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.

"Seven-Color Torus Series in Bead-Crochet: Bracelet 1," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.htmlMay 10, 2010

"Seven-Color Torus Series in Bead-Crochet: Bracelet 2," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.htmlMay 10, 2010

"Seven-Color Torus Series in Bead-Crochet: Bracelet 3," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.html May 10, 2010

"Seven-Color Torus Series in Bead-Crochet: Bracelet 4," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.html May 10, 2010

"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) May 10, 2010

"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 May 10, 2010

"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 May 10, 2010

"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 May 10, 2010

"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 May 10, 2010

"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 May 10, 2010

"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 May 10, 2010

"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 May 10, 2010

"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/ May 10, 2010

"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/ May 10, 2010