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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"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) Mar 10, 2011

"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) Mar 10, 2011

"Quasirandom Aggregation," by Tobias Friedrich (Max Planck Institute for Informatics, Saarbrücken, Germany)Digital print on glossy paper, 20" x 20", 2010

Given an arbitrary graph, a random walk of a particle is a path that begins at a given starting point and chooses the next node with equal probability out of the set of its current neighbors. Around 2000, Jim Propp invented a quasirandom analogue of random walk. Instead of distributing particles to randomly chosen neighbors, it deterministically serves the neighbors in a fixed order by associating to each vertex a "rotor" pointing to each of its neighbors in succession. The picture shows what happens when one billion particles are placed at the origin and each one runs until it reaches an unoccupied vertex. Black pixels denotes cells that never get visited by a particle; for the other cells, the color of the pixel indicates in which direction the rotor points at the end of the process. More information can be found at http://rotor-router.mpi-inf.mpg.de. --- Tobias Friedrich (http://www.mpi-inf.mpg.de/~tfried/) Mar 10, 2011

"Triad," by Mehrdad Garousi (Hamadan, Iran)Digital print on canvas, 18" x 16", 2009

This work created in TopMod comprises a uniform twisted strip with some ties and joints. The shape containing an evident three-fold rotational symmetry is composed of three similar components connected at two central joints placed back and forth. The outstanding issue is another hidden symmetry, which may not be discovered at a hasty glance. In addition to the former symmetry, condoning back and forth or up and down position of layers, as a flat plane, the whole sculpture has a three-fold mirror symmetry. The reason of such a property is the same one-fold mirror symmetry governing each of the three components. --- Mehrdad Garousi (http://mehrdadart.deviantart.com) Mar 10, 2011

"Tea for Eight," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City)Glaze on commercial ceramic, 5" x 8" x 5", 2010

The Four-Color Theorem says that we can color any map on a plane or sphere with only four colors so that no neighboring countries are the same color. On other surfaces, we may need more colors; on a two-holed torus, eight colors are sufficient, and there are maps that require all eight colors. When this tea set is stacked with the handles aligned, it forms a topological two-holed torus with a map of eight countries, each of which touches all of the others, proving that eight colors are necessary. The teapot has white, red, orange, and yellow countries, and the teacup has black, green, blue and purple countries. At the seam between the pieces, each of the top colors touches each of the bottom colors. On one-holed tori, such as the teapot and the teacup, seven colors are required for an arbitrary map. Unfortunately, a seven-color map is incompatible with the tea set's exterior pattern; when the tea set is opened, hidden colors give six-color maps of the teacup and the teapot. --- Susan Goldstine (http://faculty.smcm.edu/sgoldstine) Mar 10, 2011

"Circle Brooches," by Anansa Green (Stephen F. Austin State University, Nacogdoches, TX)Fine silver, copper, 1.5 x 1.5 x 0.25 inches each (2 brooches), 2009

These brooches were inspired by my undergraduate graph theory research into the colorability of the map created by a finite tiling of circles in the plane. I was able to prove by mathematical induction that the resulting map is 2-colorable. This result lends itself quite well to the process of married metals. Two pieces of metal were overlaid: one copper and the other fine silver. The design was pierced from both sheets at once, and alternating pieces were swapped to form the two 2-colored designs. The individual components of each image were silver-soldered together, and the sides and back of each brooch hollow constructed to create the final form. The process yields two images, each one the inverse of its partner. To emphasize the complementary nature of each image, I fabricated one brooch with a convex face and the other concave. --- Anansa Green Mar 10, 2011

"Right Angle Triangles in Flatland A," by Gary Greenfield (University of Richmond, VA)Digital print, 18" x 12", 2010

Four Flatlanders are sweeping through Flatland celebrating their discovery of how to draw right triangles. Their method is as follows: (1) pseudorandomly generate a turning angle alpha and an adjacent side length x; (2) calculate the complementary angle beta and use trigonometry to calculate the opposite side length y and hypotenuse length h; (3) then swivel right, forward x, turn alpha, forward h, turn beta, forward y, swivel left. These Flatlanders belong to the caste required to "wag" from side to side when they walk. Thus they defy convention by drawing perfectly straight thick lines when presenting their right triangle discovery. Here, Flatlanders are implemented as simulated drawing robots obeying obstacle and collision avoidance, and their wag is implemented by making one of their pens swing side to side in such a way that a sinusoidal track is drawn as they make their through Flatland. --- Gary Greenfield (http://www.mathcs.richmond.edu/~ggreenfi/) Mar 10, 2011

"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