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.

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"Still Life: Five Glass Surfaces on a Tabletop, " by Richard Palais, University of California, Irvine, and Luc BenardFirst Place, Illustration - 2006 Visualization "Vizzies" Challenge (National Science Foundation). Innumerable surfaces that we cannot touch or see or even know can be seen by mathematicians. They have long relied on their powers of imagination to picture abstract surfaces. Richard Palais of the University of California, Irvine, and graphic artist Luc Benard used the magic of computer graphics to recreate these abstract surfaces in familiar yet intriguing settings. See 2006 Vizzies Winners.

Möbius Transformations Revealed Credit: Douglas N. Arnold and Jonathan Rogness, University of Minnesota, Twin CitiesHonorable 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.

"Kuen's Surface: A Meditation on Euclid, Lobachevsky and Quantum Fields," by Richard Palais and Luc Benard, University of California at IrvineFirst Place, Illustration - 2009 Visualization "Vizzies" Challenges (National Science Foundation). Sketch a line and then draw a point off it. How many lines parallel to the first line can you draw through that point? The Greek mathematician Euclid said just one, but for more than 2,000 years after his death, mathematicians struggled to prove that he was right based on his other geometric rules. Then the 19th century Russian mathematician Nikolai Lobachevsky showed that you couldn't: In some circumstances, you can sketch an infinite number of lines through that point and not violate any of Euclid's other axioms. Mathematician Dick Palais of the University of California, Irvine, and digital artist Luc Benard wanted to convey the history of Lobachevsky's solution to this mathematical puzzle with their illustration.

In this illustration, a sheet of paper shows sketches of one of these surfaces, called Kuen's surface, and the expression, called a soliton, that describes it. "We wanted to talk about these equations in a way that non mathematicians could understand," Palais says. "So we took a symbolic approach: The surface itself stands as a symbol for that equation." See 2009 Vizzie Winners

"Exploring Complex Domain Functions Using Domain Coloring," by Konstantin Poelke and Konrad Polthier, Free University of BerlinHonorable Mention, Illustration - 2011 Visualization "Vizzies" Challenge (National Science Foundation). This illustration represents one example of a complex function. Such functions are mathematical relationships that incorporate both real and imaginary numbers, such as the square root of -1. To create this visualization, researchers at the Free University of Berlin assigned each complex number in their equation to a spot on a color wheel. The farther numbers get from zero, the brighter they are (white regions approach infinity). The result packs two dimensions of information (hue and brightness) into each point in the image. See 2011 Vizzie Winners

"The Life Cycle of a Bubble Cluster: Insight from Mathematics, Algorithms, and Supercomputers," by Robert I. Saye and James A. Sethian, UC Berkeley and Lawrence Berkeley National LaboratoryHonorable Mention, Posters & Graphics - 2013 Visualization "Vizzies" Challenge (National Science Foundation). Soap bubbles are often perceived as majestic, but the physics of popping bubbles in a foam are far from simplistic. Delving into the multi-scale phenomena underpinning bubble dynamics, one finds that there is host of challenges that need to be solved if one is to model and simulate foam behavior with computers. This poster tells part of this story, from the picturesque behavior of soap bubbles, to multi-scale physics and mathematical modeling, to simulation with powerful supercomputers. See 2013 Vizzie Winners, including a link to a video of the foam simulation.

"How origami is inspiring scientific creativity," (still from video) by Larry Howell, Julie Walker, Robert Lang, Spencer Magleby, and Brian WilcoxPeople’s Choice for the video category, and People’s Choice, Best Overall - 2015 Visualization "Vizzies" Challenge (National Science Foundation). Engineers use origami principles to design spacecraft solar panels and other devices that flex or unfurl, as in this video by a lab at Brigham Young University. Larry Howell, the team leader, says the work is just plain fun. "There's so much potential for applications. These things can really make a difference." [url= http://www.nsf.gov/news/special_reports/scivis/popup/2014/origami_video.jsp]View the video[/url].