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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"Magic Square 25 Study," by Margaret Kepner (Washington, DC)First Place Award, 2011 Mathematical Art Exhibition

Archival inkjet print, 12.5" x 12.5", 2010

Magic squares are numerical arrays that have substructures with constant sums. This design is based on a magic square of order 25, containing the numbers from 0 to 624. Each row, column, and main diagonal sums to the “magic constant” of 7800. The numbers in the magic square are represented by a visual base-5 system: four concentric squares serve as the 1, 5, 25, and 125 places, while shades of grey stand for the numerals 0 to 4. Coding the numbers into their base-5 versions yields a pattern of 625 unique, nested-squares in shades of grey. This particular magic square also has a substructure of 25 mini-squares of size 5. Each of these mini-squares is “magic” (although the numbers are not consecutive), with rows, columns, and diagonals summing to 1560. In addition, certain other groups of 5 squares add up to 1560. Examples are the quincunx and the plus-sign shapes (when fully contained in a mini-square). The colored accents are used to indicate a few of these “magic” substructures. --- Margaret Kepner

"Unknown," by Ghee Beom Kim (Sydney, Australia)Print, 300 x 250 mm, 2007

Hexagonal fractal creates this terrace-like surface that imitates exotic terrain, which has architectural implication. The triangular fractal formations occasionally occur amongst rather random surface. --- Ghee Beom Kim (http://sites.google.com/site/geometricarts/)

"Whirled Heart," by Matjuska Teja Krasek (Ljubljana, Slovenia)Digital print, 170 x 230 mm, 2010

In the mysterious world of chaos and strange attractors a seeker can find very heartful things. --- Matjuska Teja Krasek (http://tejakrasek.tripod.com)

"The Fibonacci Project," by Lindsay Lindsey (University of Alabama, Tuscaloosa)Cast aluminium, 18" x 18" x 16", 2010

The sculpture is based off the mathematical concept of the Fibonacci sequence and the spiral found in the Nautilus shell. In order to accurately construct a three-dimensional spiral that has the specifications of the Fibonacci sequence, special attention had to be paid to the size of the sculpture. At specific intervals along the sculpture, the diameter of the sculpture accurately increased with respect to the Fibonacci sequence. The intervals along the sculpture were also planned out using the sequence as a guide to the ever-increasing segments. The turns of the spiral were calculated using the Nautilus shell as a guide. Their increasing diameters are directly proportional to the diameter of the shell. Throughout the construction process, various checks were made to insure that the sequence was being preserved. The sculpture has truly become an accurate three-dimensional representation of both the sequence and the spiral. --- Lindsay Lindsey

"Equal Areas," by Susan McBurney (Western Springs, IL)Digital print, 12" x 12", 2010

This artwork was inspired by two pages from Leonardo DaVinci's notebooks. While these magnificent books are legendary for their beauty of illustration and depth of subject matter, his purely geometric diagrams have been dismissed by some as intellectual doodling. Closer inspection reveals that at least some of them highlight the equality of different-shaped areas. "Equal Areas" builds upon that concept to also include relative areas of similar figures. In particular, those areas of a certain color in the border design are equal to the same-colored areas in the central figure. All light yellow areas in the borders add up to the all the yellow areas in the center, etc. Note that in some cases the shapes of the same colors are different, yet they are still equal in area. --- Susan McBurney

"Infinite Journey," by Frank Mingrone Poster (scan of hand ink drawing on paper), 32” x 24” (original 45” x 42”), 1985

There were no computers used in the creation of this drawing. It was completely hand drawn using a pen and ruler and consists of straight, unbroken, parallel lines that extend to the outermost perimeter. If the perimeter expanded and the lines repeated and extended, the symmetrical pattern would continue infinitely. The use and placement of straight lines are not a random guess but must conform to a mathematical framework for their representation. Each group of lines is analogous to a group of integers, and it is the exact arrangement of the lines arising from balanced proportions that create the intricate patterns. The lines can flow in a successive order, or, with varied intricate combinations. The singularity of straight lines unites a complex system of multiple interrelated sections creating the illusion of curvature. The various parts relate to the whole and the patterns grasped and visualized as a whole. --- Frank Mingrone (http://www.supersymmetryart.com/)

"Paper stars," by Velichka Minkova (Bulgaria, Sofia) Digital C, 18 "x 18", 2010

Law is offered at a symmetry in proper square network and her use at making abstract constitution by a volumetric-plastic forms. --- Velichka Minkova

"Woman flower," by Marcel Morales (Institut Fourier, Université de Grenoble I, France) Digital print on canvas, 300 x 450 mm, 2010

I use hyperbolic geometry, in fact the idea of tiling the hyperbolic plane, to produce a tile such that by repeating hyperbolic rotations we can fill the plane. In this artwork a difficult point is to find the tile. My idea is to use a woman to fulfill a flower, and this flower fulfills the moon and the earth, changing colors and getting maturity. --- Marcel Morales (http://marcel.morales.perso.sfr.fr/)

"Longest and Shortest Crease-B," by Sharol Nau (Northfield, MN)Folded book, 12.5” x 9” x 6”, 2010

For this book-sculpture of several hundred pages, the shortest crease was obtained by folding the pages without separating them from the binding. Also the folding process began in the middle in an effort to achieve a symmetrical design. --- Sharol Nau (http://www.sharolnau.snakedance.org)

"DART," by Jo Niemeyer (Schluchsee, Germany)Archival inkjet print, 20" x 20", 2010

DART ... as a very simple game by Jo Niemeyer 2010 . Two dart players, A and B, are facing this "image", whose area is split 1 : 0.618.. into white and black. This two basic elements are rotated in 90° increments. The winner is, who aims first a black part. Since we have two equal partners and an uneven distribution of the "target", one would think, that this is not a fair game. But it is! Because A as the "majority", and consequently B as the "minority", transferred their inequality onto the "court". The ratio of the two playing partners is 1:1. With this harmonious proportionality there is exactly the same chance to win for both players A and B! The Swiss mathematician Hans Walser mentions for the justice condition, the formula p = 1/2*(3-sqrt(5)). And with sqrt(5), we have the golden section in this game, which ensures equity between different partners. This is also a very fair game! Or a piece of art, which ensures harmony and balance. --- Jo Niemeyer (http://www.jo.niemeyer.com)

"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/)

"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

"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/)

"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/)