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.

 "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 May 10, 2010
 "Elements," by Jeff Chyatte (Montgomery College and River Bend Studio at Water’s Edge, Washington DC)Painted High Carbon Steel, Impala Black Granite, Height 18” , Width 16” , Depth 16”, 2009. Fusing math, art and aesthetics, Elements incorporates mathematically significant dimensions that add an intriguing subtlety to its construction. Euclid studied the Golden Ratio 1 to 1.618 (Greek letter Phi) for its many interesting properties as described in his manuscript Elements. Those proportions were used by great artists and architects throughout the Renaissance in the form of the Golden Rectangle. The three intersecting planes that comprise Element’s core are Golden Rectangles. Their intersection creates 20 equilateral triangles, drawn from their points - an Icosahedron. Further, these rectangles use dimensions from the Fibonacci Sequence providing for a variety of mathematical implications. --- Jeff Chyatte (Montgomery College and River Bend Studio at Water’s Edge, Washington DC) May 10, 2010
 "Natural Cycles," by Erik Demaine (Massachusetts Institute of Technology) and Martin Demaine (Massachusetts Institute of Technology, Cambridge, MA)Elephant hide paper, 9"x9"x9", 2009. The sculpture is a modular combination of three interacting pieces. Each piece is folded by hand from a circle of paper, using a compass to score the creases and cut out a central hole.This transformation of flat paper into swirling surfaces creates sculpture that feels alive. Paper folds itself into a natural equilibrium form depending on its creases. These equilibria are poorly understood, especially for curved creases. We are exploring what shapes are possible in this genre of self-folding origami, with applications to deployable structures, manufacturing, and self-assembly. "We explore many mediums, from sculpture to performance art, video, and magic. In our artwork we look for epiphanies, challenges, and often connections and understanding to help solve problems in mathematics." --- Erik Demaine (Massachusetts Institute of Technology) http://erikdemaine.org/curved/NaturalCycles/. May 10, 2010
 "Arrangement (2)," by Adrian Dumitrescu (University of Wisconsin-Milwaukee)Digital print, 5" x 7", 2009. This arrangement was inspired by the following result from the theory of binary space partitions in computational geometry: There exists a set S of n disjoint axis-parallel line segments with the property that any axis-parallel binary space partition of S has size at least 2n-O(n^2/3). The construction is based on a shifted double grid made of disjoint segments. "Art could come from anywhere. One just wants to be careful and not overlook it." --- Adrian Dumitrescu (University of Wisconsin-Milwaukee) http://www.cs.uwm.edu/faculty/ad/ May 10, 2010
 "Three Elements 4-5-3," by Doug Dunham (University of Minnesota Duluth)Color print,11” by 11”, 2007. This pattern contains lizards, fish, and bats representing the three classical elements, earth, water, and air. The pattern is inspired by M.C. Escher's Notebook Drawing Number 85. In this hyperbolic pattern, four blue lizards meet head-to-head, five red fish meet head-to-head, and three yellow bats meet head-to-head, unlike Escher's pattern in which three of each animal meet head-to-head. The symmetry group of this pattern is generated by reflections across the lines of bilateral symmetry of each of the animals; its symmetry group is the hyperbolic kaleidoscope group *543, in orbifold notation. "The goal of my art is to create repeating patterns in the hyperbolic plane. These patterns are drawn in the Poincare circle model of hyperbolic geometry, which has two useful properties: (1) it shows the entire hyperbolic plane in a finite area, and (2) it is conformal, i.e. angles have their Euclidean measure, so that copies of a motif retain their same approximate shape as they get smaller toward the bounding circle. Most of the patterns I create exhibit characteristics of Escher's patterns: they tile the plane without gaps or overlaps, and if colored, they are colored symmetrically and adhere to the map-coloring principle that adjacent copies of the motif are different colors. My patterns are rendered by a color printer. Two challenges are to design appealing motifs and to write programs that facilitate such design and replicate the complete pattern." --- Doug Dunham (University of Minnesota Duluth) http://www.d.umn.edu/~ddunham/. May 10, 2010
 "Julia," by Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR)Photographic Paper, 20” X 20” , 2009. Julia sets are usually depicted two-dimensionally, either flat or as textures on other surfaces which themselves may have little to do with the Julia set. Here, we iterate the complex variable relation, new s = s^2 - 1.25 thirteen times to produce a polynomial in the original variable, s, of degree 8192. Now consider the three-dimensional surface, z = f(x,y) = |s^8192+ ... | where s = x+iy and | | denotes absolute value. This picture is the graph of (x,y, z) if z <= t and (x,y, t(t/z)^p) if z > t where t is a threshold value ~1.464 and p = (1/2)^13 "I am interested in applying computer graphical techniques to illuminate mathematical processes. Ideally, this can lead to a deeper understanding of the process, but even if no new insight is forthcoming, I am frequently mesmerized by the compelling beauty of the unusual shapes. I do not use 'canned' software. I wrote the code to first principles in the 'C' programming language. This particular image was constructed as a particle system made from 266 billion points and took 67 hours to compute." --- Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR) May 10, 2010
