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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"Magneto-2," by Reza Ali (Palo Alto, CA)18" 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/)

"K_7 embedded on a torus," by sarah-marie belcastro (Hadley, MA)11" 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)

"Möbius Hanging Gardens," by Tatiana Bonch-Osmolovskaya (Sydney, Australia)1276 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)

"Equivalent," by Robert Bosch and Derek Bosch (Oberlin College, OH)6" 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)

"Hyperbolic Tiling I," by Vladimir Bulatov (Corvallis, OR)20" 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)

"Creamy Blocks," by Anne Burns (Long Island University, Brookville, NY)12 " 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)24" 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)

"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)23cm 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)

"002 - Julia weaves," by Jean Constant (Santa Fe, NM)20" 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/)

"Conical panoramic view of the George Eastman House grounds," by Andrew Davidhazy (Rochester Institute of Technology, NY)Photograph, circa 1990

My area of interest is the application of mathematical concepts in technical applications of photography. Be it quantification of phenomena or the design and use of photography to visualize physical and mathematical concepts. A camera that rotated a circular piece of film past a radial slot acting as a shutter exposed the film for more than two rotations of the camera and thus recorded two plus views of the House grounds each covering a sector of about 120 degrees or so designed so that the 360 degree view of the grounds would produce a sector that could be cut and formed into a conical lampshade. Sometimes this photo is confused with those that a fisheye lens might make but the fisheye lens could only make a single image of the House per frame. Here there are two. --- Andrew Davidhazy (Rochester Institute of Technology, NY, http://people.rit.edu/andpph/)

"Sierpinski Cliffs," by Francesco De Comité (University of Sciences and Technology, Lille, France)50cm 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)

"Science/Art," by Erik Demaine (MIT, Cambridge, MA) and Martin Demaine (MIT, Cambridge, MA)22" (tall) x 28" (wide), framed poster, elephant hide paper, 2011

The crease pattern (top) folds into both SCIENCE and ART (bottom, not to scale). More precisely, the rectangular paper sheet folds into the 3D structure of the word SCIENCE, while the gray inking in the sheet (top) forms the inked ART in the background (bottom). The message is that science and art can exist on a common plane, as two different perspectives of the same object. The crease pattern was designed using an algorithm by Demaine, Demaine, and Ku (2010), which describes how to efficiently fold any orthogonal "maze" (including word outlines like SCIENCE) from a rectangle of paper. Red lines fold one way and blue lines fold the other way. --- Erik Demaine (Massachusetts Institute of Technology, Cambridge, MA, http://erikdemaine.org/art/scienceart/)

"Still Life with Magic Square," by Sylvie Donmoyer (Saumur, France)20" 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)

"Butterflies 6-4," by Doug Dunham (University of Minnesota Duluth, MN)11" 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/)

"Nueve y 220-B," by Juan G. Escudero (Universidad de Oviedo, Spain)50cm x 26 cm, Digital Print, 2011

A possible way to remove the gap between the worlds of sciences and humanities, is the search for interconnections between mathematics and physics with the sound and visual arts. This work is based on a family of algebraic surfaces with many nodal singularities. They have been introduced recently, by means of a kind of duality in the basic geometric constructions corresponding to the generation of substitution tilings ("A construction of algebraic surfaces with many real nodes". http://arxiv.org/abs/1107.3401). Here the surface is a nonic with 220 real nodes. In general, the surfaces have degrees divisible by three and cyclic symmetry. They appear as mirror pairs not necessarily topologically inequivalent (see the sextic with 59 real nodes in arXiv:1107.3401). --- Juan G. Escudero (Universidad de Oviedo, Spain)