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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"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

Mar 10, 2011

"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 Mar 10, 2011

"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) Mar 10, 2011

"Longest and Shortest Crease-B," by Sharol Nau (Northfield, MN)Folded book, 12.5” x 9” x 6”, 2010

For this book-sculpture of several hundred pages, the shortest crease was obtained by folding the pages without separating them from the binding. Also the folding process began in the middle in an effort to achieve a symmetrical design. --- Sharol Nau (http://www.sharolnau.snakedance.org) Mar 10, 2011

"DART," by Jo Niemeyer (Schluchsee, Germany)Archival inkjet print, 20" x 20", 2010

DART ... as a very simple game by Jo Niemeyer 2010 . Two dart players, A and B, are facing this "image", whose area is split 1 : 0.618.. into white and black. This two basic elements are rotated in 90° increments. The winner is, who aims first a black part. Since we have two equal partners and an uneven distribution of the "target", one would think, that this is not a fair game. But it is! Because A as the "majority", and consequently B as the "minority", transferred their inequality onto the "court". The ratio of the two playing partners is 1:1. With this harmonious proportionality there is exactly the same chance to win for both players A and B! The Swiss mathematician Hans Walser mentions for the justice condition, the formula p = 1/2*(3-sqrt(5)). And with sqrt(5), we have the golden section in this game, which ensures equity between different partners. This is also a very fair game! Or a piece of art, which ensures harmony and balance. --- Jo Niemeyer (http://www.jo.niemeyer.com) Mar 10, 2011

"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 Mar 10, 2011

"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 Mar 10, 2011

"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) Mar 10, 2011

"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