Math ImageryThe 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.

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"Torus Knot (5,3)," by Carlo H. Séquin (University of California, Berkeley)Second Place Award, 2011 Mathematical Art Exhibition

Bronze with silver patina, 10" × 8" × 16", 2010

Torus knots of type (p,q) are simple knots that wind around an invisible donut in a regular manner – p times around the hole, and q times through the hole. By using a somewhat more angular shape for the donut and a variable-size, crescent-shaped cross section for the ribbon, this mathematical construct can be turned into a constructivist sculpture. The challenge was to find a way to make a mold for casting this highly intertwined structure. The solution was to cast three identical pieces, which were then threaded together and welded to each other. --- Carlo H. Séquin (http://www.cs.berkeley.edu/~sequin/)
Mar 10, 2011
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"Hyperbolic Twistslug," by Mickey Shaw (Le Roy, KS)Fiber, 9" x 22" x 13”, 2009

This crocheted fiber soft sculpture is based on non-Euclidean geometry. It represents a variation of the hyperbolic plane ruffle effect. The piece was created using basic crochet stitches, which were increased at a rate great enough to create exponential growth. Attention was given to pushing the construction into a form of varying volume, irregular shape and an integration of pattern and color. The end result is simultaneously geometric in its basic nature and organic in its form. This creation used over two pounds of fibers. The structure is malleable, allowing the form to morph into numerous shapes. The hyperbolic soft sculpture is a further exploration of what forms can evolve in combining hard-edged geometric concepts with the fluid, textural aspects of fiber and stitches. This combination creates a three-dimensional visual and mental juxtaposition of the interconnection of the two elements. --- Mickey Shaw (http://FullLunaCreations.etsy.com/)

Mar 10, 2011
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"Quarthead," by Bob Sidenberg (Minneapolis, MN) Wood, 16" h x 16" w x 16" d, 2002

This one is trying to be a rhombic dodecahedron, but hasn't quite emerged from its tetrahedral beginnings. --- Bob Sidenberg (http://www.silkmountain.net)
Mar 10, 2011
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"Fractal," by Bradford Hansen-Smith (Chicago, IL)Folded 9" paper plates, 8" x 8" x 8", 2009

This is formed using twelve 9" paper plates all folded to the same equilateral triangular grid and reconfigured to the same design with slight variation between the four units that make the center tetrahedron pattern and the eight circles that form the outer cubic arrangement. This is one of many explorations using fifty-four creases rather than the twenty-four creases I usually work with. The higher frequency triangular grid allows greater complexity in a single circle which when combined in multiples forms designs that would not be possible otherwise. --- Bradford Hansen-Smith (http://www.wholemovement.com)
Mar 10, 2011
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"Chaos - The Movie," by Susan Happersett (Jersey City, NJ)Video, 19" flat panel monitor (about 18" by 12"),

"Chaos - The Movie" is a stop-motion animation movie in which I create a line drawing based on Chaos Theory. The drawing--and the movie--were made over a period of six months. Music is an original composition made for the movie by Max Schreier. Meejin Hong did the video editing. --- Susan Happersett (http://www.happersett.com)
Mar 10, 2011
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"Fibonacci Scroll," by Susan Happersett (Jersey City, NJ)Video, 19" flat panel monitor (about 18" by 12"), 2010

Fibonacci Scroll is a stop-motion animation of a long scroll drawing based on the Fibonacci Sequence. Susan Happersett has been creating mathematical, counted marking drawings for a number of years, but this is the first time her markings come to life. The sound track was composed specifically for the movie by Robert van Heumen, an accomplished composer and musician. --- Susan Happersett (http://www.happersett.com)
Mar 10, 2011
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"Ball and Chain," by George Hart (Museum of Mathematics, New York, NY)Nylon (selective laser sintering), 6" x 6" x 6", 2009

Ball and Chain is a ball made of triangular chain mail mesh containing twelve flexible regions in a rigid dodecahedral framework. There are 3,722 small rings, which interlock to form a sphere with chiral icosahedral symmetry. At 920 places, six triangles meet, but at 12 special points (at the center of the twelve dimples) only five triangles meet. The ball does not collapse down to a disk because the dodecahedral structure of ribs (made by having some of the rings lock to form a skeleton) is rigid. But in twelve circular regions the rings are free to hang freely. No matter how it is turned, the top six regions hang to make concavities while the lower six regions are convex and blend in with the overall spherical form. The complete structure was created as one unit in its assembled state by selective laser sintering. --- George Hart (http://www.georgehart.com)
Mar 10, 2011
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"Hyperbolic Planes Take Off!," by Vi Hart (New York, NY)Oil paint on canvas, 20" x 16", 2010

What does it look like when you crease the hyperbolic plane? This painting is an attempt at visualizing simple origami done with hyperbolic paper. Each plane has a mountain and valley fold perpendicular to each other. Done with your average Euclidean sheet of paper, it would be impossible to have both creases folded at a non-zero angle, but the hyperbolic plane can fold both ways at once. The creased plane can then be manipulated into different "birds", or so I imagine. --- Vi Hart (http://vihart.com)
Mar 10, 2011
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"Hyperbolic Twist: Forever in Memory," by Charlotte Henderson (A.K. Peters, Natick, MA)Acrylic yarn and glass beads, 6 × 6 × 4 inches (3d work), 2010

