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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"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)

"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/)

"Hyperbolic Twistslug," by Mickey Shaw (Le Roy, KS)Fiber, 9" x 22" x 13”, 2009

This crocheted fiber soft sculpture is based on non-Euclidean geometry. It represents a variation of the hyperbolic plane ruffle effect. The piece was created using basic crochet stitches, which were increased at a rate great enough to create exponential growth. Attention was given to pushing the construction into a form of varying volume, irregular shape and an integration of pattern and color. The end result is simultaneously geometric in its basic nature and organic in its form. This creation used over two pounds of fibers. The structure is malleable, allowing the form to morph into numerous shapes. The hyperbolic soft sculpture is a further exploration of what forms can evolve in combining hard-edged geometric concepts with the fluid, textural aspects of fiber and stitches. This combination creates a three-dimensional visual and mental juxtaposition of the interconnection of the two elements. --- Mickey Shaw (http://FullLunaCreations.etsy.com/)

"Quarthead," by Bob Sidenberg (Minneapolis, MN) Wood, 16" h x 16" w x 16" d, 2002

This one is trying to be a rhombic dodecahedron, but hasn't quite emerged from its tetrahedral beginnings. --- Bob Sidenberg (http://www.silkmountain.net)

"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)

"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

"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)

"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)

"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

"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

"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)

"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/)