Functions in Two-Dimensional Vector Space or Complex Plane
This Demonstration illustrates that functions in a real two-dimensional vector space are equivalent to functions in the field of complex numbers. Specifically, we consider a class of functions of the form , where and are different integers. The black arrow traces out the function , shown as a black curve, while the two component vectors are represented by red and blue arrows, tracing out red and blue circles.
The idea for this Demonstration comes from [1, p. 17].
 F. A. Farris, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, Princeton: Princeton University Press, 2015.
"Functions in Two-Dimensional Vector Space or Complex Plane"
Wolfram Demonstrations Project
Published: February 15 2016