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.

"Super Buckyball of Genus 31," by Bih-Yaw Jin (National Taiwan University, Taipei)546 views20" x 20" x 20", Plastic beads, 2011

Joining with students and teachers of the Taipei First Girls High School in November 2011, we made two bead models of super buckyball, a polyhedron of genus 31. Each vertex in this model is itself a buckyball punched with three holes and then connects to three neighbored vertices by three shortest carbon nanotubes. We can also view structure as the second level Sierpinski buckyball, which can be extended arbitrarily to infinity. -- Bih-Yaw Jin

"Crystal Morphohedron," by Stephen Wilmoth (University of California, Berkeley)522 views6" 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 Wilmoth

"Meandering Eightfold Path," by Margaret Kepner (Washington, DC)498 viewsFlat: 18" x 18"; Folded: 3" x 6" x 0.5", Archival Inkjet on Paper, 2012

This work is a modified version of an earlier design – The Eightfold Path. Colors have been added, and the flat 2D print has been deconstructed into a folding book format. The underlying design is a visual presentation of the five non-isomorphic groups of order eight: C8, C2 x C4, C2 x C2 x C2, D4, and Q8. It employs a visual vocabulary derived from a traditional quilt pattern, Drunkard’s Path. Each of the small shapes used in the design is a quarter circle in a square, scaled so that its area equals the square’s residual area. The 36-page book structure is created from a single sheet of paper through a series of cuts and folds. A continuous meander folding path is followed, with varying length fold-sequences, and no beginning nor end. When it is fully folded up, the book assumes a double-square footprint. A smaller-scale meander path, which would result in a continuous 144-page book, is expressed through color accents. -- Margaret Kepner

"540º," by Ayelet Lindenstrauss Larsen (Indiana University, Bloomington)474 views1" x 8" x 8", Cotton, silk, dissolvable muslin, 2003

This Möbius band is twisted three times, each time by 180 degrees, before its short sides are attached to each other. Topologically, it is the same as the usual Möbius band where the band is twisted only once before attaching the short sides to each other; it is just embedded in 3-space in a different way. The edge of a Möbius band consists of one long circle. If you follow the edge of this Möbius band, you will go through the colors of the rainbow, from red to purple, and then transition to red through red-purple and begin the cycle again. -- Ayelet Lindenstrauss Larsen

"White hyperbolic disk," by Gabriele Meyer (University of Wisconsin, Madison)472 views20" x 17" x 17", Polyester yarn and shaped line, 2012

This started out as a white disk crocheted in a spiral fashion. By making more than the necessary stitches, it takes on this characteristic wavy form. The hyperbolic disk is the most basic form of a hyperbolic surface, other possible shapes include blossoms and algae. I particularly like the play of light and shadow on this hyperbolic surface. That's the reason why I made this one just pure white. The holes created by double and triple stitches allow further light and shadow play. -- Gabriele Meyer

"Blue Sun," by Mojgan Lisar (Enschede, The Netherlands) and Reza Sarhangi (Towson University, MD)450 views16" x 16", combination of hand painting and computer work, 2012

The Blue Sun is a collage of two different Persian works of art, both with deep mathematical roots: Tiling and Tazhib. The mosaic design on the back is a two-level self-similar tiling that has been made based on the decagram. This structure possesses a 10-fold rotational symmetry. This symmetry can be expanded in all directions using the five Sâzeh motifs introduced in "Polyhedral Modularity in a Special Class of Decagram Based Interlocking Star Polygons" by Reza Sarhangi, the 2012 Bridges Proceedings. The tiling at the center presents the tiling of a decagram that follows the same rules as the larger tiling in the back. The front image is a decagram Tazhip. In a traditional Persian Tazhib, one can find mathematical ideas and concepts, such as symmetries, spirals, polygons and star polygons. -- Mojgan Lisar and Reza Sarhangi

"Inlaid Wooden Box of Makoto Nakamura's Tessellations," by Kevin Lee (Normandale Community College, Saint Paul, MN)447 views4" x 4" x 4", Wood: Cherry, Maple, Walnut, Oak, Butternut, Mahogany, 2012
Best textile, sculpture, or other medium, 2013 Mathematical Art Exhibition

Makoto Nakamura has created a collection of tessellation designs that rival M. C. Escher’s collection. The six designs on this box represent the asymmetric, isohedral Heesch tile types: TCCTC, TG1G2TG2G1, TCCTGG, C3C3C3C3C3C3, and CC4C4C4C4. -- Kevin Lee

"Convergence?", by Robert Spann (Washington, DC)440 views11" 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 Spann

"Starry Pines," by Charles Redmond (Mercyhurst University, Erie, PA)420 views16" X 16", Generated by code written in the Context Free Art language, no photo-editing software used, 2011

Every program written in the Context Free Art language may be considered to be a context free grammar for creating images. Thus, when one programs in this language, one is "inventing" grammars. Any image produced with such a grammar may be considered to be a legal sentence in the grammar. If randomness is introduced into the program, then there are many different legal sentences or images, and one is producing generative art. When I created Starry Pines, I was studying tree creation with Context Free Art while at the same time experimenting with a technique of mine for creating star clusters and galaxies. I put them together for this work, along with a recursive icy swirl added to the rules for the trees. -- Charles Redmond

"Pointed Planes and Bézier Beaks," by Harry Rubin-Falcone (Oberlin College, Oberlin, OH)418 views15.5" x 20", Digital Print, 2012

Each figure is created with a series of Bézier curves. Because there is space between each curve, you can see some parts of the curves behind the others, which gives each figure a translucent and three dimensional look. The curves can be thought of as lying on a plane, which means each figure is a representation of a folded-over plane that comes to a point. -- Harry Rubin-Falcone

"Double Boy Klein Bottle," by Carlo Séquin (University of Califnornia, Berkeley)417 views6" 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 Séquin

"Seifert surfaces for torus knots and links," by Saul Schleimer (University of Warwick, UK) and Henry Segerman (University of Melbourne, Australia)416 viewsFour 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 Segerman