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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"Ramanujan, in the style of Chuck Close, using wavelets," by Edward Aboufadel (Grand Valley State University, Allendale, MI), Clara Madsen (University of Oregon, Eugene) and Sarah Boyenger (Florida State University, Tallahassee)1946 viewsDigital print, 16" x 20", 2009

Both the subject of this work and the method of creation are intricately mathematical. Ramanujan is the famous 20th century Indian mathematician who established or conjectured a broad collection of results in number theory. He caught the attention of Hardy, who recognized Ramanujan's genius. To create this digital image in the style of Chuck Close, wavelet filters were used to detect the existence and orientation of edges in the original image, and other calculations were made to determine the colors in the "marks".

"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/)1945 viewsDr. 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/.

"Night Hunter, opus 469" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).1938 viewsCrease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Night Hunter" origami work in this album.

"80 circles in an icosahedron pattern," by Bradford Hansen-Smith1937 viewsYou can also see many hexagonal and pentagonal shapes in this pattern. The symbol of the circle is used as metaphor for nothing and for everything, and endless parts in-between. Folding circles appears to have little history: Somewhere in the history of origami lies the circle, unrecognized and discarded in favor of the square; Buckminster Fuller also folded the circle, with informational intent. Fuller is the inspiration for my own exploration into geometry and provided the seed for folding and joining circles-9" paper plates.

-- Bradford Hansen-Smith, Wholemovement

"Spellbinder," by Richard Spix1916 viewsThis image was formed with no less than 105 fractal layers composed of a large number of different coloring algorithms applied to a Phoenix set. It is the kind of image in which the composition enjoys much importance and in which many of the formulas are used to correct or emphasize small details, at times almost imperceptibly. The name of the image comes from the idea of representing an ancient artifact, enigmatic, bound to legend and secret mysteries. The author of this complex picture, Richard Spix, has created fractal art since 1996 in Florida (USA) where he works as a technical engineer.

"Fiddler Crab, opus 446" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).1894 viewsCrease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Fiddler Crab" origami work in this album.

"The PowerStar: Synergetic Sacred Geometry, " Warren Scott Fentress (2008)1893 viewsMagneBlocks, 20" x 20" x 20". "Platonic fundamental shapes like the tetrahedron and pyramid, which are culminated recursively into 'powered tetrahedrons & pyramids', are arranged into pentagonal forms that mimic the 5-fold geometry of flowers. I invented MagneBlocks because of the bicameral mind resonating with the fundamental consciousness waveforms that permeate spacetime." --- Warren Scott Fentress, Imaginatuer, Brookfield, CT

Circle 41871 viewsComputers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

"Lilacs--an Imaginary Inflorescence," by Anne M. Burns (Long Island University, Brookville, NY)1841 views"Inflorescence" is the arrangement of flowers, or the mode of flowering, on a plant--sometimes simple and easily distinguishable, sometimes very complex. "Lilacs" is an example of an imaginary inflorescence that I have created using computer graphics techniques. Two Java applets allow users to see and draw purely imaginary inflorescences at various stages using the recursive (repeatedly applied) functions. Download the code from either applet, and see photographs of real inflorescences several imaginary inflorescences, at http://myweb.cwpost.liu.edu/aburns/inflores/inflores.htm. --- Anne M. Burns (Long Island University, Brookville, NY)

"Magneto-2," by Reza Ali (Palo Alto, CA)1837 views18" by 24" print, 2011

This image is a snap-shot from a real-time interactive particle simulation using Lorentz's Law to define each particle's movements. The color palette, perspective, magnetic field placement, and rendering style were designed by the artist. Physics and mathematics define the piece's motion and overall pattern formation. --- Reza Ali (Palo Alto, CA, http://www.syedrezaali.com/)

"Overlapping Circles I," by Anne Burns, Long Island University, Brookville, NY (2008)1812 viewsDigital print, 13" x 12". "This is an iterated function system made up of Mobius Transformations, programmed in ActionScript. I began my studies as an art major; later I switched to mathematics. In the 1980's I bought my first computer and found that I loved programming and could combine my all of my interests: art, mathematics, computer programming and nature." --- Anne Burns, Professor of Mathematics, Long Island University, Brookville, NY

"ParaStar8," by Mary Candace Williams. Quilt copyright 2003 Mary Candace Williams; photograph by Robert Fathauer.1804 viewsThis quilt is is the third in a series of quilts based on the approximation of a parabola by drawing a series of straight lines. There were eight divisions of the orginal block which was then mapped onto a rhombus and repeated eight times for the complete quilt. The star part of the design was enhanced by the use of shades of color.

--- Mary Candace Williams

"The Vase," by Harry Benke (www.harrybenke.com)1775 views2010 Mathematical Art Exhibition Second Prize.

Giclee Print. 18" x 14.8", 2009. "The Vase" is composed of a digitally modeled vase with "Lilies" which are Dini's Surfaces. A surface of constant negative curvature obtained by twisting a pseudosphere is known as Dini's Surface. Imagine cutting the pseudosphere along one of the meridians and physically twisting it. Its parametric equations are: x=acos(u)sin(v); y=asin(u)sin (v); z=a{cos(v)+ln[tan(v/2)]}+bu, where 0<= u <= 2pi and 0< v< pi. Take a=1 and b=0.2. "I'm primarily an artist. My shadow is mathematics. I'm helpless at preventing mathematics from intruding in my work and it's delightful to have the body of mathematics to work with. My art attempts to produce a nexus between mathematical beauty and the beauty of the natural world to produce a satisfying aesthetic experience." --- Harry Benke (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.

Hilbert's Square-Filling Curve1774 views"Hilbert's Square-Filling Curve" by The
3DXM Consortium

In 1890 David Hilbert published a construction of a continuous curve whose image completely fills a square, which was a significant contribution to the understanding of continuity. Although it might be considered to be a pathological example, today, Hilbert's curve has become well-known for a very different reason---every computer science student learns about it because the algorithm has proved useful in image compression. See more fractal curves on the 3D-XplorMath Gallery.

--- adapted from "About Hilbert's Square Filling Curve" by Hermann Karcher

Hamid Naderi Yeganeh, "1,000 Line Segments (2)" (August 2014)1752 viewsThis image shows 1,000 line segments. For each i=1,2,3,...,1000 the endpoints of the i-th line segment are: (-sin(4πi/1000), -cos(2πi/1000)) and ((-1/2)sin(8πi/1000), (-1/2)cos(4πi/1000)). I created this image by running my program on a Linux operating system. --- Hamid Naderi Yeganeh