Mobius transformations

In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics (Part #10)

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For mathematicians, Möbius transformations are functions that send each point on a complex plane (with real and imaginary axes) to a corresponding point somewhere else on the plane, either by rotation, translation, inversion, or dilation (see illustrative java applets: transformation of fixed points, iterated application to a circle). Douglas N. Arnold and Jonathan Rogness have developed a widely appreciated video to illustrate these processes (Möbius Transformations Revealed, 2007), awarded in the annual Science and Visualization Challenge (Science, 2007). This took advantage of the fact that a Möbius transformation may be achieved by performing a stereographic projection from a plane to a sphere, rotating and moving that sphere to a new arbitrary location and orientation, and performing a stereographic projection back to the plane.

As helpfully discussed by Julie J. Rehmeyer (A Video That's Worth a Million Words, MathTrek, 2007), the video clearly illustrates how the processes transform a square grid, but as she says:

Next comes the video's magical step. The mathematicians move into the third dimension to provide a way of visualizing the Möbius transformations. They suspend a sphere above the plane and use it a bit like a slide projector. They put a picture onto the sphere, and a light at the top of the sphere shoots an image of the picture down onto the plane. The picture on the sphere is shaped in such a way that when the light projects the image onto the plane, it forms the original square.

The key to the success of the explanation lies in the understanding then enabled through various different movements of the sphere. The sphere is in this case a representation of the Riemann sphere, namely the complex number plane wrapped around a sphere through stereographic projection. The Rieman sphere is important in mathematics as a way of extending the plane of complex numbers with one additional point at infinity (cf David Mumford, Caroline Series and David Wright, Indra's Pearls: The Vision of Felix Klein. Cambridge University Press, 2002).

Of relevance to the above argument is whether such a video effectively provides a powerful way of understanding the subtle relationships between the features of Figure 2 (as "the square") in the light of the role of a third dimension (a "fifth perspective"), from which those features are viewed (the "sphere"). It is the particular "movements" of that sphere that transform the relations between the elements of Figure 2 -- thereby indicating how they may be variously understood. Can such transformations then be understood as cognitive transformations in understanding of the various possible relationships between "problematique", "resolutique", "imaginatique" and "irresolutique"? The sphere also then suggests interesting ways of understanding the coherence of the "identique".

Of related interest are the explorations of the relationship between such transformations and the classical studies of biologist D'Arcy Wentworth Thompson (On Growth and Form, 1917/1961) regarding animal forms (see for example Using a computer to visualise change in biological organisms, 2000). These suggest the possibility of using such transformations to distinguish between cognitive frameworks as variously perceived to be "distorted" with respect to one another.

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