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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"Self-similar Knot No. 1," by Robert Fathauer (Tessellations, Phoenix, AZ)Digital print, 13" x 16", 2009. A starting knot was created that possessed sufficient geometric regularity to allow iterative replacement of a portion of the knot with a scaled down copy of the knot. Three such iterations were carried out to obtain the knot shown here. In addition, the path of the strands was smoothed out so that strand in the final knot curves gracefully, as opposed to being a series of straight line segments that change angle abruptly. The knot was constructed using the program KnotPlot and then exported to PhotoShop for touching up. --- Robert Fathauer (Tessellations, Phoenix, AZ) http://members.cox.net/tessellations/index.html

Robert Fathauer makes limited-edition prints inspired by tiling, fractals, and knots. He employs mathematics in his art to express his fascination with certain aspects of our world, such as symmetry, complexity, chaos, and infinity. His artworks are created on a Macintosh computer.

"Coiled Figure Eight," by Nat Friedman (University at Albany, NY)3/8“ copper tubing, wood dowel, gorilla glue, 13" x 10" x 9", 2008. Copper tubing comes in a coil so is naturally “rounded”. A knot diagram is coiled if the diagram can be traced in a constant clockwise direction. A coiled diagram is also referred to as being in a braid configuration. The typical diagram of a figure eight knot in a knot table is not coiled. The diagram of the figure eight knot shown above is coiled. A coiled diagram is suitable for forming a corresponding copper tubing knot since the natural rounded curvature of the tubing can be maintained as the knot is formed. I form copper tubing knots by hand. One must be careful not to "crink" the tubing. The ends are joined using a wooden dowel and gorilla glue. Knots are ideal mathematical forms for sculptures since a knot is completely three-dimensional with no preferred top, bottom, front, or back and can look completely different from different viewpoints. --- Nat Friedman (University at Albany, NY) www.isama.org

"Fire, Water, Soil and Air," by Mehrdad Garousi (Artist and photographer, Hamadan, Iran)Digital art print, 21" x 24", 2009. This complex 3D object consists of four identical but differently colored separated tape-like shapes that have been interwoven with each other. The result of this arrangement is a six-pointed shape with some kind of unusual symmetry. Each of four separated constitutive parts and also the shapes at all six arms of the final complete form are exactly the same, but their different arrangement is the reason for the final unusual symmetry. It’s a long time that I have worked with traditional hands-on art, and I feel much interest in the probabilities and capacities of the new generation of computer aided or generated arts. However, the main field in which I am being drowned is fractal art. Recently, I have been attracted strongly by mathematical sculpting, especially by means of wonderful software such as Topmod. These kinds of software considerably ease the imagination of complex mathematical shapes, and playing with them can be done without any limitation other than the mind. Initially, I create my basic shapes in software like Topmod and do arrangements, adjustments, texturings, and final renders as final realistic sculptures in render engines like Modo. I hope to have the luck to make some of them in the real world in large scale." --- Mehrdad Garousi (Artist and photographer, Hamadan, Iran) http://mehrdadart.deviantart.com

"Cuboctahedral Symmetries to Travel," by S. Louise Gould (Connecticut State University, New Britain)Original digitized machine stitched patterns on cotton reinforced by Timtex, Five moveable pieces, collapsible each 3” × 3” ×3”, 2009. Conway enumerates the 7 spherical symmetries compatible with the uniform polyhedra in “The Symmetries of Things.” Using the symmetry types these are 332, *332, 432, 3*2, *432, 532 and *532. The simple cuboctahedron exhibits the first 5 of the symmetry patterns: *432 has 48 symmetries (the full group of symmetries), *332, 432 and 3*2 have 24 (the three subgroups of index 2=48/24) while 332 has only 12 (the ones of index 4=48/12). Coloring the faces of the models for the Archimedean solids is a natural extension of my recent work with pop-up polyhedra. "My mathematical art grows out of my experiences with my students and my explorations of mathematics, textiles, paper, and technology. I enjoy working with computer controlled machines such as the computerized embroidery sewing machine and the Craft Robo (plotter cutter) as well as traditional looms and knitting machines." --- S. Louise Gould (Connecticut State University, New Britain)

