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.

 "The Three Gates," by Benjamin Wells (University of San Francisco, CA)Inkjet on transparencies, mounted on acrylic layers in an acrylic frame hung by a laced beadchain, 11" x 9", hung by a beadchain, plus 11"x7" explanatory placard 2009 (updated from 2002). Symbols from logic map a classical aphorism about watching one's tongue into a visually recursive statement. In addition, the colors of red and green play on the binary nature of electronic gates. An accompanying placard gives the aphorism and lists the symbols used. "The art offered is a melding of symbolism from science and mysticism. The flexibility of computer-aided design and execution supports this blend of ancient and modern expression. I used to think in threes because my name ends in III. (For more about small numbers, see Michael Schneider’s 'Constructing the Universe.') Although I am now partial to 8, 17, 36, 1111, and 10^10, I wholly support only 1 alone. But three things can start a sequence, give a contrast or equivalence, or triangulate. Here they pose visual riddle. Math is fun, and art can help make that clear. When it can also take a supportive, spiritual, inspirational, cooperative color, then it is a harbinger of a new humanity. I hope to make art that way." --- Benjamin Wells (University of San Francisco, CA) May 10, 2010
 "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 May 10, 2010
 "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/ May 10, 2010
 "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/May 10, 2010
 "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/May 10, 2010
 "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/ May 10, 2010
 "Cut Space Series # 76" ©, by Clifford Singer (Artist/Teacher, Las Vegas, NV)Acrylic on Plexiglas, 36" x 48", 2008. "As a model, this painting represents a space time continuum, the singularity of beginning and end points are of an anomaly. In the painting I suggest the dynamics of a space that continually expands the structural space and geometrical fragmentation in space-time from a Big Bang. Line segments in the universe are interconnected, even when they appear to be separate from one another. In space-time curvature the geometrical strain illustrates a deformation of space. In the art one can identify a time axis with our world line. So, pursued through painting, works of art as a structured space perceived in space-time yields the image. 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." http://www.lastplace.com/EXHIBITS/VIPsuite/CSinger/ . All Copyrights Reserved to Clifford Singer. --- Clifford Singer (Artist/Teacher, Las Vegas, NV) May 10, 2010
 "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
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 Art & Music, MathArchives Geometry in Art & Architecture, by Paul Calter (Dartmouth College) Harmony and Proportion, by John Boyd-Brent International Society of the Arts, Mathematics and Architecture Journal of Mathematics and the Arts Mathematics and Art, the April 2003 Feature Column by Joe Malkevitch Maths and Art: the whistlestop tour, by Lewis Dartnell Mathematics and Art, (The theme for Mathematics Awareness Month in 2003) MoSAIC - Mathematics of Science, Art, Industry, Culture Viewpoints: Mathematics and Art, by Annalisa Crannell (Franklin & Marshall College) and Marc Frantz (Indiana University) Visual Insight, blog by John Baez