This model is a hyperbolic Möbius band. The starting “spine” consists of 20 chain stitches, and the outer single edge has over 1600 single crochet stitches. The negative curvature of the surface allows the width of the band to be much greater than if the curvature were zero. The surface can move freely through the “hole” in the center. The bead work highlights the nonorientability of the surface. In isolated sections, it looks as if the beads are on two sides of the band; but if one traces the line of beads, one will return to the chosen starting point having traced all of the beads. The same amount of yarn is used for the red and the pink, to display the exponential growth of the surface. (There is a greater amount of white yarn, to have a constant final row for aesthetic purposes.) The color scheme arose from the fact that, at an earlier construction stage, the silhouette of the model resembles a heart. --- Charlotte Henderson

Mar 10, 2011
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"Walking the Water's Edge," by Diane Herrmann (University of Chicago, IL)Needlepoint on canvas, 14" x 14", 2009

In this piece, the line imitates the edge of a wave on the shore. To make this wave look realistic, we used a mathematical curve that models the way a wave breaks on the beach. To be mathematically precise, we work with the sum of two trigonometric curves to show the action of water as it sloshes over itself in the push to get on the shore. The graph that defines the line of the Florentine Stitches is a close approximation to the curve: 
f (x) = 5 sin x + 4 cos (2x + π/3).
The technique of thread blending creates the shading of the wave. Freeform eyelet stitches mimic the foamy edge of the wave and beads add sparkle. A single starfish is added in Bullion Knots. --- Diane Herrmann
Mar 10, 2011
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"Proof Mining; The Gordian Geometric Knot," by Karl H(einrich) Hofmann (Tulane University, New Orleans, LA, and TU Darmstadt, Germany)Pencil, felt pen, tempera, 20" x 24", 2009

Two artworks from the Darmstadt Colloquium Poster Project, framed together.
 The techniques used are pencil, felt pen, tempera. The calligraphy of the posters is obtained with a Copic felt pen in a typography specifically developed for this purpose. The texts are prescribed by the departmental colloquium program determined one semester in advance.
 A complete collection of scans of the last 12 semesters can be inspected on my website. --- Karl H(einrich) Hofmann (http://www3.mathematik.tu-darmstadt.de/index.php?id=241)
Mar 10, 2011
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"Longest and Shortest Crease-B," by Sharol Nau (Northfield, MN)Folded book, 12.5” x 9” x 6”, 2010

For this book-sculpture of several hundred pages, the shortest crease was obtained by folding the pages without separating them from the binding. Also the folding process began in the middle in an effort to achieve a symmetrical design. --- Sharol Nau (http://www.sharolnau.snakedance.org)
Mar 10, 2011
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"DART," by Jo Niemeyer (Schluchsee, Germany)Archival inkjet print, 20" x 20", 2010

DART 
... as a very simple game by Jo Niemeyer 2010

. Two dart players, A and B, are facing this "image", whose area is split 1 : 0.618.. into white and black. This two basic elements are rotated in 90° increments. The winner is, who aims first a black part.
 Since we have two equal partners and an uneven distribution of the "target", one would think, that this is not a fair game. But it is! Because A as the "majority", and consequently B as the "minority", transferred their inequality onto the "court". The ratio of the two playing partners is 1:1.
With this harmonious proportionality there is exactly the same chance to win for both players A and B! The Swiss mathematician Hans Walser mentions for the justice condition, the formula p = 1/2*(3-sqrt(5)). And with sqrt(5), we have the golden section in this game, which ensures equity between different partners. 
This is also a very fair game! 
Or a piece of art, which ensures harmony and balance. --- Jo Niemeyer (http://www.jo.niemeyer.com)
Mar 10, 2011
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"Bucky Madness," by Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR)Digital print on archival paper, 20" x 20", 2010

This is my response to a request to make a ball and stick model of the buckyball carbon molecule. After deciding that a strict interpretation of the molecule lacked artistic flair, I proceeded to use it
as a theme. Here, the overall structure is a 60-node truncated icosahedron (buckyball), but each
node is itself a buckyball. The center sphere reflects this model in its surface and also recursively
reflects the whole against a mirror that is behind the observer.

I was recently surprised to read in David Richeson's book, Euler's Gem, that Legendre proved
Euler's Formula, V - E + F = 2, by projecting a polyhedron onto a sphere and then summing the
areas of the various spherical polygons. I think this fact resonates rather well with this design. --- Jeffrey Stewart Ely
Mar 10, 2011
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"Branched Manifold," by Juan G. Escudero (Universidad de Oviedo, Spain)Digital Print, 20"x20", 2010

A cell complex is defined in the analysis of the cohomology of tiling spaces. It contains a copy of every kind of tile that is allowed, with some edges identified for the 2D case, and the result is a branched surface. When the tiling does not force the border, collared tiles can be used. Here the triangles with the same shape, color and orientation represent the same tile in the complex. The manifold appears in the cohomology computations of an octagonal pattern belonging to a random tiling ensemble introduced by the author in the context of mathematical quasicrystals. --- Juan G. Escudero
Mar 10, 2011
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