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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"Borromean links cage," by Roger Bagula (Lakeside, CA)3.4" x 3.2" x 0.9", White plastic, 2012

The art work is made using Mathematica and 3d models that can be printed are obtained for most of the work. Since I was a boy I have drawn jet air planes and I have made a number of 3d models what I imagined. I have been studying ruled surfaces determined by torus knots and Möbius connecting surfaces. Here three links in a Borromean configuration are bridged by a cage surface. The bridge surface is not a Seifert surface, but a simple set of ruled Bezier surfaces. --- Roger Bagula

"Cardinal," by Harry Benke (Visual Impact Analysis LLC, Novato, CA)20" x 26.6", Giclee (pigmented archival print), 2012

I'm an artist and mathematician. My art attempts to produce a nexus between abstract mathematical beauty and the natural world to produce a satisfying aesthetic experience. I've been examining Kuen's surface for a very long time. The red shape is Kuen's surface as seen from above, looking down the z axis. Kuen's surface is well known since it has constant negative Gaussian curvature except on sets of measure 0. This surface is virtually never seen from above, which is intriguing and beautiful. -- Harry Benke (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.

"Knight to C3," by Leo Bleicher (San Diego, CA)30" x 25", Photographic print of 3D model, 2011

My mathematical art forms several large series exploring the creation of complex forms through sequences of simple operations or representations of simple relationships. The operations include geometric transformations, neighbor finding, attraction/repulsion and others. These computational processes attempt to replicate features of both geologic and organic morphogenesis. This image was created by application of a sequence of 3D geometric transformations to a collection of spheres on the faces of a hollow cube. The exact sequence to produce this image was obtained through the use of a genetic algorithm using subjective aesthetic appeal as the fitness function. -- Leo Bleicher

"Truchet from Truchet Tiles," by Robert Bosch (Oberlin College, Oberlin, OH)18" x 18", Digital print on canvas, 2012

Father Sébastien Truchet (1657-1729) was a Carmelite clergyman. He was King Louis XIV's favorite hydraulic engineer. He designed fonts. And in 1704 he published an article, "Mémoire sur les combinaisons," that described his mathematical and artistic investigations into how a simple set of square tiles, each divided by a diagonal into a white half and a black half, can be arranged to form an infinity of pleasing patterns. Today, Truchet's tiles are known as, well, Truchet tiles. To create a Truchet-tile portrait of Truchet, I started with the orientations specified by Pattern D of Plate 1 of Truchet's article. I then allowed the diagonals of the tiles to "flex" or bend at their midpoints. To make a tile darker, I would flex the diagonal into the black half. To make a tile brighter, I would flex the diagonal into the white half. With my Flexible Truchet Tiles, I can approximate any grayscale image, using the image to "warp" any initial pattern of Truchet tiles. -- Robert Bosch

"Bended Circle Limit III," by Vladimir Bulatov (Corvallis, OR)24" x 24", Digital print, 2012
Best photograph, painting, or print, 2013 Mathematical Art Exhibition

M.C. Escher's hyperbolic tessellations Circle Limit III is based on a tiling of the hyperbolic plane by identical triangles. The tiling is rigid because hyperbolic triangles are unambiguously defined by their vertex angles. However, if we reduce the symmetry of the tiling by joining several triangles into a single polygonal tile, such tiling can be deformed. Hyperbolic geometry allows a type of deformation of tiling called bending. Let's extend the tiling of the hyperbolic plane by identical polygons into tiling of hyperbolic space by identical infinite prisms. The prism's cross section is the original polygon. The shape of these 3D prisms can be carefully changed by rotating some of its sides in space and preserving all dihedral angles. Such operation is only possible in hyperbolic geometry. The resulting tiling of 3D hyperbolic space creates 2D tiling on the infinity of hyperbolic 3D space, which is a Riemann sphere. The sphere is stereographically projected to the plane. -- Vladimir Bulatov

"Inversions Five," by Anne Burns (Long Island University, Brookville, NY)12" x 12", Digital Print, 2012

Five pairwise tangent circles are all tangent to a sixth circle centered at the origin. The discs bounded by these six circles are colored in blue-green. An iterated function system is made up of repeated inversions in the six circles. -- Anne Burns

"Mathematical Game board," by Sylvie Donmoyer (Saumur, France)28" x 28", Oil paint on canvas, 2012

The design of the game board is suggested by the Archimedean spiral, divided in 62 spaces. It is played by two or more competitors and two dice. In the spaces are images relating to the History of Mathematics, in chronological order. As usual in this type of game, some spaces will bring you forward and others back, while the winner is the first to reach the stars. -- Sylvie Donmoyer

"CONTINUATIONS - Recursion Study in Wood," by Jeannye Dudley (Atlanta, GA)18" x 18" x 4", Basswood, 2012

The visual continuous curves were generated parametrically by a recursive design pattern developed based on simple a square motif and a replication rule (ratio) of 1 to 9. At first glance the piece appears flat; this effect is achieved through the black background. All the visual clues that the eye searches for to determine depth are lost in the dark monolithic background. The success in this piece is that it encourages the observer to wonder where does the pattern begin and end. The pattern becomes a path of CONTINUATIONS by providing an overlap at the initial four squares. This recursive design pattern provides the starting point for many other architectural investigations, like a stair case, a roof canopy or a wall panel system. The excitement is - integration of math - art - architecture. -- Jeannye Dudley

