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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"A cmm Pattern from Peppers on a Cutting Board," by Frank A. Farris, Santa Clara University, CAInk jet on paper.

The bright oranges in this cmm pattern come from a photograph of assorted chopped peppers, with collard greens and the glint of the knife as nice contrasts. --- Frank A. FarrisJun 21, 2013

"Tiffany Glass from a Mountain Gentian and its Negative," by Frank A. Farris, Santa Clara University, CAInk jet on paper, 2012.

Using a composite photograph of a mountain gentian juxtaposed with its negative to produce an intense contrast of purple with the original green, I used just the right waves to make wallpaper with horizontal mirrors, vertical glides, and 2-fold rotational symmetry. In the notation of the International Union of Crystallographers, the symmetry group of this pattern is pmg, but if we allow color-swapping transformations as generalized symmetries, the larger group is cmm, so this pattern type is called cmm/pmg. The "Seeing Symmetry" virtual exhibition at includes more works and details. --- Frank A. Farris Jun 21, 2013

"Turtle, opus 269," by Robert J. LangOne uncut square of Origamido paper, 5", composed 1992, folded 2002

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. LangMay 22, 2013

"Thataway, opus 570," by Robert J. LangOne uncut dollar, 2", composed and folded 2010

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. LangMay 22, 2013

"Rattlesnake, opus 539," by Robert J. LangOne uncut rectangle of Thai unryu paper, 8", composed and folded 2008

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. Lang May 22, 2013

"QuezadaPot13, opus 590," by Robert J. LangOne uncut triskadecagon of Mexican yucca paper, 8", composed and folded 2009

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. Lang May 22, 2013

"Praying Mantis, opus 416," by Robert J. LangOne uncut square of paper, 4", composed and folded 2002

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. LangMay 22, 2013

"Locust," by Robert J. LangOne uncut square of Origamido paper, 3", composed and folded 2004

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. LangMay 22, 2013

"Grizzly Bear, opus 433," by Robert J. LangOne uncut square of Korean hanji, 8", composed 2002, folded 2003

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. LangMay 22, 2013

"C. P. Snow, opus 612," by Robert J. LangOne uncut square of Korean hanji, 10", composed and folded 2009

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. LangMay 22, 2013

"Basset Hound, opus 212," by Robert J. LangOne uncut square of kozo paper with lnclusions, 8", composed 1988, folded 2012

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation. --- Robert J. LangMay 22, 2013

"Crystal Morphohedron," by Stephen Wilmoth (University of California, Berkeley)6" x 6"x 6" closed, 6" x 6"x 18" opened, Clear acrylic, 1970

My work is directed at demonstrating the amazing interrelationship of the Regular Polyhedrons (Platonic Solids plus Kepler/Poinsot), and the Golden Ratio (1.618). Blending these together creates the three dimensional projection of the Fibonacci numbers. I call this phenomenon the Morphohedron. Transparent Dodecahedron that open to reveal clear cube inside that opens to allow a tetrahedron/octahedron to come out, which opens to reveal the inner icosahedron. All the Regular Solids (Platonic Solids) are here harmoniously nested. -- Stephen WilmothMay 16, 2013

"Convergence?", by Robert Spann (Washington, DC)11" x 14" including frame, Digital Print, 2012

Computer graphics allows one to see both the numerical and aesthetic properties of dynamical systems. Recently I became interested in the properties of complex functions in which a complex variable is raised to a complex, rather than an integer exponent. I have also been analyzing complex polynomials that have no attracting fixed points. This image is produced by applying Newton's method for root finding to the complex function (z^(2+3i)-.09)*(z^(2-3i) -.09).The white areas are points in the complex plane where this function does not converge to any root. The background is produced using Perlin noise functions. -- Robert SpannMay 16, 2013

"Double Boy Klein Bottle," by Carlo Séquin (University of Califnornia, Berkeley)6" x 8" x 7", FDM Model (blue ABS plastic), 2012

Boy's surface, a compact model of the projective plane, with a small disk removed is topologically equivalent to a Möbius band. Every Klein bottle can be composed of two Möbius bands that are glued together by their edges. In this model a Klein bottle is created by gluing together two mirror images of a 3-fold symmetrical Boy surface with a disk removed from its pole. The result is a Klein bottle with S6 symmetry, showing six of the "inverted sock" openings characteristic of the classical Klein bottle. -- Carlo SéquinMay 16, 2013

"Seifert surfaces for torus knots and links," by Saul Schleimer (University of Warwick, UK) and Henry Segerman (University of Melbourne, Australia)Four pieces: 111mm x 111mm x 105mm, 125mm x 130mm x 99mm, 125mm x 125mm x 118mm, 103mm x 103mm x103mm, PA 2200 Plastic, Selective-Laser-Sintered, 2012

As elegantly discussed in Ghys' 2006 ICM plenary talk, the natural parameterization of the Seifert surface for the trefoil knot uses Eisenstein series of lattices in the plane. This was generalized by Milnor to all (p,q) torus knots; he replaces Eisenstein series by certain fractional automorphic forms. Tsanov reduces the construction of these forms to finding an analytic description of the universal covering of the orbifold S^2(p,q,infinity) by the hyperbolic plane. Mainly following Lehner, we find a Fourier series for the covering map. Combining these ideas, we obtain a map from a hyperbolic triangle T_H, with angles pi/p, pi/q, and 0, to a domain T_S in S^3; rigid symmetries of T_S in S^3 generate the Seifert surface. Using Schwarz-Christoffel theory we uniformize T_H by a Euclidean triangle T_E having angles pi/p, pi/q and pi(1-1/p-1/q). In this way we transfer decorations on T_E to the Seifert surface; for these sculptures we use a subdivision of T_E into 15 congruent triangles. -- Saul Schleimer and Henry SegermanMay 16, 2013