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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"Paths and Points," by James Mai (Illinois State University, Normal)Digital print, 15”x7.5", 2007.

All permutations (minus symmetrical redundancies) of alternating upward & downward semi-circular paths around & between 3 points. "I employ mathematically ordered shape relationships and carefully balanced color relationships in my compositions. These call for both objective logic and subjective perception in the decoding of each composition's organizing principles. My studio work is accomplished in both traditional painting media and digital prints." --- James Mai (Illinois State University, Normal)

"A 3D model of Costa’s Minimal Surface," by O. Michael Melko (Northern State University, Aberdeen, SD)Solid model of layered polymer resin created via stereolithography, 7 ” x 7” x 6”, 2005. Costa’s minimal surface is the first example of a complete, embedded minimal surface of finite total curvature to be discovered. This surface admits an explicit parameterization in terms of elliptic functions via the Weierstrass representation for minimal surfaces. The topology of the surface is that of a torus with three punctures, but its embedding is rather difficult to grasp visually from a typical graphical image. Hence we provide a rendering in the form of a solid model, the data for which was created with Mathematica. "As a differential geometer, I am interested in creating computer-generated forms of geometrical structures that are difficult to visualize. In addition to helping the viewer better grasp the underlying mathematics, the process of creating the work of art brings pleasure to the mathematical artist, who must be creative in his use of computational tools in order to achieve the desired outcome." --- O. Michael Melko (Northern State University, Aberdeen, SD) http://www3.northern.edu/melkom

"Sudoku 4B," by Kerry Mitchell (Phoenix College, Phoenix, AZ)Photographic print, 17" wide x 17" high, 2007. In this image, I brought the notion of a Sudoku puzzle to a 4 x 4 grid, where I used shapes instead of the digits 1 - 4. I retained the requirement that each element of the four-character alphabet appear once and only once in each row, column, and in each of the four 2 x 2 sub-grids. In addition, I added an element of layering: Each finished image is a composition of four layers, with each layer being its own solved Sudoku grid. "My work is composed primarily of computer generated, mathematically-inspired, abstract images. I draw from the areas of geometry, fractals, numerical analysis, and physics, and combine these ideas with image-processing technology. An overriding theme that encompasses my work is the wondrous beauty and complexity that flows from a few, relatively simple, rules. Inherent in this process are feedback and connectivity; these are the elements that generate the patterns. They also demonstrate to me that mathematics is, in many cases, a metaphor for the beauty and complexity in life. This is what I try to capture." --- Kerry Mitchell (Phoenix College, Phoenix, AZ) http://kerrymitchellart.com

"Cambridge Mathematical Sciences 200706," by Ralph Nevins (Artist, Ottawa, Ontario, Canada)Photograph, 11”x17”, 2009. The image is 12 pictures stitched into a 360 degree panorama, then a modified Rectangular to Polar transformation to produce the box. "Trained as a Computer Scientist (BCS), Work as an Engineer (MSc), and as a Professional Artist for 5 years. I create art because there is beauty in all things, and I enjoy exploring new techniques in camera and post processing. Getting people to view the world as a surprise is the fun part." --- Ralph Nevins (Artist, Ottawa, Ontario, Canada) http://ralph.ca

"401_06," by Don Relyea (Artist/Musician/Programmer/Inventor, Dallas, TX)Archival Print, 15"x15", 2009. The algorithm I used to make this image is based on the Hilbert space filling curve, discovered by mathematician David Hilbert. The image is drawn in a custom software program I wrote myself. This version of my program recursively subdivides spaces within the total space to be filled and runs the algorithm to fill the smaller spaces separately. Each smaller space is centered on a point on the larger curve causing the smaller renderings to intersect the larger one in interesting ways. "I write software to make art." --- Don Relyea (Artist/Musician/Programmer/Inventor, Dallas, TX) http://www.donrelyea.com

"Interweaving Rhythms-2," by Irene Rousseau (Artist/Art Historian, Summit, NJ)Layered drawing: Ink drawings on mylar and paper,16" X 16", framed 20" x 20", 2009. My aim in this layered drawing is to explore geometric patterns and shapes with interweaving rhythms which change spatial locations as they weave in and out from concave to convex. "My sculptures, paintings and drawings are metaphors for the complexity and underlying order reflected in many patterns in nature and the mathematical coherence found in natural forms. They represent invisible forces made visible. My works are an interpretation of how we perceive through our senses, nature and the structure of our physical world. They are also references to the idea of space and the intellectual understanding of the unseen. Technique : My hyperbolic sculptures are composed of tessellated mosaic patterns referring to the concept of infinity. They are handmade glass and hand cut tesserae. My paintings are acrylic paint on canvas and explore spatial structures found on the microscopic and macroscopic level. They are my vehicle for expressing the rhythms and energies 'found in the universe'." --- Irene Rousseau (Artist/Art Historian, Summit, NJ) irenerousseau.com

