Playing With Symmetry

This edition's Geometer's Corner is part 1 of a two-part series on symmetry and how it is used in chemistry. I write this for two reasons. First, a transformational approach to symmetry has always fascinated me, and I have put it to good use in much of my teaching. Second, I have just retired from teaching and would like to say thanks to the current Groton head of school, Temba Maqubela, for helping make my final years at Groton so enjoyable. He is a chemist and the following is written as a thank you to him.

When talking about the symmetry that a given geometric object has, it is useful to think in terms of a transformation such that, after it is applied, the object looks the same. For example, consider the symmetries that the rectangle ABCD shown in Figure 1 possesses: a clockwise rotation of 180∞ about P, a reflection about a vertical axis through P, and a reflection about a horizontal axis through P. In what follows, it is helpful to use diagrams like those in Figure 1. The colored arcs at the vertices are fixed as are the vertical and horizontal axes. The original figure has A by the red sector, B by the blue, C by the green, and D by the yellow. Using these diagrams, a reflection in the horizontal axis has D at the red sector, C at the blue sector, B at the green, and A at the yellow.