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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"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/)

"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)

"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)

"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

"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/)

"Fractal," by Bradford Hansen-Smith (Chicago, IL)Folded 9" paper plates, 8" x 8" x 8", 2009

This is formed using twelve 9" paper plates all folded to the same equilateral triangular grid and reconfigured to the same design with slight variation between the four units that make the center tetrahedron pattern and the eight circles that form the outer cubic arrangement. This is one of many explorations using fifty-four creases rather than the twenty-four creases I usually work with. The higher frequency triangular grid allows greater complexity in a single circle which when combined in multiples forms designs that would not be possible otherwise. --- Bradford Hansen-Smith (http://www.wholemovement.com)

"Chaos - The Movie," by Susan Happersett (Jersey City, NJ)Video, 19" flat panel monitor (about 18" by 12"),

"Chaos - The Movie" is a stop-motion animation movie in which I create a line drawing based on Chaos Theory. The drawing--and the movie--were made over a period of six months. Music is an original composition made for the movie by Max Schreier. Meejin Hong did the video editing. --- Susan Happersett (http://www.happersett.com)

"Fibonacci Scroll," by Susan Happersett (Jersey City, NJ)Video, 19" flat panel monitor (about 18" by 12"), 2010

Fibonacci Scroll is a stop-motion animation of a long scroll drawing based on the Fibonacci Sequence. Susan Happersett has been creating mathematical, counted marking drawings for a number of years, but this is the first time her markings come to life. The sound track was composed specifically for the movie by Robert van Heumen, an accomplished composer and musician. --- Susan Happersett (http://www.happersett.com)

"Ball and Chain," by George Hart (Museum of Mathematics, New York, NY)Nylon (selective laser sintering), 6" x 6" x 6", 2009

Ball and Chain is a ball made of triangular chain mail mesh containing twelve flexible regions in a rigid dodecahedral framework. There are 3,722 small rings, which interlock to form a sphere with chiral icosahedral symmetry. At 920 places, six triangles meet, but at 12 special points (at the center of the twelve dimples) only five triangles meet. The ball does not collapse down to a disk because the dodecahedral structure of ribs (made by having some of the rings lock to form a skeleton) is rigid. But in twelve circular regions the rings are free to hang freely. No matter how it is turned, the top six regions hang to make concavities while the lower six regions are convex and blend in with the overall spherical form. The complete structure was created as one unit in its assembled state by selective laser sintering. --- George Hart (http://www.georgehart.com)

"Hyperbolic Planes Take Off!," by Vi Hart (New York, NY)Oil paint on canvas, 20" x 16", 2010

What does it look like when you crease the hyperbolic plane? This painting is an attempt at visualizing simple origami done with hyperbolic paper. Each plane has a mountain and valley fold perpendicular to each other. Done with your average Euclidean sheet of paper, it would be impossible to have both creases folded at a non-zero angle, but the hyperbolic plane can fold both ways at once. The creased plane can then be manipulated into different "birds", or so I imagine. --- Vi Hart (http://vihart.com)

"Hyperbolic Twist: Forever in Memory," by Charlotte Henderson (A.K. Peters, Natick, MA)Acrylic yarn and glass beads, 6 × 6 × 4 inches (3d work), 2010

This model is a hyperbolic Möbius band. The starting “spine” consists of 20 chain stitches, and the outer single edge has over 1600 single crochet stitches. The negative curvature of the surface allows the width of the band to be much greater than if the curvature were zero. The surface can move freely through the “hole” in the center. The bead work highlights the nonorientability of the surface. In isolated sections, it looks as if the beads are on two sides of the band; but if one traces the line of beads, one will return to the chosen starting point having traced all of the beads. The same amount of yarn is used for the red and the pink, to display the exponential growth of the surface. (There is a greater amount of white yarn, to have a constant final row for aesthetic purposes.) The color scheme arose from the fact that, at an earlier construction stage, the silhouette of the model resembles a heart. --- Charlotte Henderson

"Walking the Water's Edge," by Diane Herrmann (University of Chicago, IL)Needlepoint on canvas, 14" x 14", 2009

In this piece, the line imitates the edge of a wave on the shore. To make this wave look realistic, we used a mathematical curve that models the way a wave breaks on the beach. To be mathematically precise, we work with the sum of two trigonometric curves to show the action of water as it sloshes over itself in the push to get on the shore. The graph that defines the line of the Florentine Stitches is a close approximation to the curve: f (x) = 5 sin x + 4 cos (2x + π/3). The technique of thread blending creates the shading of the wave. Freeform eyelet stitches mimic the foamy edge of the wave and beads add sparkle. A single starfish is added in Bullion Knots. --- Diane Herrmann

"Proof Mining; The Gordian Geometric Knot," by Karl H(einrich) Hofmann (Tulane University, New Orleans, LA, and TU Darmstadt, Germany)Pencil, felt pen, tempera, 20" x 24", 2009

Two artworks from the Darmstadt Colloquium Poster Project, framed together. The techniques used are pencil, felt pen, tempera. The calligraphy of the posters is obtained with a Copic felt pen in a typography speciﬁcally developed for this purpose. The texts are prescribed by the departmental colloquium program determined one semester in advance. A complete collection of scans of the last 12 semesters can be inspected on my website. --- Karl H(einrich) Hofmann (http://www3.mathematik.tu-darmstadt.de/index.php?id=241)

"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/)

"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/)