Complex Gaussian integers for “Gaussian graphics”

Abstract

Some recent computer languages incorporate rational numbers, complex numbers, and rational complex numbers. We extend these numeric facilities to deal properly with Gaussian integers ---i.e., complex numbers whose real and imaginary parts are both ordinary (rational) integers. In addition to their intrinsic mathematical interest, such extensions also raise interesting questions regarding polymorphism and multiple inheritance.Since Gaussian integers are the coordinates of discrete square pixels in the complex plane, complex operations can be used to implement 2-D graphics operations. Many 2-D algorithms are more elegant in complex number form---e.g., one can envision a 2-D spreadsheet for scientific applications whose coordinates are Gaussian integers.