 "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. May 10, 2010
 "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 May 10, 2010
 "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 May 10, 2010
 "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) May 10, 2010
 "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) May 10, 2010
 "Rosetta," by Edward Alonzo (Artist, University of Vermont)Acrylic on Wood, 5“ x 14.5”, 2009. Two steganographic codes, one ultilising a sculptural and one a painterly ciphertext, create a three way harmony with the encrypted data. Expressing code not solely as something visual, but also something tactile. My current avenue of investigation is Steganography and the place of Cryptography in our society. Encryption has become incredibly powerful and equally incredibly common place. The hidden nature of steganography is because either the cryptographer decides to do it, or in the more common case of "https" because the user is ignorant of its existence. The ignorance in the second case is due to the overwhelming complexity of computers and computations done by them. Which is akin to the overwhelming complexity of art and decisions made by artists. Both Computers and Art are incredibly common in our culture and yet both are incredibly overwhelming to many of the people who see them daily. Thus, stenographic painting seems the aesthetic equivalent to 'https'. To that extent, the focus has been on devising encoding systems that utilize color and orientation, and then finessing them to make them sing together. --- Edward Alonzo (Artist, University of Vermont) http://www.sirhair.com/ May 10, 2010
 "A mirror pair of (3,2) torus knots embedded on tori," by sarah-marie belcastro (freelance mathematician, Hadley, MA)Knitted bamboo yarn (Southwest Trading Company Twize, in colors twurple and twocean (seriously)), 6.5" x 14.5" x 3", 2009. A (p,q) torus knot traverses the meridian cycle of a torus p times and the longitudinal cycle q times. Exhibited here are the two chiral versions of the (3,2) torus knot, knitted into their embedding tori. One can represent a (p,q) torus knot on the standard flat torus by drawing a line of slope q/p. The designer of a knit torus must contend with thickening the line to make it visible (and appear continuous), compensating for the curvature of the spatially embedded torus, and discretizing the result onto the non-square grid formed by knit stitches. --- sarah-marie belcastro (freelance mathematician, Hadley, MA) http://www.toroidalsnark.net May 10, 2010
 "The Vase," by Harry Benke (www.harrybenke.com)2010 Mathematical Art Exhibition Second Prize. Giclee Print. 18" x 14.8", 2009. "The Vase" is composed of a digitally modeled vase with "Lilies" which are Dini's Surfaces. A surface of constant negative curvature obtained by twisting a pseudosphere is known as Dini's Surface. Imagine cutting the pseudosphere along one of the meridians and physically twisting it. Its parametric equations are: x=acos(u)sin(v); y=asin(u)sin (v); z=a{cos(v)+ln[tan(v/2)]}+bu, where 0<= u <= 2pi and 0< v< pi. Take a=1 and b=0.2. "I'm primarily an artist. My shadow is mathematics. I'm helpless at preventing mathematics from intruding in my work and it's delightful to have the body of mathematics to work with. My art attempts to produce a nexus between mathematical beauty and the beauty of the natural world to produce a satisfying aesthetic experience." --- Harry Benke (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.May 10, 2010
 "Peer Below the Surface - No. 65.270," by Leo S. Bleicher (Cepheus Information Systems, San Diego, CA)Digital print of 3D model on photographic paper, 23” x 19”, 2009. Serial coordinate transformations interleaving symmetry preserving and symmetry breaking operations yield a stunning variety of forms. A sequence of fourteen such operations in 3D create this shape from the unit square. Small spheres are initially an array of 40000 normals to the surface at a distance of 0.05. Larger spheres represent hierarchical clustering centroids of the normals in their final positions. Sequences are selected with a genetic recombination function using esthetic appeal as the fitness function. This transformation sequence begins with a cylindrical transform around the z-axis, and finishes with a spherical coordinate transform and rotation around the y-axis. These images are from 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. --- Leo S. Bleicher (Cepheus Information Systems, San Diego, CA) http://porterbleicher.g2gm.net/computed-paintings/ May 10, 2010
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