Abstract Algebra Theory and Applications - Computer Science ...

38 CHAPTER 2 GROUPSSymmetriesA symmetry of a geometric figure is a rearrangement of the figure preservingthe arrangement of its sides and vertices as well as its distances andangles. A map from the plane to itself preserving the symmetry of an objectis called a rigid motion. For example, if we look at the rectangle in Figure2.1, it is easy to see that a rotation of 180 ◦ or 360 ◦ returns a rectangle inthe plane with the same orientation as the original rectangle and the samerelationship among the vertices. A reflection of the rectangle across eitherthe vertical axis or the horizontal axis can also be seen to be a symmetry.However, a 90 ◦ rotation in either direction cannot be a symmetry unless therectangle is a square.ABidentity✲ABDADADADCBCBCBCDC180 ◦✲rotationBBreflection✲verticalaxis CDreflection✲horizontalaxis ACDAADCBFigure 2.1. Rigid motions of a rectangleLet us find the symmetries of the equilateral triangle △ABC. To find asymmetry of △ABC, we must first examine the permutations of the verticesA, B, and C and then ask if a permutation extends to a symmetry of thetriangle. Recall that a permutation of a set S is a one-to-one and ontomap π : S → S. The three vertices have 3! = 6 permutations, so the trianglehas at most six symmetries. To see that there are six permutations, observethere are three different possibilities for the first vertex, and two for thesecond, and the remaining vertex is determined by the placement of the