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.

"Floating Pentangle Construction," by Bente Simonsen (Landeryd, Sweden)Digital print, 20" x 24", 2010

Impossible pentangle construction, 2D and 3D mix-illusion. --- Bente Simonsen (http://geometric-impossibilities.blogspot.se) Mar 10, 2011

"Thorn Dice Set," by Chuck Stover (Lansing, MI)Printed stainless steel and bronze, 6" x 8" x 1", 2010

A set of polyhedral dice with edges defined by interlocking vines of steel. --- Chuck Stover

Mar 10, 2011

"Martin Gardner - Master Puzzler," by Bruce Torrence (Randolph-Macon College, Ashland, VA)Archival inkjet print, 20" x 20", 2010

This portrait of Martin Gardner (1914-2010) was made by coloring the individual tiles on a kite and dart Penrose tiling. This particular tiling exhibits fivefold rotational symmetry (can you find the center?), and was created by "deflating" a wheel of five kites eight times. Gardner's oft-cited January 1977 Scientific American column introduced the public to Penrose's aperiodic tiles. --- Bruce Torrence (http://faculty.rmc.edu/btorrenc) Mar 10, 2011

"Visualizing Abstract Quantity," by Anna Ursyn (University of Northern Colorado, Greeley)Archival print, 8" x 10", 2010

Unspoken fears. --- Anna Ursyn (http://www. Ursyn.com) Mar 10, 2011

"Möbius strip patterned by 48 different striped squares," by Anna Virágvölgyi (Budapest, Hungary)Folded paper, 100 x 100 x 100 mm, 2010

Diagonally striped tiles of this arrangement create concentrically striped squares. The number of squares is the number of all possible triplets of three symbols (no symbols are paired): 3*2*2 = 12. The surface of the Möbius strip is diced with this different 12 squares. The edge of the strip is diced with another whole set of such triplets. This arrangement would be realized on tori as well. --- Anna Virágvölgyi Mar 10, 2011

"Infinity," by Mary Wahr (Manistee Area Public Schools, MI)Pen and ink, 18" x 21", 2009

This is an abstract pen and ink rendering of a broccoflower. It is the first piece of art completed for my thesis and was the starting point of two years of research and art. Since my fractal ideas are accomplished without arithmetic, I needed to incorporate the components that define a fractal. This piece of art reflects the concepts of infinity, iteration, self-similarity and scaling. --- Mary Wahr Mar 10, 2011

"iteration," by Trygve Wastvedt (St. Olaf College, Northfield, MN)Bronze, plaster, wax, concrete, 18" x 18" x 8", 2010

"iteration" is a series of identical humanoid figures cast in bronze, plaster, wax, and concrete. The form is a near honeycomb so that the individual pieces stack together to fill space. Though geometric, the form still evokes human emotions, which allows the sculpture to ask social and relational questions. --- Trygve Wastvedt (http://www.trygvewastvedt.com) Mar 10, 2011

"Ten Triangular Prisms," by Magnus Wenninger (Saint John's Abbey, Collegeville, MN)Paper, 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/) Mar 10, 2011

"Sierpinski Theme and Variations," by Larry Riddle (Agnes Scott College, Decatur, GA)Counted cross stitch on fabric (25 count per inch), 13.5" x 13.5", 2009

The Sierpinski Triangle is a fractal that can be generated by dividing a square into four equal subsquares, removing the upper right subsquare, and then iterating the construction on each of the three remaining subsquares. That is our “Theme”, shown in the upper left. The “Variations” arise by exploiting symmetries of the square. The three variations in this piece were generated by rotating the upper left and lower right subsquares at each iteration by 90 or 180 degrees, either clockwise or counterclockwise. The self-similarity of the fractals, illustrated by the use of three colors, means that you can read off which rotations were used from the final image. Each design shows the construction through seven iterations, the limit that could be obtained for the size of canvas used. --- Larry Riddle (http://ecademy.agnesscott.edu/~lriddle/) Mar 10, 2011

