Math ImageryThe 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 ands, 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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"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
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"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
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"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
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"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
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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/
May 10, 2010
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"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
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"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
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"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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"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
May 10, 2010
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"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/
May 10, 2010
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"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
May 10, 2010
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"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/

May 10, 2010
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"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/
May 10, 2010
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"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)
May 10, 2010
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"Sculpture System No. 5 (2009)" by Richard Grimes (www.richardgrimes.net) and Edmund Harriss (University Of Leicester)Deltahedra are polyhedra where all the faces are regular triangles. Sculpture system 5 is a system to build any deltahedron using a triangular shapes that hinge together. Using just twenty of these shapes a huge variety of polyhedra can be made. The actual sculpture was built by a group of volunteers who also able to design the final shape that was actually built. --- Edmund Harriss
Feb 09, 2010
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