Greenland and Iceland are described with unusual scientific accuracy in the King's Mirror. However this thirteenth-century manuscript contains a few ‘wonders’ that appear more mythological than rational. They include the hafstramb and the margygr, commonly translated respectively as merman and mermaid. The mermaid has a long history in western civilisation. The commonly accepted theory that it evolved from the classical Greek siren is critically examined here. The margygr is shown to be a distinct creature based on independent observation in northern Europe. The characteristics of these observations actually modified the siren of the Physiologus, a bird-woman, into the fish-woman known today. Observations of hafstramb and margygr are explained as superior mirages. These are caused by atmospheric refraction, which distorts and magnifies distant objects. Computer simulations and photographs show that mirages of an orca, a walrus, or even a boulder match almost point for point the descriptions in the King's Mirror. Thus the apparently mythical components in the Greenland account are in fact careful scientific observations.