometric figure (shape) is “symmetrical” we mean that we can apply certain isometries, called symmetry operations , which leave the whole figure unchanged while permuting its parts. 6.1.1 Example. The capital letters E and A have bilateral (or mirror) symmetry , the mirror being horizontal for the former, vertical for the latter. (Bilateral symmetry is the symmetry of left and right, which is so noticeable in the structure of higher animals, especially the human body.) 6.1.2 Example. The capital letter N is left unchanged by a halfturn, which may be regarded as the result of reflecting horizontally and then ver- tically, or vice versa. (Alternatively, one may prefer to view the halturn as a rotation about the “centre” through an angle of 180 .) We can say that the capital letter N has rotational symmetry . 6.1.3 Example. Another basic kind of symmetry is translational symme- try . Several combinations of these so-called basic symmetries may occur (for instance, bilateral and rotational symmetry, glide symmetry, translational and rotational symmetry, two independent translational symmetries, etc.) Exercise 91 Find simple geometric figures (patterns) exhibiting each of the fore- going kinds of symmetry. Note : In counting the symmetry operations of a figure, it is usual to include the identity tranformation; any figure has this trivial symmetry. We make the following definitions. Let S be a set of points (in E 2 ).
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C.C. Remsing 85 6.1.4 Definition. Line L is a line of symmetry (or symmetry axis ) for S if σ L ( S ) = S . 6.1.5 Definition. Point P is a point of symmetry (or symmetry cen- tre ) for S if σ P ( S ) = S . Exercise 92 Can a figure have (a) exactly two lines of symmetry ? (b) exactly two points of symmetry ? Exercise 93 Why can’t a (capital) letter of the alphabet ( written in most symmet- ric form ) have two points of symmetry ? 6.1.6 Definition. Isometry α is a symmetry for S if α ( S ) = S . 6.1.7 Example. Find the symmetries of a rectangle R = a50 ABCD that is not a square. Solution : Without loss of generality, we may assume that A = ( h, k ) , B = ( - h, k ) , C = ( - h, - k ) , and D = ( h, - k ) ; h, k > 0 , h negationslash = k. Evidently, the x - and y -axes are lines of symmetry for the rectangle, and the origin is a point of symmetry for the rectangle. Denoting the reflection in the x -axis by σ x and the reflection in the y -axis by σ y , we have that σ x , σ y , σ O , and ι are symmetries for R . Note that ι is a symmetry for any set of points. Since the image of the rectangle is known once it is known which
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