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 ands, 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.

"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 Mar 10, 2011

"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/) Mar 10, 2011

"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) Mar 10, 2011

"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 Mar 10, 2011

"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 Mar 10, 2011

"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/)Mar 10, 2011

"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 MinkovaMar 10, 2011

"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/)Mar 10, 2011

"Gyrangle," by George W. Hart (www.georgehart.com)The sculpture is constructed from almost 500 laser-cut steel units, bolted together in a novel way that produces a gyroid surface entirely from equilateral triangles. Shapes come together to reveal a variety of different patterns in the "tunnels" of the sculpture. The first presentation
of this interesting geometry was at the USA Science and Engineering Festival in Washington DC, October 2010. The completed 42" sculpture was donated to Towson University. The work is described in detail at www.georgehart.com/DC. Oct 19, 2010

"Butterfly Effect," by Nathan Selikoff (www.nathanselikoff.com), 2007The "Butterfly Effect", or more technically the "sensitive dependence on initial conditions", is the essence of chaos. Besides the fact that this attractor looks like an abstract butterfly, the title of the piece is an homage to Edward Lorenz, a pioneer of chaos theory. It’s a quick jump from this popular understanding of chaos theory to playing with the Lorenz Attractor and learning a bit more about the math and science behind it. Read more at
http://nathanselikoff.com/236/strange-attractors/butterfly-effect. --- Nathan Selikoff Jul 23, 2010

"Circle D," by Anne M. Burns, Long Island University, Brookville, NYThe Unit Circle Group is a subgroup of the group of Mobius Transformations. Read about how this and other circle images are created and view more examples at http://myweb.cwpost.liu.edu/aburns/ --- Anne M. BurnsJul 23, 2010

"June wreath," by Anne M. Burns, Long Island University, Brookville, NYThis circle image is made by iterating systems of Mobius Transformations. Read about how this and other circle images are created and view more examples at http://myweb.cwpost.liu.edu/aburns/. --- Anne M. BurnsJul 23, 2010

"Kaleidoscope," by Anne M. Burns, Long Island University, Brookville, NYThis circle image is made by iterating systems of Mobius Transformations. Read about how this and other circle images are created and view more examples at http://myweb.cwpost.liu.edu/aburns/. --- Anne M. BurnsJul 23, 2010

"Owl King," by Nathan Selikoff (www.nathanselikoff.com), 2007What do you see? An owl, spider, space ship… something else? This artwork is a two-dimensional plot of the Lyapunov exponent of a chaotic dynamical system. The Lyapunov exponent is a measure of how chaotic the system is, and in this case, the system is a strange attractor with a four-dimensional phase space. Two of the dimensions are static, and the other two vary in the x and y directions of the image. A custom program renders four 16-bit grayscale images, which represent the different “components” of the spectrum of Lyapunov exponents. Read more at http://nathanselikoff.com/209/faces-of-chaos/owl-king. --- Nathan SelikoffJul 23, 2010

"Tile 7," by Anne M. Burns, Long Island University, Brookville, NYHere 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. BurnsJul 23, 2010