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.

"Four Right Angles: Ascent (left) and Cantilever (right)," by Nat Friedman (Albany, NY)17" L x 11" H x 7 D", Steel, 2010

A sculpture is defined as a form in a position relative to a fixed horizontal plane (base, ground). To hypersee an outdoor sculpture, one walks around it to see overall views and close up detail views from different viewpoints as well as in different light conditions at different times. If two sculptures consist of the same form in different positions, then the sculptures are said to be congruent. Congruent sculptures can look so completely different that one does not realize the sculptures are congruent. A hypersculpture is a group of congruent sculptures, and a more complete presentation of the sculptural possibilities of a form. In order to hypersee a form, one presents it as a hypersculture. This hypersculpture Four Right Angles consists of two vertical and three horizontal congruent sculptures and is discussed in an article of the same title in the Spring, 2011 issue of Hyperseeing, www.isama.org/hyperseeing/ . The two vertical sculptures Ascent and Cantilever are shown here. The form consists of four identical angle iron sections welded together. Each section is 5" x 5" x 6 ½" and ½" thick. --- Nat Friedman (Professor Emeritus, University at Albany, NY, Founder and Director of ISAMA, http://www.isama.org)May 14, 2012

"Beaded Star Weaves: Five Bracelets," by Gwen Fisher (beAd Infinitum, Sunnyvale, CA)Sizes vary from 1.5" to 2.5" wide by 5.5" to 8" long, Seed bead weaving, 2011

I weave beads to appeal to people's affinity for organization in design. I use mathematics, including geometry, symmetry, and topology, as an inspiration for the structure of my creations. In this series, I explore how tilings of the plane can be interpreted as beaded angle weaves. Tilings of the plane, especially periodic tilings, can be used as the basis for flat bead weaving patterns called angle weaves. The “star tilings” used to design these five bracelets are generated from the three regular tilings of the plane and two other Laves tilings. I converted each star tiling into a star weave by placing beads on the vertices and edges of the tiling and weaving them together with a needle and thread. Because all of the vertices in a regular tiling are similar, all of the stars are similar in the three regular star weaves (i.e., Kepler’s Star, Archimedes’ Star, and David’s Star). The other two star weaves (i.e., Night Sky and Snow Star) include stars of two types, reflecting the two types of vertices in their respective Laves tilings. --- Gwen Fisher (beAd Infinitum, Sunnyvale, CA, http://www.beadinfinitum.com)May 14, 2012

"Hybrid 101," by Michael Field (University of Houston, TX)24" x 24" (framed), Archival inkjet print, 2011

Hybrid 101 is a representation of an invariant measure for a dynamical system on a 2-torus with deterministic and random components. Deterministic dynamics is given by the product of two identical circle maps with topological degree 2 ('doubling maps') together with a random component which is a place dependent iterated function system: probabilities and direction and size of jumps depend on the position on the torus. Hybrid dynamics combining deterministic dynamics with an iterated function system was first studied mathematically by Kobre and Young in the context of extended dynamical systems on the line. In Hybrid 101, dynamics is defined by doubly 1-periodic maps on the plane and we reduce mod the integer lattice to obtain dynamics on a torus. We lift the measure back to the plane to obtain a repeating pattern. Appearances can be deceptive: the only symmetries of the repeating pattern are translations (the pattern is of type p1) and all the lines are straight. --- Michael Field (University of Houston, TX, http://www.math.uh.edu/~mike)May 14, 2012

"Fractal Tessellation of Spirals," by Robert Fathauer (Tesselations, Phoenix, AZ)16" x 16", Archival inkjet print, 2011

This artwork is based on a fractal tessellation of kite-shaped tiles I discovered several years ago. Grouping of the kite-shaped tiles into spirals allowed a fractal tessellation to be created in which two colors were sufficient to ensure that no two adjacent tiles have the same color. All of the spirals in the print have the same shape (more precisely, they are all similar in the Euclidean plane). --- Robert Fathauer (Tessellations, Phoenix, AZ, http://www.robertfathauer.com)May 14, 2012

"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)May 14, 2012

"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/)May 14, 2012

"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)May 14, 2012

"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/)May 14, 2012

"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)May 14, 2012

"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/)May 14, 2012

"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/)May 14, 2012

"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) May 14, 2012

"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)May 14, 2012

"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/)May 14, 2012

"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)May 14, 2012