Let a and b be reflections in parallel lines in the Euclidean plane (as in Figure 2.1). If we think of the line of reflection for a as being x = 0 and the line of reflection for b as being x = 1, then we may express a as the function a[(x, y)] = (-x, y) and b as b[(x, y)] = (2 - x, y). It follows that ab[(x, y)] = (x - 2, y) and ba[(x, y)] = (x + 2, y). Moreover, for any n ∈ ℝ, (ab)n is a horizontal translation to the left through a distance of 2n and (ba)n is a horizontal translation to the right through a distance of 2n. Thus the reflections a and b generate an infinite group.

If the two lines of reflection actually met, at an angle of π/n, then the group generated by a and b would be the dihedral group of order 2n, and the product ab would be a rotation through an angle of 2π/n. Thus the group we are considering is something like a dihedral group, except that the two reflections generate a translation, not a rotation. As the group generated by two parallel reflections is infinite, it is referred to as the infinite dihedral group, denoted D∞.