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Möbius transformation are almost fondamental mappings in geometry
with application from brain mapping to relativity theory.
A Möbius transformation sends each point of the plane to a corresponding point.
There are four basic types:
the simple translations
dilations
rotations
and invertions, which turns plane inside out.
Lines on the plane and the remain lines are transformed into circles
and right angles stay true.
In general Möbius transformation can be a complicated combination of all four effects.
True unity of the Möbius transformations is revealed by moving into the next dimension.
Taking a cue from Bernhard Riemann, we place a sphere above the plane.
A light at the top shines through the spherical surface illuminating the plane.
As the sphere moves the points on the plane follow.
When the sphere translate so does the plane.
Raise of the sphere means in dilation.
Spin the sphere at the top and the plane rotates.
Rotation around an horizontal axis corrisponds to an inversion.
Even the most complicated Möbius transformation
are revealed by simple motions of the sphere.
Transcript: Libera Scienza
http://www.facebook.com/liberascienza