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 ands, 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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"Ramanujan, in the style of Chuck Close, using wavelets," by Edward Aboufadel (Grand Valley State University, Allendale, MI), Clara Madsen (University of Oregon, Eugene) and Sarah Boyenger (Florida State University, Tallahassee)Digital print, 16" x 20", 2009

Both the subject of this work and the method of creation are intricately mathematical. Ramanujan is the famous 20th century Indian mathematician who established or conjectured a broad collection of results in number theory. He caught the attention of Hardy, who recognized Ramanujan's genius. To create this digital image in the style of Chuck Close, wavelet filters were used to detect the existence and orientation of edges in the original image, and other calculations were made to determine the colors in the "marks". Mar 10, 2011

"Gaussian Wave Packet Sculpture," by Chet Alexander (University of Alabama, Tuscaloosa)Wood (birch, walnut, maple, ebony), 9" x 11" x 10", 2006

Mathematics of the Wave -Packet Sculpture: In this sculpture, mathematics was used to calculate the Gaussian wave-packet model of a particle in quantum mechanics. This is accomplished by forming a linear combination of plane waves of different wave-numbers, k. A particle with mass and momentum p can have wave properties as described by the de Broglie wavelength relation λ=h/p. The Gaussian wave packet model is a way to combine the wave and particle properties of a particle of momentum p=hk localized at position x_0. The probability of finding the particle at position x_0 is given by the probability density of the particle as ІΨ(x,0) І^2~exp[-(x-x_0)^2/2(∆x)^2] , and by a Fourier transform the probability density of the particle's momentum can be written ІΨ(k) І^2~exp[-(k-k_0)^2/2(∆k)^2]. The wave packet sculpture presents a Gaussian wave packet envelope and an electromagnetic wave enclosed in the envelope. --- Chet Alexander Mar 10, 2011

“CORPI INCROCIATI • DUE CORPI,” by Cristina Besa (European Society for Math & Art, Islas Baleares, Spain)6 x 7 negative, print on Ilford glossy paper, 12” x 16” (framed 20” x 24”), 2004

Bilateral symmetry—a combination of identical exposures intersecting with their own reflections. The intersection of the two shapes A (original) and B (reflection of A) are positioned at the same angle to create a new form bound by its central line reflection. --- Cristina Besa

Mar 10, 2011

"The Sphere and the Labyrinth," by Robert Bosch (Oberlin College, Oberlin, OH)Wood (maple) and steel, 3" in diameter, 2010

Inspired by my favorite childhood toy, a ball labyrinth game made by Brio, I hand carved a symmetric simple closed curve into the surface of a 3'' diameter ball of maple. The curve is a channel that is deep enough and wide enough to hold a 0.5" diameter ball of steel. It is possible to pick up the ball of wood and maneuver it so that the ball of steel will roll through the entire channel and end up back where it started. --- Robert Bosch (http://www.dominoartwork.com) Mar 10, 2011

"Ideal quilt, slightly imperfect," by Andrzej K. Brodzik (Mitre Corporation, Bedford, MA)Digital print, 24'' x 20'', 2010

Ideal quilts are Zak space representations of families of ideal sequences. Ideal sequences are sequences with certain special group-theoretical properties. In particular, ideal sequences satisfy the Sarwate bound, having both zero out-of-phase autocorrelation and minimum cross-correlation sidelobes. Construction of ideal sequences was described in the recent book, Ideal sequence design in time-frequency space. Ideal quilts are (p-1)p by (p-2)!p images, where p is a prime. As these images tend to be long and narrow, to facilitate display, they are usually divided into columns. Geometrically, an ideal quilt is a sequence of distinct permutations of the canonical image of a diagonal line. Both the overall structure of the image and the association with ideal sequences convey a strong sense of symmetry, predictability, and uniqueness. To counter-balance these qualities, the corrupting effect of tiff data compression, manifested as pixel distortion, is embedded into the image. --- Andrzej K. Brodzik Mar 10, 2011

"Hyperbolic Tiling I," by Vladimir Bulatov (Corvallis, OR)Rapid prototyping sculpture, 200mm x 60mm x 60mm, 2010

This is a visualization of a tiling of the hyperbolic space. The tiling is generated by reflections in the faces of Lambert cube (Coxeter polyhedron), which becomes the fundamental polyhedron of the symmetry group of the tiling. Only 4 out of 6 sides are used, which results in sub-tiling (subgroup) filling only part of the space. It let us see the internal structure of the tiling. We use a cylinder model of the hyperbolic space--a 3D generalization of 2D band model. In this model the Poincare ball is stretched into infinite cylinder. Cylinder's axis becomes one of hyperbolic geodesics. The tiling is oriented to make one it's plane to be orthogonal to the cylinder's axis to have a feet to stand on. The cylinder's axis is close to the axis of a loxodromic transformation of the group, which gives the pieces its spiral twist. The sharp boundary of the piece corresponds to the limit set of the group. The limit set is fractal Jordan curve at the infinity of the hyperbolic space. --- Vladimir Bulatov (http://bulatov.org) Mar 10, 2011

"Circles on Orthogonal Circles," by Anne Burns (Long Island University, Brookville, NY)Third Place Award, 2011 Mathematical Art Exhibition

