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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"Ellipse Lace," by Susan McBurney (2005)Computer-generated graphic art; digital print, 14" x 14". "This complex and intricate design is created very simply from just one element--the ellipse. The width and height are varied and the sets are rotated, but the result is not at all what one might expect. In particular, the interior circles are generated entirely by the interaction of the parts. There are no circles drawn at all and the complexity of the design is entirely natural and unpredicted. It is my intention to use the computer as a tool to generate designs that are not only aesthetically pleasing, but that also reveal the order, structure and beauty inherent in mathematical objects. Additionally, if an attractive design can be made from the simplest of elements, then the generating process itself becomes an object of beauty as well. Complexity from a simple beginning via an elemental algorithm is a common, fascinating and universal process."--- Susan McBurney, Western Springs, IL

"Uniqueness--and Infinity," by Jo Niemeyer (2008)Prints, 20" x 20". "To realize the concept of 'uniqueness' in art is a true challenge. And no easy task for an artist. This could be said for the concept 'infinity' as well. As a visual experiment with both of them, this problem is shown systematically in two steps in the following two graphics: The top image shows the overlapping of two geometrical grids. The size of the mesh corresponds to the relation 1 : 0.625. Or the Fibonacci numbers 5 and 8. There are nine grid elements, which overlap accurately. Furthermore the constellations of overlappings reiterate themselves. The two grids behave periodically. There is infinity - but no uniqueness. The bottom image shows the overlapping of two grids as well. Very similar to the top image, but the size of their meshes correspond here exactly to the relation of the golden section. 1 : 0.6180339... As the last number is an irrational number, the two grids behave aperiodically. Only the upper left two grid elements overlap accurately. Each overlapping constellation of the elements is unique, even if the size of the grid would be extended to infinity!" --- Jo Niemeyer, Freelance artist, Schluchsee, Germany

"The Path Crumpled Paper Takes," by Jeanette Powers, Rockhurst University, Kansas City, MO (2008)Ink and paper, 11" x 15". "A classic example to explain fractal dimension is the piece of crumpled paper. In this example, one takes a sheet of paper to be 2 dimensional (ignoring the very thin thickness). This then is a good representation of the mathematical plane. However, if we crumple the paper into a ball, as seen below the frame, it seems to take on a volume, or third dimension. Now, there is a meta-level to the inter-dimensionality of this system. If one flattens the paper back into the two dimensional sheet of paper, then one can draw a continuous line ( in blue) of all the folds that happened during the crumpling process. Now a line is considered to be one dimensional, but is the space this line takes up really best described with only one dimension?" --- Jeanette Powers, Student, Physics and Math Department, Rockhurst University, Kansas City, MO

"Pulse," by Jeanette Powers, Rockhurst University, Kansas City, MO (2008)Acrylic, 6" x 24". "This piece explores Hausdorff Dimension. Chaos and dynamical systems collapse in ordered ways. A nebula coalescing into a galaxy, a frozen molecule tossing through the tumult and falling as a six-sided crystal, the Mandelbrot Set. As an artist, I've tried to use chaotic interactions as a tool to express the limitations of our control and the beauty of chaos. This painting uses cellophane crushed into wet pigment to create the random patterning of the surface. The result is a chaotic landscape reminiscent of leaves, cells, rivulets, the cracked dirt of arid lands. All chaotic processes which leave a recognizable mark. The pattern is not exact, but exhibits self-similarity." --- Jeanette Powers, Student, Physics and Math Department, Rockhurst University, Kansas City, MO

"Caught in a Dual Net," by Radmila Sazdanovic, The George Washington University, Washington, DC (2008)Digital print, 16" x 16". "This computer graphic represents three superimposed tessellations. The edges of a tessellation (6,6,7) are hidden below two nets consisting of tessellations (7,7,7) and (3,3,3,3,3,3,3), both dual to the original one. My inspiration stems from the rich geometric structures found in tessellations of the hyperbolic plane. Mathematical objects can be manipulated in many ways (superimposing, dualizing, breaking symmetry) to create aesthetically pleasing computer graphics brought to life by the unusual combination of colors." --- Radmila Sazdanovic, Graduate student, The George Washington University, Washington, DC

