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.

"Magneto-2," by Reza Ali (Palo Alto, CA)2006 views18" by 24" print, 2011

This image is a snap-shot from a real-time interactive particle simulation using Lorentz's Law to define each particle's movements. The color palette, perspective, magnetic field placement, and rendering style were designed by the artist. Physics and mathematics define the piece's motion and overall pattern formation. --- Reza Ali (Palo Alto, CA, http://www.syedrezaali.com/)

"Still Life with Magic Square," by Sylvie Donmoyer (Saumur, France)1414 views20" x 26", Oil paint on canvas, 2011
First Place Award, 2012 Mathematical Art Exhibition

It all arose from a sense of wonder when seeing the formal beauty of mysterious objects called polyhedra. Since then, I have joyfully played with geometric shapes and it led me to explore the possible representation of Geometry in classical painting. From Durer's magic square to strange cubes, painted by the precise brush of a would-be 17th century Dutch artist. --- Sylvie Donmoyer (Saumur, France, http://www.illustration-scientifique.fr/index-A.html)

"Möbius Hanging Gardens," by Tatiana Bonch-Osmolovskaya (Sydney, Australia)1377 views1276 x 1800 pixels, computer graphics, 2011

I use 2D and 3D computer graphics as well as photographs made by myself or my friends, to show the beauty of these objects, thus uniting the intellectual wonder of perceiving a mathematical concept with the aesthetical pleasure of viewing a beautiful image. Hanging gardens of Babylon were built in the desert as a wonder of land amelioration and engineering. In our era humanity continues to perform such wonders, e.g. in desert Australia. While the flowers on my picture, which have grown on the Möbius strip over the Australian plain, were placed there by computer graphics, it is the hard work of those who make the Red Continent green that invokes our admiration. Photographs of Australian views and the Florida festival in Canberra were used in this image. --- Tatiana Bonch-Osmolovskaya (Sydney, Australia, http://antipodes.org.au)

"Hyperbolic Tiling I," by Vladimir Bulatov (Corvallis, OR)1358 views20" x 20", Digital print, 2011

This is a tiling at the infinity of hyperbolic space. The tiling is generated by reflections in 4 planes. The planes arrangement is obtained from faces of hyperbolic tetrahedron by truncating one vertex and one of opposite edges and moving points of truncation to infinity. The interplay of reflections forms circular area with infinitely many circular holes filled with two dimensional hyperbolic triangle tilings (2 3 24). To color the tiling we use different subgroups of the total symmetry group. --- Vladimir Bulatov (Corvallis, OR, http://bulatov.org)

"K_7 embedded on a torus," by sarah-marie belcastro (Hadley, MA)1347 views11" x 11" x 4.5", Knitted cotton (Reynolds Saucy), 2010

I am a mathematician who knits as well as a knitter who does mathematics. It has always seemed natural to me to combine mathematics and knitting, and it is inevitable that sometimes the results will be artistic rather than functional. The Heawood bound shows that K_7 is the largest complete graph that can embed on the torus. This is an embedding of K_7 on the torus with all vertices centered on the largest longitude. It is the second knitted instantiation of this embedding; this version is larger and has a larger face-to-edge proportion than the first, which was exhibited at Gathering for Gardner 7 in 2006. --- sarah-marie belcastro (Hadley, MA, http://www.toroidalsnark.net)

"Hyperbolic Coasters," by Mikael Vejdemo-Johansson (University of St. Andrews, Scotland)1347 viewsLaser-etched glass, 14 items, 12cm diameter each, 2011

The advent of accessible automated tools opens up a number of new approaches to art: especially algorithmic and mathematical art works. The computational control allows us to write algorithms to generate concrete physical art; and their precision allows a higher resolution than what the eye can discern. These pieces highlighting and reifying different mathematical concepts, giving them physical presence and accessibility and turning abstract geometry into hands-on displays and objects. Among the most successful of the reified mathematics art-pieces I produced where these--hyperbolic disk tilings with the Poincare disk model were etched onto glass disks, producing a collection of reified hyperbolic geometries and symmetries. --- Mikael Vejdemo-Johansson (University of St. Andrews, Scotland, http://mikael.johanssons.org)

"Great Ball of Fire," by Eve Torrence (Randolph-Macon College, Ashland, VA)1330 viewsCraft Foam, 2010

I love the symmetric beauty of polyhedra and enjoy creating models to study. Through the process of building a model I am able to truly understand its form. I like to use color to help reveal the structure and patterns of an object. This sculpture is based on the third stellation of the dodecahedron. A stellation of a regular polyhedron is formed by extending the faces until they intersect and enclose a region of space. The faces of the dodecahedron will intersect three times as they are extended, forming the small stellated dodecahedron, the great dodecahdron, and the great stellated dodecahedron. Twelve identical pieces of craft foam were slotted at the edge of each stellation and then tightly woven. This open skeleton allows one to follow each face to view the intersections and the outline of the dodecahedron and the three stellations. Six colors of foam are used and parallel faces are the same color. Each of the five arms of each face intersects three others to form 20 colorful "flames" in an icosahedral arrangement. --- Eve Torrence (Randolph-Macon College, Ashland, VA)

"Lisbon Oriente Station," by Bruce Torrence (Randolph-Macon College, Ashland, VA)1224 viewsPanoramic Photograph, 2011

