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.

"NeuralNet," by Mike Field (University of Houston)1691 views"NeuralNet" is is part of the generating tile of a planar repeating pattern of type pgg. Repeating patterns of this type have no reflection symmetries but do have many glide reflection symmetries as well as translational symmetries and two-fold centers of rotation. The absence of reflectional symmetries often leads to very fluid and dynamic patterns. The coloring reflects the density of the invariant measure. --- Mike Field

8-torus.jpg1685 views"The 8-Crossing Torus Knot T(4,3)," by Dror Bar-Natan (University of Toronto, Canada)

This is an example of a torus knot. A torus is a surface best described as a doughnut. A torus knot can be thought of as looping around and through the torus. The symbol T(4,3) means that the string making the knot loops through the hole of the torus 4 times, making 3 revolutions. This knot is drawn with TubePlot.

--- Dror Bar-Natan

Circle 11674 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.

"Snowflake Model 13," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1666 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

"2 Circles in a bi-pentagon pattern," by Bradford Hansen-Smith1665 viewsThe 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

"P-357" Acrylic on canvas1664 viewsThere are 11 petal forms in center of this painting. The grid or plaid in the background follows a number sequence of 3 while there are 22 circles interlocking around the edge of the painting. There are also soft concentric circles which radiate from the center of the painting.

"Snowflake Model 3," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1657 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.jpg1637 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

"Sa'odat (Happiness)," by Nathan Voirol (2007)1634 viewsHand-made ceramic tile, 15" diameter. "Islamic star pattern based on a tessellation of 18 and 12 pointed stars in a hexagonal repeat. My primary artistic interest is in designing repeatable patterns--I particularly enjoy creating geometric star and floral designs, which stem from my fascination with Islamic art." --- Nathan Voirol, CAD Drafter / Freelance Artist, Santa Barbara, CA

"Imaginary Garden," by Anne M. Burns (Long Island University, NY)1622 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)

"SA_1188475827," by Nathan Selikoff1615 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/.

"Wave (32)," by Goran Konjevod, Arizona State University, Tempe, AZ (2006)1610 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

ashley.jpg1605 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

"Ten Triangular Prisms," by Magnus Wenninger (Saint John's Abbey, Collegeville, MN)1598 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/)

symmetry3.jpg1596 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.