"It's Not That Simple, I," by Gary R. Greenfield (University of Richmond, VA)Digital Print, 10" x 10", unframed, 2009. This op art sequence shows visualizations from a simulation based on cellular morphogenesis. Concentrations of three of four cell transcription factors are interpreted as RGB values. These cell substances diffuse, interact, and express and inhibit the genes within cells that are responsible for producing such substances. Here there are two types of cells and the image was captured after the gene activation equation had been evaluated 400 times. "Many of my computer generated algorithmic art works are based on simulations that are inspired by mathematical models of physical and biological processes. In exploring the space of parameters that govern the simulation, I try to focus the viewer's attention on the complexity underlying such processes." --- Gary R. Greenfield (University of Richmond, VA)

"Cubic Iteration," by Bradford Hansen-Smith (Geometer, artist, author, Chicago, IL) Eight 9" folded paper plates, 6"x6"x6", 2009. Eight 9" circles are folded to a 32-frequency diameter grid (102 creases in an equilateral triangular grid matrix.) Each circle is reconfigured to a bi-tetrahedron pattern with some variations in iterations to the folding algorithm, and joined in multiples forming eight corners of the cube. "For the last twenty years I have been exploring folding the circle for information to understand what is revealed through a simple and principled process of systematic folding, reconfiguring and multiple joining circles. The technique is simply touching points and creasing; to attach using tape, glue, and bobby pins. There is no cutting or measuring. Knowing math is not necessary to fold circles; math functions are just some of the information generated in the process. The circle functions simultaneously as both Whole and part, which is not discernable by looking at the image of a circle. My work is not about what I can do with the circle as much as to play with and discover the nature of what it is and what it will generate. Art and math are only two areas of demonstration about the implicate order inherent in the circle." --- Bradford Hansen-Smith (Geometer, artist, author, Chicago, IL) www.wholemovement.com

"Dragonflies," by George W. Hart (Stony Brook University, Stony Brook, NY)Wood, 8”x8”x8”, 2008. The form of Dragonflies consists of twelve congruent parts arranged with octahedral symmetry in a form based on the third stellation of the rhombic dodecahedron. The laser-cut wood components are each a subset of the complete face of the stellation, designed so that it does not intersect with the other eleven identical copies of itself. Interweaving and assembling the rigid physical parts was an interesting challenge. "As a sculptor of constructive geometric forms, my work deals with patterns and relationships derived from classical ideals of balance and symmetry. Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic." --- George W. Hart (Stony Brook University, Stony Brook, NY) http://www.georgehart.com

"Hyperbolic Cube," by Thomas C. Hull (Western New England College, Springfield, MA)Single sheet of Canford paper, wet-folded, 9" x 9" x 9", 2006. A Hamilton cycle on the cube has eight edges. Therefore, a regular octagon could be folded to mimic the path such a cycle traces on the cube. This piece represents a solution using folded concentric octagons, producing the illusion (?) of negative curvature. The piece was folded from a large regular octagon, approximately two feet in diameter. Concentric octagons were pre-creased, alternating mountain and valley folds. Then the model was collapsed and wet-folded to hold the cube Hamilton cycle shape. "I've been practicing origami almost as long as I've been doing math. Part of the charm of paper folding is its capacity for simple, elegant beauty as well as stunning complexity, all within the same set of constraints. This mirrors the appeal of mathematics quite well. Geometric origami, which is where most of my artwork lives, strives to express in physical form the inherent beauty of mathematical concepts in geometry, algebra, and combinatorics. The constraints that origami provides (only folding, no cutting, and either one sheet of paper or further constraints if more than one sheet is allowed) challenges the artist in a way similar to being challenged by a mathematical problem." --- Thomas C. Hull (Western New England College, Springfield, MA) http://mars.wnec.edu/~thull

