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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 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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"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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"Flora #1 (time slice)," by Brian Evans (University of Alabama, Tuscaloosa)Archival inkjet print, 9" x 6" (14" x 11" framed), 2010

How much is lost in the reduction of reality to human sensation? The infinite detail there in front of us is reduced to 100 million discrete measurements made with the rods and cones on the retina of the eye. Infinity reduced to 100 million, which is reduced another ninety-nine percent as the signal is compressed to travel only 1 million pathways on the optic nerve. It’s a wonder we can make sense of the world at all. 

These little photos are also reductions, slit-scans of flowers rotating on a tabletop—2D slices of time. The four dimensions of our reality (x, y, z, t) are reduced to two (x, t) showing a different aspect of the real. The temporal is mapped into the static and new forms and structures are seen. These works are metaphors for the language of mathematics. What wonders we can discover through the processes of abstracting, reducing, mapping, and finally looking in new ways at the little slices of information we receive from all the surrounds us. --- Brian Evans (http://brianevans.net)
Mar 10, 2011
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"Fractal Tree No. 13," by Robert Fathauer (Tesselations Company, Phoenix, AZ)Digital print, 13" x 16", 2010

"Fractal Tree No. 13" is a black-and-white digital artwork constructed by graphically iterating a photographic building block. Several photographs of a small portion of a palo verde tree were combined and digitally altered to create the building block, which allows smooth joining of smaller copies to larger copies. In addition to being scaled down, the three smaller copies, added with each iteration are rotated by varying angles, and one is reflected as well. A sufficiently large number of iterations were performed so that the image is indistinguishable to the eye from the image that would result after an infinite number of iterations. In this particular tree, the iteration rules result in considerable overlap of the branches, leading to a complex collection of small features reminiscent of pencil marks. The fractal shape of the envelope of these features didn't emerge until approximately one dozen iterations were performed. --- Robert Fathauer (http://www.tessellations.com)
Mar 10, 2011
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"Blue Ionic Polyhedra: 12 Beaded Beads in Two Sizes," by Gwen Fisher (bead Infinitum, Sunnyvale, CA)Bead weaving with crystal, glass and metal beads, and thread, 12 beads, longest diameter ranges from 11 mm to 33 mm, 2010

The Ionic Polyhedra are part of my larger body of mathematical artwork in beaded beads. A beaded bead is a cluster of smaller beads, woven together with a needle and thread, to form a composite cluster with one or more holes running though the center of the finished beaded bead. To make these beads, I started with inner cluster of beads in two sizes where the larger beads are aligned on the edges of polyhedra. For the larger six beads, I added layers of seed beads, which emphasize the edges of the underlying polyhedra. Although they appear different, the miniature version of each Ionic Polyhedron maintains the same structure and thread path as its larger version, but the beads are smaller, and the embellishment is simpler. Represented are the octahedron, cube, pentagonal dipyramid, pentagonal antiprism, cuboctahedron, and rhombic dodecahedron. This coloring of the miniature cuboctahedron and rhombic dodecahedron illustrates the dual relationship between these two polyhedra. --- Gwen Fisher (http://www.beadinfinitum.com)
Mar 10, 2011
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"Quasirandom Aggregation," by Tobias Friedrich (Max Planck Institute for Informatics, Saarbrücken, Germany)Digital print on glossy paper, 20" x 20", 2010

Given an arbitrary graph, a random walk of a particle is a path that begins at a given starting point and chooses the next node with equal probability out of the set of its current neighbors. Around 2000, Jim Propp invented a quasirandom analogue of random walk. Instead of distributing particles to randomly chosen neighbors, it deterministically serves the neighbors in a fixed order by associating to each vertex a "rotor" pointing to each of its neighbors in succession. The picture shows what happens when one billion particles are placed at the origin and each one runs until it reaches an unoccupied vertex. Black pixels denotes cells that never get visited by a particle; for the other cells, the color of the pixel indicates in which direction the rotor points at the end of the process. More information can be found at http://rotor-router.mpi-inf.mpg.de. --- Tobias Friedrich (http://www.mpi-inf.mpg.de/~tfried/)
Mar 10, 2011
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"Triad," by Mehrdad Garousi (Hamadan, Iran)Digital print on canvas, 18" x 16", 2009

This work created in TopMod comprises a uniform twisted strip with some ties and joints. The shape containing an evident three-fold rotational symmetry is composed of three similar components connected at two central joints placed back and forth. The outstanding issue is another hidden symmetry, which may not be discovered at a hasty glance. In addition to the former symmetry, condoning back and forth or up and down position of layers, as a flat plane, the whole sculpture has a three-fold mirror symmetry. The reason of such a property is the same one-fold mirror symmetry governing each of the three components. --- Mehrdad Garousi (http://mehrdadart.deviantart.com)
Mar 10, 2011
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"Tea for Eight," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City)Glaze on commercial ceramic, 5" x 8" x 5", 2010

