Complex ArtGraphs

 
Martin Hansen
 

ArtGraph produces art from equations. Examples are shown to the right. At RISC OS computer shows, ArtGraph can be seen drawing these images in real time. What's not generally known is that ArtGraph can also produce contour plots of the complex plane. 
 

 
Consider the complex function:

 
It's quadratic and so has two complex roots. As points, they're (1,1) and (-1,-1).
 
The complex number, z, can be written:

 
Which gives:

where i is the square root of -1.

 
To plot this, apply Pythagoras' theorem:

 
Typing this into ArtGraph results in the screen shot above. The grey-scale part of the colour scheme shows the height reducing around the red zeros at (1,1) and (-1,-1). The red area has been set to be more than a point to emphasise the location of the zeros.

*INFO ArtGraph

 
ArtGraph is one of a suite of six programs sold by The MathMagical Software Company for £65.
 
The latest version is v.0.764.
 
However, the contour plots in this article are possible on all versions of the software going back to 2004.