"The {3,8} Polyhedron with Fish," by Doug Dunham (University of Minnesota, Duluth)18" x 18" x 18", Color printed cardboard, 2012

The goal of my art is to create aesthetically pleasing repeating patterns related to hyperbolic geometry. This is a pattern of fish (inspired by M.C. Escher's Circle Limit III) on the regular triply periodic polyhedron composed of equilateral triangles meeting 8 at each vertex, which can be denoted by the Schläfli symbol {3,8}. It is formed from octahedral hubs which have octahedral struts connecting the hubs; the struts are on alternate faces of the hubs. This polyhedron approximates Schwarz' D-Surface which is the boundary between two congruent, complementary solids, both in the shape of a "thickened" diamond lattice (the hubs are the carbon atoms and the struts are the atomic bonds). There are fish of four colors. The blue fish all swim around the "waists" of the struts. The yellow, green, and red fish swim along lines that approximate the set of Euclidean lines that are embedded in Schwarz' D-Surface. In the image, the yellow fish swim right to left, the green fish swim from lower left to upper right, and the red fish swim from upper left to lower right. -- Doug Dunham

"Mandelbrot's Chandelier," by Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR)24" x 18", Digital print on archival paper, 2012

The spherical chandelier is composed of squarish lenses. Inside the chandelier is a cubical object that has been painted with the Mandelbrot set. Each of the lenses gives us a different view of this object. This interior object and the individual lenses are all variations of the quartic surface, x^4 + y^4 + z^4 = 1. The image was constructed using the ray tracing technique and required the solution of over a billion quartic equations, At^4 + Bt^3 + Ct^2 + Dt + E = 0, as the individual rays through each pixel were followed into this mathematical world of quartic surfaces. Snell's law was used to correctly model the refraction of the rays as they passed through the lenses. Finally, the background also shows a portion of the Mandelbrot set. -- Jeffrey Stewart Ely

"Duals," by Robert Fathauer (Tessellations, Phoenix, AZ)12" x 6" x 6", Ceramics, 2012

The cube and octahedron are duals of each other. In these two pieces, an octahedral frame encloses a cube and a cubic frame encloses an octahedron. The contrasting colors in the frames and enclosed polyhedra are due to the fact that they are made from two different types of clay. Neither was glazed, so the natural appearance of the fired clays is seen. -- Robert Fathauer

"Tessellation Evolution," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City) 18" x 15", Glass beads, gold-plated glass beads, onyx beads, gold-plated clasp, thread, 2012
Honorable Mention, 2013 Mathematical Art Exhibition

From one end of this necklace to the other, the design evolves through 16 different tessellations of the cylinder by congruent tiles in four colors. The strips of beads along the top and bottom of the frame, woven out of larger beads for clarity, exhibit the 16 tiles underlying the bead tessellations. The body of the necklace is a bead crochet rope. To construct the design, I manually colored a planar hexagonal grid of beads using the symmetry constraints imposed by crocheting the beads into a spiral. To make the necklace, I strung 4307 beads in the order dictated by the design onto five spools of thread, then crocheted the bead rope using a 1.1 mm hook. The caps at the end of the tube are woven with an additional 210 beads. -- Susan Goldstine

"Window into Infinity: A Sample of the Murhombicuboctahedron," by S. Louise Gould (Central Connecticut State University, New Britain)6.5" x 6.5" x 6.5", Embroidery on cotton fabric, 2012

This is an extension of the work I have done with fabric polyhedra. "The Symmetries of Things" written by Conway, Burgiel and Goodman-Strauss has provided many pathways to explore and it was the inspiration for this piece. This Archimedean infinite polyhedra is constructed of regular hexagons and squares. It illustrates the symmetry *642. Each vertex has 4 squares and one hexagon surrounding it. This model uses squares of four colors to show how the model moves into space. Each of the modules is constructed from a "net" of embroidered squares and hexagons. The nets were designed on Geometer's Sketchpad then digitized using 4-D Professional Software and stitched on a Viking Topaz embroidery sewing machine. The pictures show the object from three different directions. -- S. Louise Gould

"Quejido Design III," by Gary Greenfield (Prof Emeritus, University of Richmond, VA)10.5" x 10.5" (unframed), Digital Print, 2012

Between 1968-1972 Spanish artist Manuelo Quejido collaborated with computer programmers at the University of Madrid in order to execute a series of state-of-the-art computer generated “sequence” designs consisting of patterns of disks. As an homage, Quejido Design III realizes one of these patterns using state-of-the-art agent based methods. Here, agents are virtual ants modeled after the species T. albipennis that collect dispersed grains of sand in order to form circular nest walls. By using two different colors for sand grains and by assigning to each virtual ant a center, radius, and color, a uniform density grid of sand grains self-organizes into a pattern which, up close, has no color symmetry, but from a distance is perceived of as being color preserving under various symmetry operations. -- Gary Greenfield

Fractal Cylinder 2, by Daniel Gries (Hopkins School, New Haven, CT)11.5" x 22", Giclee print from JavaScript-generated digital image, 2012

In Fractal Cylinder 2, curves are drawn along waist curves, colored by a radial gradient. Low alpha values create differing densities and transparency. -- Daniel Gries