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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"Snowflake Model 3," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1580 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

symmetry2.jpg1570 views"Symmetry Energy Image II," by Rob Scharein (Centre for Experimental and Constructive Mathematics, Simon Fraser University, B.C., Canada)

This example illustrates the SE rendering mode in KnotPlot, which visualizes the symmetric energy distribution. KnotPlot is a program to visualize and manipulate mathematical knots in three and four dimensions, and the website includes a wealth of resources and pictures. This picture is a direct screen capture from KnotPlot, rendered entirely in OpenGL, an environment for portable, interactive graphics applications.

--- Rob Scharein

"Imaginary Garden," by Anne M. Burns (Long Island University, NY)1567 views"Mathscapes" are created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector. See the Gallery of Mathscapes and find citations for my articles on modeling trees, plants and mountains, and on "blending and dithering" at http://myweb.cwpost.liu.edu/aburns/gallery/gallery.htm. --- Anne M. Burns (Long Island University, Brookville, NY)

"Wave (32)," by Goran Konjevod, Arizona State University, Tempe, AZ (2006)1567 viewsFirst Prize, 2009 Mathematical Art Exhibition. One folded square sheet of paper, 10" x 10" x 5". "The wave is one of the pleat tessellations that continues to amaze me even years after I first folded it. The peculiar symmetry and the tension caused by locking the edges causes two of its corners to bulge in opposite directions, while the remaining two corners remain fairly flat. As in the simple bowl, the pleat sequences all begin at the edges and proceed towards the center of the sheet, but the difference is that all horizontal pleats are oriented the same way, and similarly all the vertical pleats." --- Goran Konjevod, Assistant Professor of Computer Science and Engineering, Arizona State University, Tempe, AZ

"SA_1188475827," by Nathan Selikoff1551 viewsAnother strange attractor, this one existing in three dimensions, comes to life with rich fiery colors that enhance the eastern Asian feel of the swirling lines. See more images at www.nathanselikoff.com/.

"Ten Triangular Prisms," by Magnus Wenninger (Saint John's Abbey, Collegeville, MN)1548 viewsPaper, 9" x 9" x 9", 2010

Robert Webb's Stella program is now the computer program I use for the construction of all the polyhedron models I have recently been making. It is the program par excellence I now use for the discovery of any new polyhedra, especially any I have never made before. The photo shows a model of Ten Triangular Prisms, recently made by me. I found the Stella version on a web page called '75 Uniform Polyhedra' done by Roger Kaufman. It is #32 on this web page. The Stella version gives me a 3D computer view in 10 colors and allows me to choose the size of the model and thus also the size and shape of the net to be used for the construction of the model. However, I wanted my model to be done using only 5 colors. This is where the artwork comes into play. The model now shows each prism with its faces in one color of the five. Thus it becomes uniquely artistic in appearance. --- Magnus Wenninger (http://www.saintjohnsabbey.org/wenninger/)

ashley.jpg1538 views"Ashley Knot," by by Rob Scharein (Centre for Experimental and Constructive
Mathematics, Simon Fraser University, B.C., Canada)

This example illustrates the SE rendering mode in KnotPlot, which visualizes the symmetric energy distribution. KnotPlot is a program to visualize and manipulate mathematical knots in three and four dimensions, and the website includes a wealth of resources and pictures. This picture is a direct screen capture from KnotPlot, rendered entirely in OpenGL, an environment for portable, interactive graphics applications.

--- Rob Scharein

symmetry3.jpg1532 views"Symmetry Energy Image III," by Rob Scharein (Centre for Experimental and Constructive Mathematics, Simon Fraser University, B.C., Canada)

This example illustrates the SE rendering mode in KnotPlot, which visualizes the symmetric energy distribution. KnotPlot is a program to visualize and manipulate mathematical knots in three and four dimensions, and the website includes a wealth of resources and pictures. This picture is a direct screen capture from KnotPlot, rendered entirely in OpenGL, an environment for portable, interactive graphics applications.

--- Rob Scharein

"Spirolateral_{6}^{1,3,4} ," by Mary Candace Williams. Quilt copyright 2005 Mary Candace Williams; design is copyright Robert Krawczyk; photograph by Robert Fathauer.1513 viewsThe challenge of doing a quilt that has only rotational symmetry and uncommon angles lies in a technique called partial seaming. The printed fabric has a lot of pursuit curves overlaide on top of one another and shows a lot of contrast with the black of the spirolateral.

--- Mary Candace Williams

Circle 31494 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.

"Crane," by Zdenka Guadarrama, Rockhurst University, Kansas City, MO (2008)1491 viewsMobile--Gauze, papyrus, silver and wood, 10" x 10" x 15". "'Crane' represents the continuous dimensional transition from a point, represented by a silver sphere, to a line, a plane and finally a crane. This transition is depicted in parallel to the evolution of the creative process which starts with an idea, represented by the same silver sphere, and which through refinements and trials culminates in the bird as well. [My] projects consist in artistic explorations that happen in parallel to the teaching/learning of mathematics (measure theory or complex analysis, for example). I search to generate art using mathematics and art inspired in the mathematics that I share with my students in order to motivate them to learn more mathematics, to make some extra connections, and to create some art of their own." --- Zdenka Guadarrama, Assistant Professor of Mathematics, Department of Mathematics and Physics, Rockhurst University, Kansas City, MO

"Scherk's First Surface," copyright Andrew Lipson. Made of Lego®1488 viewsThis is a nice example of a saddle point. The model shows (most of) one cell of a doubly-periodic Scherk surface. Actually Scherk discovered more than one minimal surface in 1835, but this one has the particularly simple parametrisation given by exp(z) = cos(x)/cos(y). This model shows the surface in the region |x|, |y| < p/2 - 0.01. As with most of my mathematical surfaces, I made use of some computer assistance. On my website you can find more pictures and an LDRAW .DAT file generated by my program for this sculpture. Beware--the .DAT file builds it out of 1x1 bricks. Actually constructing this out of larger bricks so that it holds together is a (non-trivial) exercise! Lego ® is a trademark of The Lego Group. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)

"Tile 7," by Anne M. Burns, Long Island University, Brookville, NY1481 viewsHere is a fractal tiles created with Geometer's Sketchpad. I start with a single "tile" designed using Geometer's Sketchpad. Then, using Flash Actionscript I place that "tile" in the center of the screen and surround it with 12 copies of the tile that are half the size of the original, then surround those with 36 "tiles" half the size of the second set of "tiles"; the process is continued until the tiles are too small to see. Thus we obtain a "fractal" tiling. See more fractal tiles at http://myweb.cwpost.liu.edu/aburns/. --- Anne M. Burns

"Flow 4," by Elizabeth Whiteley (2008)1477 viewsMuseum board and acrylic paint, 7.5" x 13.5" x 10.5". "'Flow 4' is created by the close proximity of two Golden Triangles (base angles of 72 degrees and vertex angle 36 degrees). The plane of each triangle is curved in opposing directions to create an aesthetically pleasing form. The sculpture changes our perception of a static and planar geometric shape and makes for a dynamic visual experience. The curves move the eye around the form and suggest multiple points of view." --- Elizabeth Whiteley, Studio artist, Washington, DC

"Snowflake Model 6," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1473 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