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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"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".

"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

“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

"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)

"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

"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)

"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)

"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/)

"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

"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

"Bucky Madness," by Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR)Digital print on archival paper, 20" x 20", 2010

This is my response to a request to make a ball and stick model of the buckyball carbon molecule. After deciding that a strict interpretation of the molecule lacked artistic flair, I proceeded to use it as a theme. Here, the overall structure is a 60-node truncated icosahedron (buckyball), but each node is itself a buckyball. The center sphere reflects this model in its surface and also recursively reflects the whole against a mirror that is behind the observer. I was recently surprised to read in David Richeson's book, Euler's Gem, that Legendre proved Euler's Formula, V - E + F = 2, by projecting a polyhedron onto a sphere and then summing the areas of the various spherical polygons. I think this fact resonates rather well with this design. --- Jeffrey Stewart Ely

"Branched Manifold," by Juan G. Escudero (Universidad de Oviedo, Spain)Digital Print, 20"x20", 2010

A cell complex is defined in the analysis of the cohomology of tiling spaces. It contains a copy of every kind of tile that is allowed, with some edges identified for the 2D case, and the result is a branched surface. When the tiling does not force the border, collared tiles can be used. Here the triangles with the same shape, color and orientation represent the same tile in the complex. The manifold appears in the cohomology computations of an octagonal pattern belonging to a random tiling ensemble introduced by the author in the context of mathematical quasicrystals. --- Juan G. Escudero

"Flora #1 (time slice)," by Brian Evans (University of Alabama, Tuscaloosa)Archival inkjet print, 9" x 6" (14" x 11" framed), 2010

How much is lost in the reduction of reality to human sensation? The infinite detail there in front of us is reduced to 100 million discrete measurements made with the rods and cones on the retina of the eye. Infinity reduced to 100 million, which is reduced another ninety-nine percent as the signal is compressed to travel only 1 million pathways on the optic nerve. It’s a wonder we can make sense of the world at all. These little photos are also reductions, slit-scans of flowers rotating on a tabletop—2D slices of time. The four dimensions of our reality (x, y, z, t) are reduced to two (x, t) showing a different aspect of the real. The temporal is mapped into the static and new forms and structures are seen. These works are metaphors for the language of mathematics. What wonders we can discover through the processes of abstracting, reducing, mapping, and finally looking in new ways at the little slices of information we receive from all the surrounds us. --- Brian Evans (http://brianevans.net)

"Fractal Tree No. 13," by Robert Fathauer (Tesselations Company, Phoenix, AZ)Digital print, 13" x 16", 2010

"Fractal Tree No. 13" is a black-and-white digital artwork constructed by graphically iterating a photographic building block. Several photographs of a small portion of a palo verde tree were combined and digitally altered to create the building block, which allows smooth joining of smaller copies to larger copies. In addition to being scaled down, the three smaller copies, added with each iteration are rotated by varying angles, and one is reflected as well. A sufficiently large number of iterations were performed so that the image is indistinguishable to the eye from the image that would result after an infinite number of iterations. In this particular tree, the iteration rules result in considerable overlap of the branches, leading to a complex collection of small features reminiscent of pencil marks. The fractal shape of the envelope of these features didn't emerge until approximately one dozen iterations were performed. --- Robert Fathauer (http://www.tessellations.com)

"Blue Ionic Polyhedra: 12 Beaded Beads in Two Sizes," by Gwen Fisher (bead Infinitum, Sunnyvale, CA)Bead weaving with crystal, glass and metal beads, and thread, 12 beads, longest diameter ranges from 11 mm to 33 mm, 2010

The Ionic Polyhedra are part of my larger body of mathematical artwork in beaded beads. A beaded bead is a cluster of smaller beads, woven together with a needle and thread, to form a composite cluster with one or more holes running though the center of the finished beaded bead. To make these beads, I started with inner cluster of beads in two sizes where the larger beads are aligned on the edges of polyhedra. For the larger six beads, I added layers of seed beads, which emphasize the edges of the underlying polyhedra. Although they appear different, the miniature version of each Ionic Polyhedron maintains the same structure and thread path as its larger version, but the beads are smaller, and the embellishment is simpler. Represented are the octahedron, cube, pentagonal dipyramid, pentagonal antiprism, cuboctahedron, and rhombic dodecahedron. This coloring of the miniature cuboctahedron and rhombic dodecahedron illustrates the dual relationship between these two polyhedra. --- Gwen Fisher (http://www.beadinfinitum.com)