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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.

[Link] Z-Squared Tetraview in SB3D

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Created: 13 Oct 2003
Last modified: Oct 15, 2003 11:30:09 AM
Comments to: dpvc@union.edu
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