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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"Three (2k+2, 2k) links," by sarah-marie belcastro (Hadley, MA)Knitted hand-dyed wool, 2013
A (p,q) torus link traverses the meridian cycle of a torus p times and the longitudinal cycle q times; when p and q are coprime, the result is a knot, and when not (ha!) the result is a gcd(p,q)-component link with each component a (p/gcd(p,q), p/gcd(p,q)) torus knot. Here we have (in increasing order of complexity) a (4,2) torus link, a (6,4) torus link, and an (8,6) torus link. Each is knitted so that both the knotting and the linking are intrinsic to the construction (rather than induced afterwards via grafting). They were made as proof-of-concept for the methodology for knitting torus knots and links that the artist introduced at the 2014 JMM. --- sarah-marie belcastro (http://www.toroidalsnark.net)

"Basilica," by Anne Burns (professor emerita, Long Island University, Brookville, NY)Digital print, 2014
The Julia Sets of z^n + c are familiar objects to mathematicians. In particular c=-1 yields the well-known "Basilica". Adding a term d/z(z^2-1) introduces three poles: z=0, z=-1, z=1. The orbits of initial points near the poles rapidly diverge to ∞; for very "small" (real, positive) d, amazingly, the boundary of the set of points whose orbit escapes (the Julia Set) contains an infinite number of tiny decorations resembling the decorations on the original "Basilica". --- Anne Burns

"Intrinsic Transformation III," by Conan Chadbourne (San Antonio, TX)Archival inkjet print, 2014
This work is part of a series of visual meditations on the structure of the alternating group on 5 elements, also known as the icosahedral group. This image explores the structure of the icosahedral group through a particular presentation by two generators. The group's elements, which appear as yellow disks in this image, are arranged at the vertices of a rhombicosidodecahedron, shown here in stereographic projection, while the group's generators, of orders 3 and 5, correspond to the regions between the disks, colored green and blue, respectively. The image is composed of multiple hand-drawn images which are digitally composited and output as an archival digital print. --- Conan Chadbourne (http://www.conanchadbourne.com)

"19 x 31," by Larry Crone (associate professor emeritus, American University, Alum Bank, PA)Print, 2014
In addition to its beauty, this image is of interest because the underlying quadratic rational function has an attracting fixed point cycle of order 19, and another of order 31. Just as a mountain presents many targets to a photographer, this function can be viewed from different perspectives, and it was hard to decide which one to use. The windows program Gplot, the camera which took this picture, is available for free download at http://www.american.edu/cas/faculty/lcrone.cfm. --- Larry Crone (http://www.american.edu/cas/faculty/lcrone.cfm)

"The Lost Art of Cyclides Islands Weavers," by Francesco De Comité (University of Lille, France)Digital print on cardboard, 2014
Dupin cyclides are the images of tori by sphere inversion. Since sphere inversion preserves circles, the set of Villarceau circles one can draw on a torus is transformed in a set of circles on the cyclide. The game is then to find nice images illustrating this fact, together with some story 'à la Raymond Roussel' to reinforce the magic. Manipulation of digital images, and use of ray-tracing software can help you to concretize mathematical concepts. Either for giving you an idea of how a real object will look or to represent imaginary landscapes only computers can handle. Things become yet more interesting, when you can transform your two-dimensional dream objects in real three dimensional sculptures. You can then handle your creations, and look at them from an infinity of view angles. --- Francesco De Comité ([url=http://www.lifl.fr/~decomite]http://www.lifl.fr/~decomite]/url])

"Constructing the Inner Apollonian," by Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR)Digital print on archival paper, 2014
Unlike pictures of two-dimensional Apollonian gaskets, most renderings of the three-dimensional analogue, Apollonian sphere packing, tend to be disappointing because they do not reveal the interior structure the way that their two-dimensional
cousins do. This image tries to reveal the inner structure in several ways. First, some of the larger spheres that obstruct the view have been removed. The negative spaces caused by their removal are plain to 'see'. Second, the observer has been located in one of these negative spaces, affording a more intimate view. Finally, the process has been deliberately left incomplete, giving a sense of both the coarser and finer stages of the construction. --- Jeffrey Stewart Ely

"Hope in Base 8," by Sally Eyring (Watertown, MA)Woven cotton, 2013
Weaving technology is closely related to the computer industry - Hollerith cards were a direct inspiration from dobby looms. In this piece the word HOPE is translated into a weaving pattern using an 8 shaft loom. Using ANSII codes - A is represented as 101, B as 102, etc. up to Z represented as 132. First 100 was subtracted from each code to create a workable weave structure. Next, 1 was added to each code (using base 10) because weaving software programs number the shafts from 1 to 8. That resulted in representing A by 12, B by 13, etc. with Z represented as 43. Thus, the word "HOPE" is represented by 8 threads. Rotating the set of numbers by 1, 7 times, created a twill weave with a repeat of 64 threads, producing "HOPE", woven in base 8. The colors depict the colors of a sunrise; the red and orange raising out of the black of night. --- Sally Eyring (http://sallyglassdreams.blogspot.com/)

