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.

Kleinian PearlsPeople have long been fascinated with repeated patterns that display a rich collection of symmetries. The discovery of hyperbolic geometries in the nineteenth century revealed a far greater wealth of patterns, some popularized by Dutch artist M. C. Escher in his Circle Limit series of works.

This cover illustration portrays a pattern which is symmetric under a group generated by two Möbius transformations. These are not distance-preserving, but they do preserve angles between curves and they map circles to circles. The image accompanies "Double Cusp Group," by David J. Wright (Notices of the American Mathematical Society, December 2004, p. 1322).

"(1,3,5) Pretzel Knot," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.The yellow tube is a (1, 3, 5) pretzel knot. Such a pretzel knot or link consists of a sequence of angles, where each tangle has a number of twists. The brown surface is a Seifert surface: an orientable surface bounded by the knot. Here the surface is easy to understand; for arbitrary knots such surfaces often have strange and difficult shapes. However, for any knot or link such surfaces can be found, as shown by Herbert Seifert in the 1930's. This image was made with a tool called SeifertView.

--- Jarke J. van Wijk

"Borromean Rings," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.The Borromean Rings consist of three links. Take one link away and the other links fall apart, but together they are inseparable. Because of this, they are popular as a symbol for strength in unity. Here they are shown from an unusual point of view, and also a Seifert surface is shown. This is an orientable surface, bounded by the links. This image was made with a tool called SeifertView.

--- Jarke J. van Wijk

"Three Link Chain," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.This knot consists of three similar links, and is threefold-symmetric. The surface shown is a Seifert surface, an orientable surface bounded by the links. Considering only the links, it is hard to imagine that such a surface does exist. However, in the 1930's, the German mathematician Herbert Seifert presented an algorithm to find such surfaces for any knot or link. This image was made with a tool called SeifertView.

--- Jarke J. van Wijk

"Overwrought," by Damien JonesDamien Jones is a respected artist and fractal expert. His Internet domain fractalus is one of the most complete sources to start with for fractal art. Through years of explorations of the mathematics for aesthetic reasons, Damien’s work has appeared in numerous books, magazines, posters, calendars, and international exhibitions. Born in the United Kingdom, he currently resides in Florida (USA) with his wife Michelle, whom he married while collaborating on the organization of this exhibition. The image "Overwrought" belongs to the Mandelbrot set, although it is difficult to see because of the use of "turbulence," which distorts the calculations before the application of the fractal coloring. After the image is colored, the turbulence is removed and the calculation continues. The process produces a cloudy texture but keeps the underlying shapes unaltered. The coloring—austere, mournful, and at times apocalyptic—often produces an emotional response in the viewer of the art.