 
Summary: www.sciencemag.org SCIENCE VOL 317 28 SEPTEMBER 2007 1863
ANY REAL NUMBER CAN BE PLOTTED ON A LINE THAT RUNS FROM NEGATIVE TO POSITIVE
infinity, but throw in an imaginary component and the line becomes a plane, where
complex numbers are plotted on both the real and the imaginary axes. Möbius
transformations are mathematical functions that send each point on such a plane
to a corresponding point somewhere else on the plane, either by rotation, translation,
inversion, or dilation. It may sound confusing, but after watching this simple and elegant
explanation of Möbius transformations created by Douglas N. Arnold and Jonathan
Rogness of the University of Minnesota, Minneapolis, everything becomes clear. Set to
classical music, the video demonstrates the transformations in two dimensions but then
backs away and adds a thirdplacing a sphere above the plane and shining light through
it. As the sphere moves and rotates above the plane, suddenly all the transformations
become linked, in a way that conveys visually in minutes what would otherwise take
"pages of algebraic manipulations" to explain, says Rogness.
H O N O R A B L E
M E N T I O N ( T I E )
MOBIUS TRANSFORMATIONS
REVEALED
Douglas N. Arnold and
Jonathan Rogness, University
