The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.
Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.
Möbius Transformations Revealed Credit: Douglas N. Arnold and Jonathan Rogness, University of Minnesota, Twin Cities
Honorable Mention, Noninteractive Multimedia (screen shots) - 2007 Visualization "Vizzies" Challenge (National Science Foundation). Any real numbers 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 UNM, everything becomes clear. Set to classical music, the video demonstrates the transformations in two dimensions but then backs away and adds a third--placing 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. See 2007 Vizzie Winners.