Complex functions of a single complex variable involve four unknowns, two independent and two dependent variables, and thus cannot be adequately represented in two- or three- dimensional space. Various geometric constructions in both two and three dimensions have been devised in the past, however, in attempts to illuminate complex function theory. The standard and most useful, of these representations is that developed by Gauss and Riemann employing two complex planes simultanesously. These show the correspondence between a particular curve or region in the object plane and its image, as mapped by a given transformation, in the image plane
