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.

"Rosetta," by Edward Alonzo (Artist, University of Vermont)2480 viewsAcrylic 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/

"Embrace," by Robert Bosch (Oberlin College, Oberlin, OH)2148 views2010 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

"The Vase," by Harry Benke (www.harrybenke.com)1937 views2010 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.

"Julia's Loops," by Jennifer Ziebarth (California College of the Arts, Oakland, CA)1526 viewsDigital print, 16" x 13", 2009. This fractal image is based on a Julia set, visible in dark blue along the intersections of the loops. The loops, which all begin and end on the Julia set, also exhibit self-similarity, and hint at the existence of more small loops hidden behind the larger loops. "I have always been fascinated with repetition, abstraction, and the search for pattern, and this is what drew me to mathematics. As a mathematical artist, this love of repetition and detail has lead me to fractal art. As a mathematician teaching at an art college, some of my work is pedagogical in the sense of illustrating mathematical concepts in aesthetically pleasing ways; some of it is purely visual play." --- Jennifer Ziebarth (California College of the Arts, Oakland, CA)

"Meditations on f(x,y)= (x^2)/2 + xy/2 – (y^4)/8," by Richard Werner (Santa Rosa Junior College, Santa Rosa, CA)1441 views2010 Mathematical Art Exhibition Third Prize.

Plastic and wood, two pieces, each 6”x7”x7”, 1998. The two pieces give alternate views of the same three-dimensional surface. The sculpture has been used for classroom illustrations of the concept of partial derivatives as well as integration of functions of two variables. Since the construction is with clear plastic, a myriad of delightful views of intersecting curves can be found allowing the viewer to hypersee the surface. "I have been a recreational wood worker and sculptor for much of my life. As a mathematics teacher, I have always been captivated by the beauty of the subject and have wanted to enhance the visual concepts in whatever way I can. The two activities were destined to meet. The first mathematical art that I made was intended mainly for classroom demonstrations. The response was very positive and I began to branch out. New materials, especially metal, have captured my interest. The work that I do now is becoming a blend of my interest in math and my love of nature, with a little bit of steam-punk influence creeping in as well." --- Richard Werner (Santa Rosa Junior College, Santa Rosa, CA)

"Monarch Safye," by Safieddine Bouali (University of Tunis, Tunisia)1425 viewsDigital 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)1416 viewsStainless 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

"A mirror pair of (3,2) torus knots embedded on tori," by sarah-marie belcastro (freelance mathematician, Hadley, MA)1390 viewsKnitted 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

"Three Elements 4-5-3," by Doug Dunham (University of Minnesota Duluth)1380 viewsColor 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/.

"Geodesic Cuboctahedron 7 frequency," by Magnus Wenninger (Saint John’s Abbey, Collegeville, MN)1356 viewsPapercraft, 12 inches in diameter, 2009. "Geodesic domes are well known as architectural structures, but generally they exhibit only triangular grids. My main interest, however, has been in having geometric patterns projected onto a spherical surface. The icosahedron is most frequently used for this purpose, but other polyhedrons can serve just as well for the same purpose. 'Geodesic Cuboctahedron 7 frequency' is the cuboctahedron in a 7 frequency basket weave pattern with 6 squares of one color and 12 rectangles of 6 other colors projected onto the surface of the cuboctahedron’s circumsphere." More information about the techniques I use to produce my artistic patterns on a spherical surface can be found in the Dover publication of my book Spherical Models (1999), originally the Cambridge University Press publication of Spherical Models (1979). Robert Webb’s Stella program is now my computer program par excellence. --- Magnus Wenninger (Saint John’s Abbey, Collegeville, MN) http://www.saintjohnsabbey.org/wenninger/

"Peer Below the Surface - No. 65.270," by Leo S. Bleicher (Cepheus Information Systems, San Diego, CA)1350 viewsDigital 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/

"Elements," by Jeff Chyatte (Montgomery College and River Bend Studio at Water’s Edge, Washington DC)1349 viewsPainted 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)1325 viewsElephant 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/.

"Seven-Color Torus Series in Bead-Crochet: Bracelet 2," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) 1237 viewsBead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.html

"Quilt," by Galina Bulatova (Artist, Corvallis, OR)1202 viewsGlass, 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/