"Traveling Ribbons," by Irene Rousseau (Irene Rousseau Art Studio, Summit, NJ)Painted wood and paper collage with gestural expression, 17"x17"x5", 2010

"Sierpinski's Doughnut," by Ian Sammis (Holy Names University, Oakland, CA)Digital print on canvas, 15" x 12", 2010

A Sierpinksi curve is a space-filling curve that fills a triangle. Sierpinski curves may be chained together to construct a continuous path from triangle to triangle. The correct arrangement of triangles allow the construction of a single path that fills the unit square while following an Eulerian path along a graph with the topology of a torus. Mapping the square onto the torus in the usual way gives us a space-filling closed circuit on the surface of a torus. The image is a render of a tube following such a circuit. --- Ian Sammis Mar 10, 2011

"Calm," by Reza Sarhangi (Towson University, Towson, MD)Digital print, 16" x 20", 2008

"Calm" is an artwork based on the “Modularity” concept presented in an article “Modules and Modularity in Mosaic Patterns” (Reza Sarhangi, Journal of the Symmetrion, Volume 19, Numbers 2-3, 2008/. Another article in this regard is “Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical, and Theoretical Considerations” (Sarhangi, R., S. Jablan, and R. Sazdanovic, Bridges Conference Proceedings, 2004). The set of modules with extra cuts used to create this artwork is presented in this figure: http://gallery.bridgesmathart.org/sites/gallery.bridgesmathart.org/files/Reza/Calm_figure.jpg. --- Reza Sarhangi (http://pages.towson.edu/gsarhang/) Mar 10, 2011

"Tryptique," by Radmila Sazdanovic (University of Pennsylvania) and Aftermoon studio (Paris, France)Ink/brush, 24" x 8", 2010

Tryptique is a drawing of three different kinds of diagrams used in categorifications of the one-variable polynomial ring with integer coefficients. These diagrams are elements of three distinct algebras: on the level of Grothendieck rings, projective modules spanned by these diagrams correspond to Chebyshev polynomials, integer powers of x and (x-1), and Hermite polynomials. Asgar Jorn's comment about Pierre Alechinsky's work could as well apply to the signs Aftermoon studio created based on our diagrams.

"L'image est écrite et l'écriture forme des images... on peut dire qu'il y a une écriture, une graphologie dans toute image de même que dans toute écriture se trouve une image." --- Radmila Sazdanovic (http://www.math.upenn.edu/~radmilas/) Mar 10, 2011

"Æxploration (Aesthetic Exploration)," by Nathan Selikoff (Digital Awakening Studios, Orlando, FL)Real-time Video Projection, variable, 2009

Æxploration (Aesthetic Exploration) is a real-time, interactive video projection. This custom software visualizes a variety of two- and three-dimensional strange attractors, allowing the viewer to control the coefficients, color, and translation of the attractor. Until recently, my goal has been to generate high quality still images of strange attractors, and my interactive software has been geared towards that purpose alone - an artist's tool that is a byproduct of the process, viewable only by myself. But recently, in the course of a single day, I made some changes to my code that completely revolutionized what I was seeing on the screen while using my software, and I am excited to share the results. The image above is a screen capture. Video is available at http://nathanselikoff.com/251/strange-attractors/aesthetic-exploration. --- Nathan Selikoff (http://www.nathanselikoff.com) Mar 10, 2011

"Torus Knot (5,3)," by Carlo H. Séquin (University of California, Berkeley)Second Place Award, 2011 Mathematical Art Exhibition

Bronze with silver patina, 10" × 8" × 16", 2010

Torus knots of type (p,q) are simple knots that wind around an invisible donut in a regular manner – p times around the hole, and q times through the hole. By using a somewhat more angular shape for the donut and a variable-size, crescent-shaped cross section for the ribbon, this mathematical construct can be turned into a constructivist sculpture. The challenge was to find a way to make a mold for casting this highly intertwined structure. The solution was to cast three identical pieces, which were then threaded together and welded to each other. --- Carlo H. Séquin (http://www.cs.berkeley.edu/~sequin/) Mar 10, 2011