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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"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/

"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

"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.

"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/

"Embrace," by Robert Bosch (Oberlin College, Oberlin, OH)2010 Mathematical Art Exhibition, First Prize.

Stainless steel and brass, Diameter = 6 inches, thickness = 0.25 inches, 2009. 2010 Mathematical Art Exhibition, First Prize. I began by converting a drawing of a two-component link into a symmetric collection of points. By treating the points as the cities of a Traveling Salesman Problem and adding constraints that forced the salesman's tour to be symmetric, I constructed a symmetric simple-closed curve that divides the plane into two pieces: inside and outside. With a water jet cutter, I cut along this Jordan curve through quarter-inch thick, six-inch diameter disks of steel and brass. By swapping inside pieces I obtained two copies of the sculpture. Here, steel is inside and brass is outside. All artists are optimizers. All artists try to perform a task--creating a piece of artwork--at the highest level possible. The main difference between me and other artists is that I use optimization explicitly. Here's how I work: After I get an idea for a piece, I translate the idea into a mathematical optimization problem. I then solve the problem, render the solution, and see if I'm pleased with the result. If I am, I stop. If not, I revise the mathematical optimization problem, solve it, render its solution, and examine it. Often, I need to go through many iterations to end up with a piece that pleases me. I do this out of a love of mathematical optimization--the theory, the algorithms, the numerous applications. --- Robert Bosch (Oberlin College, Oberlin, OH) www.dominoartwork.com

"Monarch Safye," by Safieddine Bouali (University of Tunis, Tunisia)Digital print, 20" x 24", 2009. Deterministic 3D strange attractor built with the dynamical system:

dx/dt = 0.02 y + 0.4 x ( 0.2 - y2 ) (1)

dy/dt = - x + 35 z (2)

dz/dt = 10 x - 0.1 y (3)

Initial Condition (x0, y0, z0 ) = ( 0, 0.01, 0 ), fifth-order Runge Kutta method of integration, and accuracy = 10-5. Euclidian coordinates representation : ( y, - x, z). I have always been fascinated by the Lorenz Attractor. I like to create and simulate systems of ordinary differential equations on my computer. A simple raylight formed by a 3D model follows intricate dynamics. Visualizing an infinite trajectory drawing elegant attractors within a limited phase of space unravels the aesthetics appeal of the Deterministic Theory of Chaos. Indescriptible happiness when new strange attractors emerge in my computer screen ! These are sculptures of motion. Derived from the Sensitive Dependency on Parameters , an unique chaotic model displays an unpredictable class of attractors. Indeed, from theoretical viewpoint, no relationship between mathematical equations and attractor shapes has ever been found. Chaotic attractors are mysterious figures but reproducible in various media by everyone if mathematical formulas are clearly expressed, I think discovering unexpected strange attractors by the exploration of 3D dynamical models constitutes a full artistic principle. By unconventional ways, I search beauty. --- Safieddine Bouali (University of Tunis, Tunisia) http://chaos-3d.e-monsite.com/

"Origami I," by Vladimir Bulatov (Artist, Corvallis, OR)Stainless steel and bronze. Direct metal print, 4"x 4"x 4", 2008. The sculpture is inspired by 3 dimensional origami construction. 30 identical ribbons bent around the surface of a cylinder are joined together to form the shape with the rotational symmetry of an icosahedron. My artistic passions are purely mathematical images and sculptures, which express a certain vision of forms and shapes, my interpretations of distance, transformations and space. In my opinion, mathematics is not simply a profession, but rather a way of thinking, a way of life. My images and sculptures are like photographs of interesting mathematical ideas. I have always been intrigued by the possibility of showing the intrinsic richness of the mathematical world, whose charm and harmony can really be appreciated by everyone. --- Vladimir Bulatov (Artist, Corvallis, OR) http://bulatov.org

"Quilt," by Galina Bulatova (Artist, Corvallis, OR)Glass, 7.5” x 7.5”, 2009. This tessellation is the traditional coloring and layout of the Churn Dash pattern. The Churn Dash quilt block is an interesting pattern that is composed of right triangles, rectangles, and a single square. This symmetrical design is based on a square and can be recolored in this manner to produce a tessellating motif. Geometric design and especially geometric patterns always intrigued me. I was excited by seeing how patterns changed as various symmetrical arrangements were applied to them. My pieces are created by layering and melting together various shapes and colours of glass in a fusing kiln, which heats the glass to 1500 degrees Fahrenheit. --- Galina Bulatova (Artist, Corvallis, OR) http://bulatov.org/

"Overlapping Circles #25," by Anne Burns (Long Island University, Brookville, NY)Digital print, 19" X 13", 2009. The subgroup of Möbius Transformations that maps the unit circle onto itself is composed with Möbius Transformations that map the unit circle into itself to produce overlapping circles. The program was written in Actionscript. I am fascinated by art, mathematics and nature and try to combine all three in my work. --- Anne Burns (Long Island University, Brookville, NY) http://myweb.cwpost.liu.edu/aburns/

"Ready to Fly High," by Mingjang Chen (National Chiao Tung University, Hsinchu, Taiwan)Digital print by PowerPoint, 17" x 22", 2008. Complete r-partite graph is the graph with vertices set consisting of r disjoint sets such that any two vertices in different sets are connected by an edge and not for vertices in the same set. The work is a complete bipartite graph, following by a rotation on each line segments. One part of vertices is positioned on two adjacent line segments with equal distance; another part of vertices is positioned on an oval. There are 27 vertices on one part and 24 vertices on the oval. Hence, there are 27*24 line segments in this work. The transparency of these line segments is high up 95%. Structural Cloning Method (SCM) implemented on PowerPoint is a Human Computer Interface which handles geometry transformations on a huge number of drawing elements; it can be used to arrange complicate elements. Based on SCM, we can explore symmetry patterns and fractal patterns in different ways, and so math art becomes interesting. Complete r-partite graph is a common used graph, the number of edges in a complete r-partite graph is very large, and its edges always cover the space among vertices when visualizing. Once we tune up the transparency of these line segments, the transparency of overlapped areas become various degrees, this effect makes overlapped areas look like light-spots or rays if the background is in dark color, however the line segments are in light color. Parts of vertices can be arranged in various structures such that the overlapped transparent line segments can generate various and amazing patterns. --- Mingjang Chen (National Chiao Tung University, Hsinchu, Taiwan)

"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)

"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/.

"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/

"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/.

"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)