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.

"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)2067 viewsDigital 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".

"Ten Triangular Prisms," by Magnus Wenninger (Saint John's Abbey, Collegeville, MN)1598 viewsPaper, 9" x 9" x 9", 2010

Robert Webb's Stella program is now the computer program I use for the construction of all the polyhedron models I have recently been making. It is the program par excellence I now use for the discovery of any new polyhedra, especially any I have never made before. The photo shows a model of Ten Triangular Prisms, recently made by me. I found the Stella version on a web page called '75 Uniform Polyhedra' done by Roger Kaufman. It is #32 on this web page. The Stella version gives me a 3D computer view in 10 colors and allows me to choose the size of the model and thus also the size and shape of the net to be used for the construction of the model. However, I wanted my model to be done using only 5 colors. This is where the artwork comes into play. The model now shows each prism with its faces in one color of the five. Thus it becomes uniquely artistic in appearance. --- Magnus Wenninger (http://www.saintjohnsabbey.org/wenninger/)

"The Sphere and the Labyrinth," by Robert Bosch (Oberlin College, Oberlin, OH)1321 viewsWood (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)

"Gaussian Wave Packet Sculpture," by Chet Alexander (University of Alabama, Tuscaloosa)1286 viewsWood (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

"iteration," by Trygve Wastvedt (St. Olaf College, Northfield, MN)1218 viewsBronze, plaster, wax, concrete, 18" x 18" x 8", 2010

"iteration" is a series of identical humanoid figures cast in bronze, plaster, wax, and concrete. The form is a near honeycomb so that the individual pieces stack together to fill space. Though geometric, the form still evokes human emotions, which allows the sculpture to ask social and relational questions. --- Trygve Wastvedt (http://www.trygvewastvedt.com)

"Infinity," by Mary Wahr (Manistee Area Public Schools, MI)1212 viewsPen and ink, 18" x 21", 2009

This is an abstract pen and ink rendering of a broccoflower. It is the first piece of art completed for my thesis and was the starting point of two years of research and art. Since my fractal ideas are accomplished without arithmetic, I needed to incorporate the components that define a fractal. This piece of art reflects the concepts of infinity, iteration, self-similarity and scaling. --- Mary Wahr

"Fractal Tree No. 13," by Robert Fathauer (Tesselations Company, Phoenix, AZ)1146 viewsDigital 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)

"Drosted Philadelphia," by Francesco De Comité (University of Sciences and Technology of Lille, France)1139 viewsDigital 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

"Hyperbolic Tiling I," by Vladimir Bulatov (Corvallis, OR)1138 viewsRapid 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)1136 viewsThird 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)

"Loopy Love," by Barry Cipra (Northfield, MN)1107 viewsLetterpress 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

"Blue Ionic Polyhedra: 12 Beaded Beads in Two Sizes," by Gwen Fisher (bead Infinitum, Sunnyvale, CA)1103 viewsBead 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)

"Ideal quilt, slightly imperfect," by Andrzej K. Brodzik (Mitre Corporation, Bedford, MA)1085 viewsDigital 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

"MacDuff's Shield," by Robert A. Chaffer (Central Michigan University, Mount Pleasant, MI)1043 viewsGiclee 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/)

"Bucky Madness," by Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR)1019 viewsDigital 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