The Four-Color Theorem says that we can color any map on a plane or sphere with only four colors so that no neighboring countries are the same color. On other surfaces, we may need more colors; on a two-holed torus, eight colors are sufficient, and there are maps that require all eight colors.
 When this tea set is stacked with the handles aligned, it forms a topological two-holed torus with a map of eight countries, each of which touches all of the others, proving that eight colors are necessary. The teapot has white, red, orange, and yellow countries, and the teacup has black, green, blue and purple countries. At the seam between the pieces, each of the top colors touches each of the bottom colors.
 On one-holed tori, such as the teapot and the teacup, seven colors are required for an arbitrary map. Unfortunately, a seven-color map is incompatible with the tea set's exterior pattern; when the tea set is opened, hidden colors give six-color maps of the teacup and the teapot. --- Susan Goldstine (http://faculty.smcm.edu/sgoldstine)
Mar 10, 2011
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"Circle Brooches," by Anansa Green (Stephen F. Austin State University, Nacogdoches, TX)Fine silver, copper, 1.5 x 1.5 x 0.25 inches each (2 brooches), 2009

These brooches were inspired by my undergraduate graph theory research into the colorability of the map created by a finite tiling of circles in the plane. I was able to prove by mathematical induction that the resulting map is 2-colorable. This result lends itself quite well to the process of married metals. Two pieces of metal were overlaid: one copper and the other fine silver. The design was pierced from both sheets at once, and alternating pieces were swapped to form the two 2-colored designs. The individual components of each image were silver-soldered together, and the sides and back of each brooch hollow constructed to create the final form. The process yields two images, each one the inverse of its partner. To emphasize the complementary nature of each image, I fabricated one brooch with a convex face and the other concave. --- Anansa Green
Mar 10, 2011
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"Right Angle Triangles in Flatland A," by Gary Greenfield (University of Richmond, VA)Digital print, 18" x 12", 2010

Four Flatlanders are sweeping through Flatland celebrating their discovery of how to draw right triangles. Their method is as follows: (1) pseudorandomly generate a turning angle alpha and an adjacent side length x; (2) calculate the complementary angle beta and use trigonometry to calculate the opposite side length y and hypotenuse length h; (3) then swivel right, forward x, turn alpha, forward h, turn beta, forward y, swivel left. These Flatlanders belong to the caste required to "wag" from side to side when they walk. Thus they defy convention by drawing perfectly straight thick lines when presenting their right triangle discovery. Here, Flatlanders are implemented as simulated drawing robots obeying obstacle and collision avoidance, and their wag is implemented by making one of their pens swing side to side in such a way that a sinusoidal track is drawn as they make their through Flatland. --- Gary Greenfield (http://www.mathcs.richmond.edu/~ggreenfi/)
Mar 10, 2011
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"Ramanujan, in the style of Chuck Close, using wavelets," by Edward Aboufadel (Grand Valley State University, Allendale, MI), Clara Madsen (University of Oregon, Eugene) and Sarah Boyenger (Florida State University, Tallahassee)Digital print, 16" x 20", 2009

Both the subject of this work and the method of creation are intricately mathematical. Ramanujan is the famous 20th century Indian mathematician who established or conjectured a broad collection of results in number theory. He caught the attention of Hardy, who recognized Ramanujan's genius. To create this digital image in the style of Chuck Close, wavelet filters were used to detect the existence and orientation of edges in the original image, and other calculations were made to determine the colors in the "marks".
Mar 10, 2011
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"Gaussian Wave Packet Sculpture," by Chet Alexander (University of Alabama, Tuscaloosa)Wood (birch, walnut, maple, ebony), 9" x 11" x 10", 2006

Mathematics of the Wave -Packet Sculpture: 
In this sculpture, mathematics was used to calculate the Gaussian wave-packet model of a particle in quantum mechanics. This is accomplished by forming a linear combination of plane waves of different wave-numbers, k. A particle with mass and momentum p can have wave properties as described by the de Broglie wavelength relation λ=h/p. The Gaussian wave packet model is a way to combine the wave and particle properties of a particle of momentum p=hk localized at position x_0. The probability of finding the particle at position x_0 is given by the probability density of the particle as 
ІΨ(x,0) І^2~exp[-(x-x_0)^2/2(∆x)^2]
, and by a Fourier transform the probability density of the particle's momentum can be written 
ІΨ(k) І^2~exp[-(k-k_0)^2/2(∆k)^2]. 
The wave packet sculpture presents a Gaussian wave packet envelope and an electromagnetic wave enclosed in the envelope. --- Chet Alexander
Mar 10, 2011
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“CORPI INCROCIATI • DUE CORPI,” by Cristina Besa (European Society for Math & Art, Islas Baleares, Spain)6 x 7 negative, print on Ilford glossy paper, 12” x 16” (framed 20” x 24”), 2004

Bilateral symmetry—a combination of identical exposures intersecting with their own reflections.
 The intersection of the two shapes A (original) and B (reflection of A) are positioned at the same angle to create a new form bound by its central line reflection. --- Cristina Besa

Mar 10, 2011
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