Z-Squared Tetraview
The "tetraview" is an attempt to understand graphs of complex functions
better. We treat them as objects in real 4-space, and view the graph by
looking down the positive coordinate axes (corner views). These views
correspond to the graphs in 3-space of the real and imaginary parts of the
function, and the real and imaginary parts of the inverse relation.
The various viewpoints in 4-space can be thought of as points on the
3-sphere; the four canonical viewpoints form the corners of a (spherical)
tetrahedron. Moving the viewpoint along one of the edges of the
tetrahedron effectively rotates from one view to another, helping to
understand the interconnection between the views. The central view in the
tetraview is the view from the center of the tetrahedron and represents in
a real sense the "average" of the other four.
In this image, we view the complex squaring function. The graphs of the
real and imaginary parts are the saddles in the lower left and upper
right. The other two are the real and imaginary parts of the square-root
relation; these illustrate different ways of representing the Riemann surface
for the square root, and show the ramification point at the origin.
This image appeared as part of the art show Surfaces Beyond the Third Dimension.
This surface is also the subject of Z-Squared Necklace.
|