"Art of Fourier Space," by Ian Sammis (University of California, Davis)Print of digital art, 24”x20” (framed), 2008. This is the computed Fourier transform of a constant linear measure placed on a piecewise-linear approximation to the space-filling Sierpinski Curve. The curve itself is shown in the lower-left corner. The reduced art appears gray, but in the original each pixel has a hue determined by its complex phase. The transformation was computed by the Geometric Nonuniform Fast Fourier Transform. "Over the course of earning my Ph.D., I've become fascinated by the fact that in generating images for the most utilitarian of purposes (debugging, testing hypotheses, and the like) the most useful images are usually also the most aesthetically pleasing." --- Ian Sammis (University of California, Davis) http://math.ucdavis.edu/~isammis

"Hope," by Reza Sarhangi (Towson University, Towson, MD)Digital print, 16" X 20", 2008. "Hope" is an artwork based on the "Modularity" concept using triangles and rhombuses as its motifs in three colors. The "Modularity" concept has been presented in an article by Reza Sarhangi, Modules and Modularity in Mosaic Patterns, the Journal of the Symmetrion (Symmetry: Culture and Science), Volume 19, Numbers 2-3, 2008. Another article in this regard would be Sarhangi, R., S. Jablan, and R. Sazdanovic, Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical, and Theoretical Considerations, 2004 Bridges Proceedings. In the following figure, except for the corners with constant color, the two compound triangles (modules) are in a positive-negative color relationship with respect to each other. Using these two modules in a rotational fashion, results in the pattern in the artwork. "I am interested in Persian geometric art and its historical methods of construction, which I explore using the computer software Geometer's Sketchpad. I then create digital artworks from these geometric constructions primarily using the computer software PaintShopPro." --- Reza Sarhangi (Towson University, Towson, MD) http://geometricarts.googlepages.com/home

"Poincare berries," by Radmila Sazdanovic (Mathematical Sciences Research Institute, Berkeley, CA) Digital print, 20”x20”, 2009. The pattern consisting of triangles and circles introduced into the fundamental domain emphasizes four and six fold rotational symmetry of the (4,4,4,6) tessellation. The interplay of the white weave and the pattern reinforces the underlying structure. "My inspiration stems from the rich geometric structures found in tessellations of the hyperbolic plane and my area of research- knot theory. 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 (Mathematical Sciences Research Institute, Berkeley, CA) http://home.gwu.edu/~radmila/

"Rational Circles," by Stephen Schiller (Adobe Systems Inc., Oakland, CA)Digital Print, 24” by 15.6”, 2009. This image consists of a large number of circles. To describe the set of circles let [a,b,c,d] represent the circle whose points are the zeros of the bivariate polynomial p(x,y)=a(x^2+y^2)+bx+cy+d. If a, b, and c are relatively prime integers then I call the circle a "reduced rational" circle. The drawing then consists of reduced rational circles such that a^2+b^2+c^2 <= 9^2, as viewed through a rectangle whose lower left is (0.01,0.21667) and whose upper right is (0.395,0.46667). (The view box was mostly chosen for aesthetic reasons.) The darkness of each circle depends inversely on its radius and on the term a^2+b^2+c^2. Most of my mathematical art has its origins in images I make to help me understand the solution to some problem I am facing in my work as a computer scientist. There is great power in mathematical theorems that help us understand a complex set of objects. But sometimes such theorems hide, or at least allow us to temporarily ignore, the true complexity of a subject. This duality often comes up when one tries to actually implement a mathematical idea. Thus, I find myself interested in images that are a manifestation or rediscovery of the complexity that is inherent in even simple mathematical areas." --- Stephen Schiller (Adobe Systems Inc., Oakland, CA) http://stephenschiller.imagekind.com/

"K12 embedded in Genus 6," by Carlo Sequin (University of California, Berkeley)Plaster model, hand painted, 5" tall, 2004. The complete graph K12 connects 12 vertices with 66 lines. Of course, in the plane this graph cannot be drawn without many crossings. A surface needs to be at least of genus 6 to allow a crossing-free embedding. With this model I have attempted to find the embedding of highest possible symmetry. The model has the 12-fold symmetry of the oriented tetrahedron. The 44 countries bounded by the 66 lines, and colored differently, are all 3-sided. "My professional work in computer graphics and geometric design has also provided a bridge to the world of art. In 1994 I started to collaborate with Brent Collins, a wood sculptor, who has been creating abstract geometrical art since the early 1980s. Our teamwork has resulted in a program called “Sculpture Generator 1” which allows me to explore many more complex ideas inspired by Collins’ work, and to design and execute such geometries with higher precision. Since 1994, I have constructed several computer-aided tools that allow me to explore and expand upon many great inspirations that I have received from several other artists. It also has resulted in many beautiful mathematical models that I have built for my classes at UC Berkeley, often using the latest computer-driven, layered-manufacturing machines. My profession and my hobby interests merge seamlessly when I explore ever new realms of 'Artistic Geometry'." --- Carlo Sequin (University of California, Berkeley) http://www.cs.berkeley.edu