"Chinese Dragon," by Nathan Selikoff (2007)Lightjet print, 18" x 24". "Underlying this artwork is a two-dimensional plot of the 'typical behavior' of a chaotic dynamical system, a strange attractor. The base image is computed with a set of iterated functions, which serve as a numerical approximation to integrating the underlying differential equations. The iterated functions contain four coefficients, which are controlled by sliders in interactive custom software and control the appearance of the attractor. Once the particular form is chosen, it is rendered as a high-resolution 16-bit grayscale image, colorized using gradient mapping and edited to enhance contrast, control composition, and add special effects. I love experimenting in the fuzzy overlap between art, mathematics, and programming. The computer is my canvas, and this is algorithmic artwork--a partnership mediated not by the brush or pencil but by the shared language of software. Seeking to extract and visualize the beauty that I glimpse beneath the surface of equations, I create custom interactive programs and use them to explore algorithms, and ultimately to generate artwork. " --- Nathan Selikoff, Artist, Orlando, FL

"Star Birth," by Nathan Selikoff (2007)Lightjet print, 24" x 18". "Underlying this artwork is a two-dimensional plot of the 'typical behavior' of a chaotic dynamical system, a strange attractor. The base image is computed with a set of iterated functions, which serve as a numerical approximation to integrating the underlying differential equations. The iterated functions contain four coefficients, which are controlled by sliders in interactive custom software and control the appearance of the attractor. Once the particular form is chosen, it is rendered as a high-resolution 16-bit grayscale image, colorized using gradient mapping and edited to enhance contrast, control composition, and add special effects. I love experimenting in the fuzzy overlap between art, mathematics, and programming. The computer is my canvas, and this is algorithmic artwork--a partnership mediated not by the brush or pencil but by the shared language of software. Seeking to extract and visualize the beauty that I glimpse beneath the surface of equations, I create custom interactive programs and use them to explore algorithms, and ultimately to generate artwork." --- Nathan Selikoff, Artist, Orlando, FL

"Figure-8 Knot," by Carlo H. Sequin, University of California, Berkeley (2007)Second Prize, 2009 Mathematical Art Exhibit. Bronze, 9" tall. "The Figure-8 Knot is the second simplest knot, which can be drawn in the plane with as few as four crossings. When embedded in 3D space it makes a nice constructivist sculpture. This particular realization has been modeled as a B-spline along which a crescent-shaped cross section has been swept. The orientation of the cross section has been chosen to form a continuous surface of negative Gaussian curvature." --- Carlo H. Sequin, Professor of Computer Science, EECS Computer Science Division, University of California, Berkeley

"Chinese Button Knot," by Carlo H. Sequin, University of California, Berkeley (2007)Bronze, 8" tall. "The Chinese Button Knot is a nine-crossing knot, number 9-40 in the knot table. It actually has more symmetries than one would infer from the usual depiction in these tables. This has been brought out in this 3D sculpture, which has one 3-fold and three 2-fold rotational symmetry axes. It has been implemented as an alternating over-under path on the surface of a sphere, realized by a ribbon of continuous negative Gaussian curvature." --- Carlo H. Sequin, Professor of Computer Science, EECS Computer Science Division, University of California, Berkeley

"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, MO

"Universe Before Big Bang," by Clifford Singer, Clark County School District, Las Vegas, NV copyright 1989Acrylic on plexiglass, relief, 36" x 36" x 2". "This painting as a model entitled, Universe Before Big Bang, 1989, is intended to reconstruct the universe prior to the Big Bang. My concept in 1989 was to take a snapshot of the universe encapsulated in a non-Euclidean square. Thus, matter is then present before the Big Bang. Origins of the cosmos are found in supersymmetries and further understanding of concepts for their elucidation. As an artist and geometer 'infinity' has taken an important place in my life in terms of abstraction. My art combines both ancient and modern mathematical foundations ranging from Pythagoras to Einstein." --- Clifford Singer, Artist/Fine Art Teacher, Clark County School District, Las Vegas, NV

"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, Australia

"Equinox," by Anna Ursyn, University of Northern Colorado, Greeley (2008)Fortran, photosilkscreens, photolithographs, photographs, etc., 8" x 10". "I explore dynamic factor of line. I find computers to be a perfect tool to explore the regularity of nature. I use the computer on different levels. First I draw abstract geometric designs for executing my computer programs. Then I add photographic content using scanners and digital cameras. The programs that produce two-dimensional artwork serve as a point of departure for photolithographs and photo silkscreened prints on canvas and paper. All of these approaches are combined for image creation with the use of painterly markings." --- Anna Ursyn, Professor, University of Northern Colorado, Greeley

"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 Kong

"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, Hungary