Digital print, 12" x 16", 2010

A loxodromic Möbius transformation has two fixed points, one attracting and the other repelling. Starting with a small circle around the repelling fixed point, and repeatedly applying the Möbius transformation, results in a family of circles that grow at first, each containing the previous one. Successive images eventually pass over the perpendicular bisector of the line connecting the fixed points and shrink as they are attracted to the other fixed point. Each circle in a second family of circles passes through the fixed points and is mapped to another circle in that family. Each circle in the second family is orthogonal to every circle in the first family. --- Anne Burns (http://www.anneburns.net) Mar 10, 2011

"MacDuff's Shield," by Robert A. Chaffer (Central Michigan University, Mount Pleasant, MI)Giclee print on watercolor smooth paper, 12" x 15", 2007

MacDuff's Shield depicts the shield and spirit of the character MacDuff in Shakespeare's play, "Macbeth." The shield is constructed from two superimposed copies of a texture taken from a Julia set and styled for this purpose. This composite was then laid upon a surface of revolution to complete the shield. The ethereal veil in which the shield floats presents the idea that MacDuff is "not of woman born." This veil was created by orbit-trapping in the interior of the Mandelbrot set. --- Robert A. Chaffer http://people.cst.cmich.edu/chaff1ra/Art_From_Mathematics/) Mar 10, 2011

"Loopy Love," by Barry Cipra (Northfield, MN)Letterpress print on paper, 11" x 6" x 5", 2010

"Loopy Love" was composed for a workshop on Creative Writing in Mathematics and Science, held at the Banff International Research Station (BIRS), May 2-7, 2010. The idea was to explore the implications of writing a short story on a möbius strip. The story, a dialog presenting both sides (or is there only one side?) of a twisting love/hate relationship between two characters named Daniel and Danielle, was letterpress printed by Red Dragonfly Press in Red Wing, Minnesota, on Fabriano paper using the font FF Quadraat. Assembled by hand with tape, the resulting scrollable sculpture retains its shape yet remains flexible, so that the reader can easily read the story without ever having to turn the page. Viewers are invited to pick it up, play with the paper, and read the story from start to finish--except there is no start, nor any finish! --- Barry Cipra Mar 10, 2011

"Drosted Philadelphia," by Francesco De Comité (University of Sciences and Technology of Lille, France)Digital picture, 30x30 cm, 2009

Drosting, a hyperbolic tiling. Manipulation of digital images and use of ray-tracing software can help you to concretize mathematical concepts, either for giving you an idea of how a real object will look, or to represent imaginary landscapes. --- Francesco De Comité Mar 10, 2011

"Jones," by Slavik Jablan (The Mathematical Institute, Belgrade, Serbia)Digital print on paper, 420 x 420 mm, 2010

The graphics shows the plot of the zeros of the Jones polynomials of rational knots and links up to n=17 crossings, where zeros of knots are red, and zeros of links are yellow. --- Slavik Jablan (http://math.ict.edu.rs/) Mar 10, 2011

"Three Strange Dreams," by Karl Kattchee (University of Wisconsin-La Crosse)Digital print, 24" x 12", 2010

This is derived from one of my other works---"Rings and Monoids"---by tiling the plane with it, capturing three close-ups, and weaving them together to create these three images. The viewer is invited to attempt reconstructing "Rings and Monoids" or to visualize a looping animation composed of these three frames. Or not, if you prefer. --- Karl Kattchee (http://www.uwlax.edu/faculty/kattchee/) Mar 10, 2011

"Magic Square 25 Study," by Margaret Kepner (Washington, DC)First Place Award, 2011 Mathematical Art Exhibition

Archival inkjet print, 12.5" x 12.5", 2010

Magic squares are numerical arrays that have substructures with constant sums. This design is based on a magic square of order 25, containing the numbers from 0 to 624. Each row, column, and main diagonal sums to the “magic constant” of 7800. The numbers in the magic square are represented by a visual base-5 system: four concentric squares serve as the 1, 5, 25, and 125 places, while shades of grey stand for the numerals 0 to 4. Coding the numbers into their base-5 versions yields a pattern of 625 unique, nested-squares in shades of grey. This particular magic square also has a substructure of 25 mini-squares of size 5. Each of these mini-squares is “magic” (although the numbers are not consecutive), with rows, columns, and diagonals summing to 1560. In addition, certain other groups of 5 squares add up to 1560. Examples are the quincunx and the plus-sign shapes (when fully contained in a mini-square). The colored accents are used to indicate a few of these “magic” substructures. --- Margaret Kepner Mar 10, 2011

"Unknown," by Ghee Beom Kim (Sydney, Australia)Print, 300 x 250 mm, 2007

Hexagonal fractal creates this terrace-like surface that imitates exotic terrain, which has architectural implication. The triangular fractal formations occasionally occur amongst rather random surface. --- Ghee Beom Kim (http://sites.google.com/site/geometricarts/) Mar 10, 2011

"Whirled Heart," by Matjuska Teja Krasek (Ljubljana, Slovenia)Digital print, 170 x 230 mm, 2010

In the mysterious world of chaos and strange attractors a seeker can find very heartful things. --- Matjuska Teja Krasek (http://tejakrasek.tripod.com) Mar 10, 2011