Even Isometries

A product of two reflections is a translation or a rotation. By considering the fixed points of each, we see that neither a translation nor a rotation can be equal to a reflection. Thus, σ_nσ_m ≠ σ_l for lines l,m,n When a given isometry is expressed as a product of reflections, the number of reflections is not invariant; indeed, we can always add 2 to the number of reflections in a given product by inserting “σ_lσ_l” into the product. Although a product of two reflections cannot be a reflection, we know that in some cases a product of three reflections is a reflection. We shall see this is possible only because both 3 and 1 are odd integers. In mathematics, parity refers to the property of an integer being either even or odd. An isometry that is a product of an even number of reflections is said to be even; an isometry that is a product of an odd number of reflections is said to be odd. Since an isometry is a product of reflections, then an isometry is even or odd. However, the definition will be useful only if we show that no isometry is both even and odd. Of course no integer can be both even and odd, but is it not conceivable some product of ten reflections could be equal to some product of seven reflections? To show this is impossible, we first show that a product of four reflections is always equal to a product of two reflections.