1 - Introduction to Symmetries

Summary

I know it when I see it.

(Justice Potter Stewart: Jacoblellis v. Ohio, 378 U.S. 184, 197 [1964])

Symmetries of Planar Objects

In order to understand symmetries of differential equations, it is helpful to consider symmetries of simpler objects. Roughly speaking, a symmetry of a geometrical object is a transformation whose action leaves the object apparently unchanged. For instance, consider the result of rotating an equilateral triangle anticlockwise about its centre. After a rotation of 2π/3, the triangle looks the same as it did before the rotation, so this transformation is a symmetry. Rotations of 4π/3 and 2π are also symmetries of the equilateral triangle. In fact, rotating by 2π is equivalent to doing nothing, because each point is mapped to itself. The transformation mapping each point to itself is a symmetry of any geometrical object: it is called the trivial symmetry.

Symmetries are commonly used to classify geometrical objects. Suppose that the three triangles illustrated in Fig. 1.1 are made from some rigid material, with indistinguishable sides. The symmetries of these triangles are readily found by experiment. The equilateral triangle has the trivial symmetry, the rotations described above, and flips about the three axes marked in Fig. 1.1(a). These flips are equivalent to reflections in the axes. So an equilateral triangle has six distinct symmetries. The isoceles triangle in Fig. 1.1(b) has two: a flip (as shown) and the trivial symmetry.