"flat end of an arm of an anchor," 1560s, perhaps from fluke (3) on resemblance of shape, or from L.Ger. flügel "wing." Meaning "whale's tail" (in plural, flukes) is 1725.
"flatfish," O.E. floc "flatfish," related to O.N. floke "flatfish," flak "disk, floe" (see flake). The parasite worm (1668) so called from resemblance of shape.