"El Nido Fractal," by Karl Kattchee (University of Wisconsin-La Crosse)Digital Print, 10" x 16", 2009. The boundary between land and sea is complex, like a fractal. At the bottom of this drawing we have land, represented by rigid lines and shapes. At the top, there is the sea, swirling around. In between is the boundary, where the right angles gradually give way to curves. There is self-similarity, as one would expect in a fractal. "What is mathematical art? This question not only begs for criteria to make the judgment, but it also asks how math and art interact. That strange interaction is what makes math art fun for me. I almost always start with sketches on paper, but I recently began transferring them to the computer and carrying on the work electronically. As such, I can spend time experimenting with different ideas and change my mind often about what I'm doing. While I try to render mathematical ideas in my art, I also realize that the artistic process is itself a lot like the mathematical process. Sometimes the original 'problem' needs to be modified after careful 'research'. To me, the final product is a lot like a theorem." --- Karl Kattchee (University of Wisconsin-La Crosse) http://www.uwlax.edu/faculty/kattchee/

"Composition No 7," by Nuria Juncosa (Artist, Amsterdam, The Netherlands)Oil on linen, 23.6 x 23.6 inches, 2009. This painting is a partial projection of a wireframe deltoidal icositetrahedron. A deltoidal icositetrahedron dual polyhedron which looks a bit like an overinflated cube which has all of its edges bisected. The 24 faces are deltoids. The short and long edges of each kite are in the ratio 1.00:1.29. The deltoidal icositetrahedron is a crystal structure formed by the minerals analcime and garnet. After painting the projection of the Deltoidal Icositetrahedron wireframe on the canvas, I took the artistic freedom of creating new subfaces, occupying complementary locations by colouring up the spaces obtained by the intersections of the wireframe. I like to say that as an artist, I discover mathematics through art. --- Nuria Juncosa (Artist, Amsterdam, The Netherlands) www.la-nuria.com

"Jellyfish," by Kendra Lockman (Photographer, artist, and teacher, Oakland, CA)Digital print, 20" x 24" , 2009. Two fractals are combined to mimic the shape of the jellyfish used to create this image. The "head" fractal uses the famous dragon curve iteration. Here, the first iteration maps the negative-sloped diagonal of the starting photograph to the lower edge, and also maps the same diagonal to the left edge. The "tail" fractal uses a double spiral iteration. The original photograph was taken at Monterey Bay Aquarium by the artist. "I began iterating photographs into fractals after watching a video on fractals, in which the point was made that whether you started with a single segment or a 2-dimensional photograph, the resulting fractal was the same. I explored this on my own and learned that it can be more visually interesting to expose each step of the iteration. Photographs interact with themselves at each iteration level to reveal new shape and structure. Fractals are appealing for their seemingly complex structures which bloom from often simple iteration rules. I find that using photographs in the iterations can make the fractals much more captivating than if they were created with abstract geometry. I work intensely between Photoshop and The Geometer's Sketchpad to create these images." --- Kendra Lockman (Photographer, artist, and teacher, Oakland, CA) kendralockmanphoto.com

"Gong Gan," by Ghee Beom Kim (Artist, Sydney, Australia)Digital Print, 12 x12 inches, 2009. “Gong Gan” employs a dodecahedron as its base form. I saw polyhedra as a form of tessellation on a sphere (spherical tessellation) and just by replacing each pentagon face on the dodecahedron with a module that tessellates within the pentagon and with the adjacent ones as well I was able to create this pleasant looking sculpture. "As for me my art is a channel through which I communicate with the higher entity. It’s a form of profound prayer on my part. Geometry has, so far, given me the best means to explore universe and reveal His secrets. By exploring into geometry from purely aesthetic perspective I can see clear relationship between the pure beauty of mathematics and God. My art is devoid of any human feelings and the focus is solely upon revealing unknown mathematical and geometrical order leading to visual essence. I create most of my artwork using Autocad and touch up in Photoshop afterward." --- Ghee Beom Kim (Artist, Sydney, Australia) http://geometricarts.googlepages.com/home

