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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"Snowflake Model 9," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In 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 GriffeathMay 11, 2009

"Snowflake Model 10," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)In 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 GriffeathMay 11, 2009

"Crocheted Lorenz manifold, black background," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.

The black background in the photograph brings out the separating properties of the Lorenz manifold: points on one side of the surface can never cross to the other side, even though they will visit both left and right wings of the butterfly attractor in a seemingly unpredictable manner.

For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/. May 05, 2009

"Crocheted Lorenz manifold, white background," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.

The white background in the photograph brings out the rotational symmetry of the Lorenz manifold and gives an idea of the structure of the mesh.

For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/. May 05, 2009

"Crocheted Lorenz manifold, detail," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.

The photograph shows a particularly nice detail of the intriguing geometry of the Lorenz manifold. The wire running through the crocheted work illustrates one of the paths on the surface that end at the origin.

For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/. May 05, 2009

"Summertime," by Anne M. Burns, Long Island University, Brookville, NY"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/. --- Anne M. Burns May 05, 2009

"Infinite Curl 7," by Matjuska Teja Krasek in collaboration with Dr. Clifford Pickover (2006)Digital print, 9.9" x 10.1". Kraskek's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. Her artworks also illustrate properties such as golden mean relations, self similarity, ten- and fivefold symmetry, Fibonacci sequence, inward infinity and perceptual ambiguity. She employs contemporary computer technology as well as classical painting techniques.

"The image represents the behavior of mathematical feedback loops, and more particularly the iteration of a complex function. The figure is our rendition of a visually interesting quartic variant of a Ushiki Phoenix Julia set. As with other fractals, the image exhibits a wealth of detail upon successive magnifications. The image �Infinite Curl 7� has been made in collaboration with Dr. Clifford Pickover, the author of more than thirty books about mathematics, art, and science." --- Matjuska Teja Krasek, Freelance artist, Ljubljana, Slovenia, EU Apr 28, 2009

"Flow 4," by Elizabeth Whiteley (2008)Museum board and acrylic paint, 7.5" x 13.5" x 10.5". "'Flow 4' is created by the close proximity of two Golden Triangles (base angles of 72 degrees and vertex angle 36 degrees). The plane of each triangle is curved in opposing directions to create an aesthetically pleasing form. The sculpture changes our perception of a static and planar geometric shape and makes for a dynamic visual experience. The curves move the eye around the form and suggest multiple points of view." --- Elizabeth Whiteley, Studio artist, Washington, DCApr 14, 2009

"Flow 1," by Elizabeth Whiteley (2008)Laminated canvas and acrylic paint, 7.5" x 14" x 7.5". "'Flow 1' is created by intersecting two Golden Triangles (base angles of 72 degrees and vertex angle 36 degrees). The plane of each triangle is partially bisected and then curved to create an aesthetically pleasing form. One triangle is smooth; the other has a textured surface. The sculpture changes our perception of a static and planar geometric shape and makes for a dynamic visual experience. The curves move the eye around the form and suggest multiple points of view." --- Elizabeth Whiteley, Studio artist, Washington, DCApr 14, 2009

Ulfah (Harmony)," by Nathan Voirol (2008)Silkscreen print on paper, 20" x 24". "Islamic star pattern based on a tessellation of a 54 pointed star surrounded by 9 and 18 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, CAApr 14, 2009

"Sa'odat (Happiness)," by Nathan Voirol (2007)Hand-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, CAApr 14, 2009

"A Pattern of 48 Different Squares," by Anna Viragvolgyi (2008)Digital print, 20" x 20". "This is a pattern of the 48 different squares, where the square sheets are striped diagonally, the stripes are colored by three colors such that the adjacent stripes are different color. Albeit the arrangement of the squares is not regular, since all the elements are different, the whole surface is symmetrical. Changing the neighborhoods of the elements engenders a different shape. There are innumerable patterns possible. (For example rectangles may be made--with matching opposite borders--which form tori.) The almost limitless solution patterns enhance cognitive skills." --- Anna Viragvolgyi, Mathematician, Budapest, HungaryApr 14, 2009

"Monge's Theorem," by Sumon Vaze, King George V School, Hong Kong (2008)Acrylic on Canvas, 18" x 24". "The external tangents to three circles, taken in pairs, meet at three points, which are collinear. I seek to depict interesting mathematical truths, curiosities and puzzles in simple, visually descriptive ways. Mathematical amusements inspire the color and form in my paintings, and I try to strike a balance between the simplicity of the concepts and their depiction in art." --- Sumon Vaze, High School Teacher of Mathematics, King George V School, English Schools Foundation, Hong KongApr 14, 2009

"Swarming Pentaplex," by Paul Stacy (2004)Giclee print on canvas (mounted) scanned from original artwork, acrylic paint on board, 20" x 20". "'Swarming Pentaplex' is a representation of the seven Penrose rhomb vertex groups, which I inadvertently "discovered" while experimenting with various matching rules. Of course the Penrose vertex groups have been long-known, however this exploded arrangement results from a very simple underlying tile decoration, with a gradual feathering out of the basic pattern. The resultant picture has great beauty inherent to pentagonal geometry with its aesthetic revelations of the "golden mean". The title refers to the fact that in the right half-light and standing at the right distance the painting comes alive with movement in waves across the canvas, like swarming bees! " --- Paul Stacy, Landscape Architect, Sydney, New South Wales, AustraliaApr 14, 2009

"Fractaled Fire," by Christopher Shaver, Rockhurst University, Kansas City, MO (2008)Digital photography, 11" x 14". "This work is a collage of photos taken during the fireworks display at Fair St. Louis on July 4, 2008. Each firework is somewhat self-similar and recursive in nature, with a common pattern appearing at both the center and the outer edges, and each piece having almost the same appearance. The shape is complex even on a small scale. The dimension of a firework is difficult to comprehend since its shape is constantly changing over time, but is a three-dimensional display. The change over time can be viewed and even is part of the overall image because of the appearance of the smoke left behind in the same shape as the colored flame. These art pieces are the product of a student research project I was a part of, exploring the relationship between art and math by a study of fractals." --- Christopher Shaver, Student, Department of Mathematics and Physics, Rockhurst University, Kansas City, MOApr 14, 2009