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.

"Enmpperaltta," by Inigo Quilez2320 viewsInigo Quilez is an engineer born in the Basque Country, Spain, who actually works in Belgium designing virtual reality tools. The word that titles the picture, Enmpperaltta, signifies nothing; it is simple a permutation of the French word "L'Appartement." The reason is the obsession shown by the author while trying to buy the perfect apartment in Brussels; that goal was finally achieved and he celebrated with this image. Enmpperaltta is in fact a still frame from an animation calculated by means of proprietary software written in the C language from a variant of the well-known Pickover algorithm, a formula that generates shapes resembling those produced by mixing fluids, for example liquids of different colors. To generate the image, the formula was repeated three times with slightly altered parameters, each in a separate process, and applied to the three basic components of color in the image: red, green and blue, that are combined together to produce the final result.

"Snowflake Model 8," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)2306 viewsIn nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David Griffeath

"Totem," Harry Benke, Visual Impact Analysis LLC (2008)2257 viewsArchival digital print, 19" x 13.6". "'Totem' represents the frontier, the uncharted, the often surprising and almost mystic nature of mathematical discovery. The totem is composed of ellipsoids ((x2/a2)+(y2/b2)+(z2/c2)) = 1, ray-trace rendered over an algorithmically generated fractal skyscape. Atmospheric effects were calculated as well such as scattering, moisture etc. The totem signifying the last guidepost to the unknown." --- Harry Benke, freelance artist/mathematician, Novato, CA (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.

"A Mathematician's Nightmare," by JoAnne Growney (2008)2255 viewsLaser print on paper, 15 1/2" x 17 1/2" . "The poem, 'A Mathematician's Nightmare,' introduces a version of the unsolved Collatz Conjecture which asserts that when prescribed operations are iterated on any positive integer, the sequence produced will eventually reach 1. The prescribed operations are these, for any starting positive integer n: if n is even, replace n by n/2 (i.e., decrease n by half); if n is odd, replace n by (3n+1)/2 (i.e., increase n by half and round up to the next integer); my exhibit-entry displays both the poem and a graph of the sequence of iterations applied to the integer 27." --- JoAnne Growney, Poet, Professor Emerita, Department of Mathematical Sciences, Bloomsburg University. Residence: Silver Spring, MD

"Embrace," by Robert Bosch (Oberlin College, Oberlin, OH)2237 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

"Poincare," by Mary Candace Williams. Quilt copyright 2005 Mary Candace Williams; photograph by Robert Fathauer.2234 viewsThis is a hyperbolic design so it is as if a sphere was mapped onto a plane. The printed fabric has distorted spheres. This quilt is unusual in that it is pieced from the outside to the center.

--- Mary Candace Williams

"Ramanujan, in the style of Chuck Close, using wavelets," by Edward Aboufadel (Grand Valley State University, Allendale, MI), Clara Madsen (University of Oregon, Eugene) and Sarah Boyenger (Florida State University, Tallahassee)2196 viewsDigital print, 16" x 20", 2009

Both the subject of this work and the method of creation are intricately mathematical. Ramanujan is the famous 20th century Indian mathematician who established or conjectured a broad collection of results in number theory. He caught the attention of Hardy, who recognized Ramanujan's genius. To create this digital image in the style of Chuck Close, wavelet filters were used to detect the existence and orientation of edges in the original image, and other calculations were made to determine the colors in the "marks".

"Irregular hyperbolic disk as lamp shade," by Gabriele Meyer (University of Wisconsin, Madison)2186 viewsPhotograph, 2013
I like to crochet hyperbolic surfaces. More recently I am interested in irregular hyperbolic shapes and how they interact with light. This is an image of a crocheted hyperbolic surface used as a lamp shade. The object itself is about 1 yard in diameter. On one side it has a more negative curvature than on the other, an irregularity, which makes it appear more interesting. The surface is created with white yarn, so that nothing detracts from the shape. --- Gabriele Meyer (http://www.math.wisc.edu/~meyer/airsculpt/hyperbolic2.html)

"Crocheted Lorenz manifold, detail," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)2175 viewsDr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.

The photograph shows a particularly nice detail of the intriguing geometry of the Lorenz manifold. The wire running through the crocheted work illustrates one of the paths on the surface that end at the origin.

For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/.

"Night Hunter, opus 469" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).2169 viewsCrease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Night Hunter" origami work in this album.

Hamid Naderi Yeganeh, "1,000 Line Segments (2)" (August 2014)2169 viewsThis image shows 1,000 line segments. For each i=1,2,3,...,1000 the endpoints of the i-th line segment are: (-sin(4πi/1000), -cos(2πi/1000)) and ((-1/2)sin(8πi/1000), (-1/2)cos(4πi/1000)). I created this image by running my program on a Linux operating system. --- Hamid Naderi Yeganeh

Circle 42161 viewsComputers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

"80 circles in an icosahedron pattern," by Bradford Hansen-Smith2160 viewsYou can also see many hexagonal and pentagonal shapes in this pattern. The symbol of the circle is used as metaphor for nothing and for everything, and endless parts in-between. Folding circles appears to have little history: Somewhere in the history of origami lies the circle, unrecognized and discarded in favor of the square; Buckminster Fuller also folded the circle, with informational intent. Fuller is the inspiration for my own exploration into geometry and provided the seed for folding and joining circles-9" paper plates.

-- Bradford Hansen-Smith, Wholemovement

Hamid Naderi Yeganeh, "1,000 Line Segments (1)" (August 2014)2151 viewsThis image shows 1,000 line segments. For each i=1,2,3,...,1000 the endpoints of the i-th line segment are: (-sin(2πi/1000), -cos(2πi/1000)) and ((-1/2)sin(8πi/1000), (-1/2)cos(12πi/1000)). I created this image by running my program on a Linux operating system. --- Hamid Naderi Yeganeh

"Fiddler Crab, opus 446" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).2141 viewsCrease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Fiddler Crab" origami work in this album.