"Spirals in Chaos - The Dance in Four Movements: Design Drawing 1," by Mickey Shaw (Artist, Le Roy, KS)Design drawing (not computer generated ), 21" x 22" ( framed ), 2009. Spirals are curves emanating from central points, progressively growing further away as they revolve around the point. These drawings are unique, one of a kind renditions of spirals, but created in reverse direction from outer edges into a central point. Some variations resembling Sinusoidal, Archimedean and Hyperbolic spirals and even an occasional pseudospheres are created. Drawings are created on a weighted drawing board suspended from a pole with an attached arm holding a pen. The board is set in motion by hand. Drawings are manipulated by changing the motion of the drawing board. Finished drawings are scanned and printed. "My inspirations are drawn from nature, mathematics and science. These inspirations are combined with my own experiences and emotions creating a marriage between what is seen, what is known and what is felt internally. My goal, as artist, is to create for the viewer, visually, the concept that art, mathematics and science display a fundamental connection conveying the idea that all three encompass more than what can just be seen. I believe that art is an intrinsic aspect of all visual experiences and mathematics can provide a basis for understanding and recreating those same experiences. The spiral design drawings convey a two-dimensional visualization and exploration of this interconnection." --- Mickey Shaw (Artist, Le Roy, KS) http://FullLunaCreations.etsy.com/

"Spirals in Chaos - The Dance in Four Movements: Design Drawing 2," by Mickey Shaw (Artist, Le Roy, KS)Design drawing (not computer generated ), 21" x 22" ( framed ), 2009. Spirals are curves emanating from central points, progressively growing further away as they revolve around the point. These drawings are unique, one of a kind renditions of spirals, but created in reverse direction from outer edges into a central point. Some variations resembling Sinusoidal, Archimedean and Hyperbolic spirals and even an occasional pseudospheres are created. Drawings are created on a weighted drawing board suspended from a pole with an attached arm holding a pen. The board is set in motion by hand. Drawings are manipulated by changing the motion of the drawing board. Finished drawings are scanned and printed. "My inspirations are drawn from nature, mathematics and science. These inspirations are combined with my own experiences and emotions creating a marriage between what is seen, what is known and what is felt internally. My goal, as artist, is to create for the viewer, visually, the concept that art, mathematics and science display a fundamental connection conveying the idea that all three encompass more than what can just be seen. I believe that art is an intrinsic aspect of all visual experiences and mathematics can provide a basis for understanding and recreating those same experiences. The spiral design drawings convey a two-dimensional visualization and exploration of this interconnection." --- Mickey Shaw (Artist, Le Roy, KS) http://FullLunaCreations.etsy.com/

"Spirals in Chaos - The Dance in Four Movements: Design Drawing 3," by Mickey Shaw (Artist, Le Roy, KS)Design drawing (not computer generated ), 21" x 22" ( framed ), 2009. Spirals are curves emanating from central points, progressively growing further away as they revolve around the point. These drawings are unique, one of a kind renditions of spirals, but created in reverse direction from outer edges into a central point. Some variations resembling Sinusoidal, Archimedean and Hyperbolic spirals and even an occasional pseudospheres are created. Drawings are created on a weighted drawing board suspended from a pole with an attached arm holding a pen. The board is set in motion by hand. Drawings are manipulated by changing the motion of the drawing board. Finished drawings are scanned and printed. "My inspirations are drawn from nature, mathematics and science. These inspirations are combined with my own experiences and emotions creating a marriage between what is seen, what is known and what is felt internally. My goal, as artist, is to create for the viewer, visually, the concept that art, mathematics and science display a fundamental connection conveying the idea that all three encompass more than what can just be seen. I believe that art is an intrinsic aspect of all visual experiences and mathematics can provide a basis for understanding and recreating those same experiences. The spiral design drawings convey a two-dimensional visualization and exploration of this interconnection." --- Mickey Shaw (Artist, Le Roy, KS) http://FullLunaCreations.etsy.com/

"Spirals in Chaos - The Dance in Four Movements: Design Drawing 4," by Mickey Shaw (Artist, Le Roy, KS)Design drawings (not computer generated ), 21" x 22" ( framed ), 2009. Spirals are curves emanating from central points, progressively growing further away as they revolve around the point. These drawings are unique, one of a kind renditions of spirals, but created in reverse direction from outer edges into a central point. Some variations resembling Sinusoidal, Archimedean and Hyperbolic spirals and even an occasional pseudospheres are created. Drawings are created on a weighted drawing board suspended from a pole with an attached arm holding a pen. The board is set in motion by hand. Drawings are manipulated by changing the motion of the drawing board. Finished drawings are scanned and printed. "My inspirations are drawn from nature, mathematics and science. These inspirations are combined with my own experiences and emotions creating a marriage between what is seen, what is known and what is felt internally. My goal, as artist, is to create for the viewer, visually, the concept that art, mathematics and science display a fundamental connection conveying the idea that all three encompass more than what can just be seen. I believe that art is an intrinsic aspect of all visual experiences and mathematics can provide a basis for understanding and recreating those same experiences. The spiral design drawings convey a two-dimensional visualization and exploration of this interconnection." --- Mickey Shaw (Artist, Le Roy, KS) http://FullLunaCreations.etsy.com/