"Trifurcation," by Robert Fathauer (Tessellations, Phoenix, AZ)Ceramics, 2014
This sculpture is a fractal tree carried through five generations. With each iteration, the number of branches is tripled. The scaling factor from one generation to the next is the inverse of the square root of 3, approximately 0.577. As more and more branches are added, the top surface begins to display the classical fractal known as the Sierpinski triangle. More information: http://mathartfun.com/shopsite_sc/store/html/Art/Trifurcation.html. --- Robert Fathauer (http://robertfathauer.com)

"Seven Sided Seven Color Torus," by Faye E. Goldman (Ardmore, PA)Strips of polypropylene ribbon, 2014
This toroid shape is made from over 3200 strips of ribbon. I love the fact that there needs to be as many heptagons making the negative curvature in the center as there are pentagons around the outside. It is the fourth torus I've made and the most interesting. When I decided to create a seven sided torus, it was obvious that it needed to have seven colors to show the seven color map problem on a torus. --- Faye E. Goldman (http://www.FayeGoldman.com)

"Hyperbolic Constellation," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City, MD)Glass beads, crochet cotton thread, 2014
Hyperbolic Constellation is inspired by Daina Taimina's innovative technique for crocheting hyperbolic surfaces. Her breakthrough is that if you crochet with an increase (made by stitching twice into the same stitch) every n stitches for a fixed number n, the result has constant negative curvature. I have always been curious about how these increases are arranged. While many artists have woven hyperbolic surfaces with beads, I have yet to see other examples of hyperbolic bead crochet, which moves more organically. In this pseudospherical beaded surface, the gold beads (every 6th bead on the thread) mark the locations of the crochet increases. The initial round contains 6 beads, while the outer edge contains 6 x 64 = 384 beads. --- Susan Goldstine (http://faculty.smcm.edu/sgoldstine)

"Map Coloring Jewelry Set," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City, MD)Glass beads, gold-filled beads, thread, ear wires, 2014
Best textile, sculpture, or other medium
2015 Mathematical Art Exhibition
While every map on a plane can be colored with four colors so that no two adjacent countries are the same color, maps on other surfaces may require more colors. This jewelry set displays maps requiring the maximum number of colors for three surfaces. The bracelet, bead crochet with a bead-woven closure, is a double torus in eight colors, each of which touches all the others. The gold bead in the center of the pink and blue spiral is strictly ornamental. The pendant is a bead-crochet torus with seven colors, and all of the color contacts are visible from the front side. The bead-woven earrings are each four-color maps in the plane. With over 5300 beads in total, the entire set is wearable topology at its finest. --- Susan Goldstine (http://faculty.smcm.edu/sgoldstine)

"SA Labyrinth #5223," by Gary Greenfield (professor emeritus, University of Richmond, VA)Digital Print, 2014
A point, realized as an autonomous drawbot, traces a curve parametrized by arc length by constantly adjusting its tangent angle and curvature. The drawing method was first introduced by Chappell. When it encounters itself, it strives to match its current curvature with its previous curvature. In non-degenerate cases this behavior yields labyrinths. Feathering the curve using a normal vector helps accentuate the drawbot's directional and curvature changes. --- Gary Greenfield

"Three studies in CNC milling," by Edmund Harriss (University of Arkansas, Fayetteville, AR)Oak and red cedar, 2014
Mathematician, Teacher, Artist, Maker. I like to play with the ways that the arts can reveal the often hidden beauty of mathematics and that mathematics can be used to produce interesting or beautiful art. These three designs show patterns that can be obtained running a CNC mill along lines. For the first two pieces the lines are defined purely mathematically. For the third the lines are constructed by an algorithm to lie at right angles to the grain. --- Edmund Harriss (http://maxwelldemon.com)

"Orb," by George Hart (Stony Brook University, NY)Laser-cut wood, 2014
As a sculptor of constructive geometric forms, my work deals with patterns and relationships derived from classical ideals of balance and symmetry. Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic. Sixty identical laser-cut and laser-etched wood components assemble easily with small cable ties, illustrating chiral icosahedral symmetry. Designed to be repeatedly reconstructed and disassembled in public workshops at MoSAIC math/art festivals, the separate parts travel conveniently in a small package. Orb has been assembled and disassembled multiple times by many groups of people. --- George Hart (http://georgehart.com)

"More fun than a hypercube of monkeys," by Henry Segerman (Oklahoma State University Stillwater, OK) and Will Segerman (Brighton, UK)PA 2200 Plastic, Selective-Laser-Sintered, Computer Animation (on a tablet computer), 2014
This sculpture was inspired by a question of Vi Hart. This seems to be the first physical object with the quaternion group as its symmetry group. The quaternion group {1,i,j,k,-1,-i,-j,-k} is not a subgroup of the symmetries of 3D space, but is a subgroup of the symmetries of 4D space. The monkey was designed in a 3D cube, viewed as one of the 8 cells of a hypercube. The quaternion group moves the monkey to the other seven cells. We radially project the monkeys onto the 3-sphere, the unit sphere in 4D space, then stereographically project to 3D space. The distortion in the sizes of the monkeys comes only from this last step -- otherwise all eight monkeys are identical. The animation shows the result of rotating the monkeys in the 3-sphere. More information: http://www.segerman.org/pics/monkeys_128_frames_left_mult_rev_5_400.gif. --- Henry Segerman (http://segerman.org) and Will Segerman (http://willsegerman.com)