I've been exploring recent developments in digital imagery which allow me to utilize mathematics and computer programming to solve visual problems. This is a projection made from a panorama of 13 photographs. The individual photos were shot from precisely the same point in space, and when stitched together they comprise the entire "viewable sphere" centered at that vantage point. That is, the panorama has complete coverage of the scene---360 degrees around, and 180 degrees from top to bottom. Stereographic projection was then applied to the spherical panorama, with the projection taken from the North Pole so that the point directly overhead becomes the point at infinity. This produces a lovely "little planet" effect, with the geometry of the roof structure framing the scene. The panorama was shot at Oriente Station in Lisbon, Portugal. --- Bruce Torrence (Randolph-Macon College, Ashland, VA, http://www.flickr.com/photos/thebrucemon/, http://faculty.rmc.edu/btorrenc/)

"Equivalent," by Robert Bosch and Derek Bosch (Oberlin College, OH)1201 views6" x 6" x 2", Nylon (selective laser sintering), 2011

The mathematician in me is fascinated with the various roles that constraints play in optimization problems: sometimes they make problems much harder to solve; other times, much easier. Equivalent is a three-piece, 3D-printed sculpture that consists of three topologically equivalent variations of the Borromean rings. In the Borromean rings, no two of the three rings are linked, so if any one of them is destroyed, the remaining two rings will come apart. For the photograph, we positioned Equivalent on a piece of Lenox china (a wedding gift). --- Robert Bosch (Oberlin College, OH, http://www.dominoartwork.com)

"Butterflies 6-4," by Doug Dunham (University of Minnesota Duluth, MN)1172 views11" x 11", Color printer, 2009

This is a hyperbolic pattern of butterflies, six of which meet at left front wing tips and four of which meet at their right rear wings. The pattern is inspired by M.C. Escher's Euclidean image Regular Division Drawing Number 70, and is colored similarly. Disregarding color, the symmetry group of this pattern is generated by 6-fold and 4-fold rotations about the respective meeting points of the wings, and is 642 in orbifold notation (or [4,6]+ in Coxeter notation). This pattern exhibits perfect color symmetry and its color group is S3, the symmetric group on three objects. --- Doug Dunham (University of Minnesota Duluth, MN, http://www.d.umn.edu/~ddunham/)

"Beaded Fullerene of Schwarz's D Surface," by Chern Chuang (MIT, Cambridge, MA), Bih-Yaw Jin (National Taiwan University), Wei-Chi Wei (The Beaded Molecules)1143 views23cm x 21cm x 18cm, Faceted plastic beads and fish thread, 2008

Geometry is an essential ingredient of chemistry. The functionality of molecules depends heavily on their geometries. Here is the conjugate surface of the P surface. We chose to construct this surface in a tetrahedral form to avoid unconnected component. In contrast to the P surface, one can find this surface comprising helicoid units of two opposite chiralities, lining up along C2 axes. Octagonal rings are represented by green beads. --- Chern Chuang, (MIT, Cambridge, MA)

"Creamy Blocks," by Anne Burns (Long Island University, Brookville, NY)1117 views12 " X 16", Digital print, 2011

I began life as an art major. Much later I became interested in mathematics. When I bought my first computer I found that I could combine my love of art with my love of mathematics. The possibilities are endless. Here, attached to each point in a sequence of points along the lines y = ±x is a vector whose length and direction are determined by a complex function h(x+iy). The color and transparency of the vector are functions of arctan(Im(h)/Re(h)). --- Anne Burns (Long Island University, Brookville, NY http://www.anneburns.net/)

"A5, Variation I," by Conan Chadbourne (San Antonio, TX)1114 views24" x 24", Archival Inkjet Print, 2011

This work is an exploration of the structure of the alternating group on five elements, and its particular presentation by two generators of orders 2 and 5. A stylized Cayley graph of this presentation of the group is shown over its dual graph. The regions in the dual image are colored according to the order of the element in the group. The image is constructed from multiple hand-drawn elements and natural textures which are scanned and digitally manipulated to form a composite image and subsequently output as an archival digital print. --- Conan Chadbourne (San Antonio, TX, http://www.conanchadbourne.com)

"Sierpinski Cliffs," by Francesco De Comité (University of Sciences and Technology, Lille, France)1107 views50cm x 50cm, Digital print, 2011

Seeking ways to illustrate mathematical concepts and constructions is an endless game. Jumping from one idea to another, mixing techniques and computer code, and then waiting for the image to appear on my screen, leads often to surprising results. Playing around with Apollonian gaskets, recursivity and circle inversion can give rise to landscapes no one has seen before. --- Francesco De Comité (University of Sciences and Technology, Lille, France, http://www.lifl.fr/~decomite)

"002 - Julia weaves," by Jean Constant (Santa Fe, NM)1076 views20" x 20", Mixed media on canvas, 2011

This is a combination of Julia set fractal and droste effect. Julia Sets are one of the most famous types of fractals formed using formula iteration. The Droste effect depicts a smaller version of itself in a place where a similar picture would realistically be expected to appear. Combining the two effects brings visually significant occurrences explored sometimes more intuitively in medieval architecture, stained glass windows and weaving work . --- Jean Constant (Santa Fe, NM, http://hermay.org/)