"Nonsimple," by Goran Konjevod (Arizona State University, Tempe, and Livermore, CA)One uncut square of paper, 8" by 8" by 8", 2009. This piece uses intersecting pleats to create tension within the folded sheet and encourage three-dimensionality. In addition to sharp points created by stretching pleats close to the four corners of the original sheet, it also features a joint where the centers of two opposite sides of the square are held together by a folded lock mechanism, creating the appearance of a non-simply connected surface. "I fold (mostly flat and mostly paper) surfaces into interesting shapes. To do this, I use sequences of pleats to arrange layers so that they create tension that forces the material towards a curved surface. The simplest of these pieces are more appropriately described as discovered than created, but in others I build on the basic equilibrium shape to bend and curve the pleated surface further. The mathematics show up in many ways, but the two of my favorite are the combinatorics in the arrangement of pleats and the mathematical physics in understanding the forms preferred by the paper when folded." --- Goran Konjevod (Arizona State University, Tempe, and Livermore, CA) http://organicorigami.com

"Perspicuous," by Matjuska Teja Krasek (Artist, Ljubljana, Slovenia)Digital print, 1995/2008. "Perspicuous" is a geometrical composition where a basic decagonal shape exhibits fivefold symmetry. The straight lines inside reveal the richness and interconnectedness of the geometrical shapes (pentagons, triangles, Penrose rhombs and kites), we can observe golden mean relations. Krasek’s theoretical, as well as practical, work is especially focused on symmetry as a linking concept between art and science, on filling a plane with geometrical shapes, especially those constituting Penrose tilings (rhombs, kites, and darts). The author's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. The artworks among others illustrate certain properties, such as golden mean relations, selfsimilarity, fivefold symmetry, Fibonacci sequence, inward infinity, and perceptual ambiguity… Krasek’s work concentrates on melding art, science, mathematics and technology. She employs contemporary computer technology as well as classical painting techniques. Her artworks and articles are exhibited and published internationally. Krasek’s artworks are among the winners of the 2nd and the 3rd International NanoArt competition. --- Matjuska Teja Krasek (Artist, Ljubljana, Slovenia) http://www.tejakrasek.com

"Ribbon weaving no. 1," by Christine Liu, Ergun Akleman, Qing Xing (Texas A&M University, College Station, TX)Digital print, 13” x 9”, 2009. This plain-weaving model is automatically generated from the algorithm in the paper Cyclic plain-weaving on polygonal mesh surfaces with graph rotation systems. Cycles of the plain-woven object are created from any manifold-mesh surface by twisting every edge of the manifold mesh and convert the plain-weaving cycles to 3D thread structures. Users interactively control the shape of the threads and the size of gaps with a set of parameters creating a wide variety of unique plain- weaving patterns. This generated weaving model has 16 identical closed cycles with user control over the width, displacement, and curvature of the weaving yarns. "Inspired by the detail of that which is supple, Christine Liu continually updates her mathematical techniques with a self-driven craft refinement in classical training. Paint, sculpture and architectural drafting have all formulated a longstanding basis for an appropriate representation in her personal form of rational digital expression. Here, a timeless balance of scale, form and simulated vernacular methods are evident in the otherwise conventional mathematical art expression. Inspiration for the selected style and palette come from classic vibrant colored ribbon fabric and various types of ribbon weaving crafts. This piece is rendered and produced through Maya and Photoshop." --- Christine Liu, Ergun Akleman, Qing Xing (Texas A&M University, College Station, TX)