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.

"Tetra-Tangle of Four Bow-Tie Links," by Carlo Séquin (University of California, Berkeley)ABS plastic, printed on an FDM machine, 2014
Four sets of three mutually parallel, 3-sided prisms, pointing in 4 different tetrahedral directions, form the core of the TETRAXIS® puzzle. When two triangular prism-end-faces that share a common vertex are closed off with a connecting sweep, a loose "bow-tie" is formed. If all twelve pairs of adjoining triangular end-faces are joined in this manner, the result is a link of 4 mutually interlocking, twisted, prismatic bow-tie loops. This represents an alternating 12-crossing link that has the same connectivity as the “Tetra-Tangle,” which I constructed from 4”-diameter card-board tubes in 1983. The new geometry is has been realized as 4 differently colored sets of 6 tubular snap-together parts each, fabricated on an FDM machine. --- Carlo Séquin (http://www.cs.berkeley.edu/~sequin/) Apr 06, 2015

"Dance of Stars II," by Reza Sarhangi (Towson University, Towson, MD)Heavy paper, 2014
Dance of Stars II is a decorated Great Stellated Dodecahedron, with Schläfli Symbol (5/2, 3), which has been patterned by the sâzeh module tiles, that are used in the majority of tiling that conforms to local fivefold symmetries. In an article that appeared in Science, the authors proposed the possibility of the use of a set of tiles, girih tiles, by the medieval craftsmen, for the composition of the underlying pattern. I used girih tiles and left the dashed outlines in the final tessellation. I also included rectilinear patterns that appear as additional small-brick pattern in the decagonal Gunbad-i Kabud tomb tower in Maragha, Iran. --- Reza Sarhangi Apr 06, 2015

"15 Irregular Hexahedra," by Aaron Pfitzenmaier (student)Paper, 2014
Honorable Mention, 2015 Mathematical Art Exhibition
This model was made from 180 units of four different types. It consists of 15 irregular hexahedra interlocked together. Each hexahedron has 2-fold dihedral symmetry and the positioning of each hexahedron is based on a pair of opposite edges on an icosahedron. This compound has the most complex weaving pattern out of anything I have designed, and is an example of a model where I extensively used the ray tracer POV-Ray as well as a computer program I wrote to aid in the design and folding/assembly process. --- Aaron Pfitzenmaier (http://bit.ly/aaronsorigami)

Apr 06, 2015

"Hyperbolic Catacombs," by Roice Nelson (Austin, TX) and Henry Segerman (Oklahoma State University, Stillwater)Digital Print, 2014
This picture visualizes the regular, self-dual {3,7,3} honeycomb in the upper half space model of hyperbolic 3-space. The cells are {3,7} tilings and the vertex figure is a {7,3} tiling. The cells have infinite volume: the vertices are "ultra-ideal", living beyond the boundary of hyperbolic space. The intersection of each cell with the boundary is an infinite collection of heptagons, together with a disk. The white ceiling and each red "creature" are isometric cells; for all other cells we only show the intersection with the boundary of hyperbolic space, on the floor of the catacombs. Every disk on the floor containing a {7,3} tiling is associated with an ultra-ideal vertex of the honeycomb. --- Roice Nelson (http://google.com/+roicenelson) and
Henry Segerman (http://segerman.org) Apr 06, 2015

"Longest Crease/Perfect Shuffle-1," by Sharol Nau (Northfield, MN)Folded Book, 2014
A classical calculus problem, the so-called Paper Creasing Problem is essential to the design of these sculptures. Pages in a book provide a series of rectangular sheets of paper which are creased by matching one corner; say the lower right-hand corner to a point on the opposite edge where the sheets have been bound. Waves are obtained through incremental changes in the length of the crease from page to page. Two sets of points have been used for these new examples. In each case every other page begins its sequence at a different point. The result of the two series interleaved is a so-called perfect shuffle. --- Sharol Nau (http://www.sharolnau.com) Apr 06, 2015

"Penrose Pursuit 2," by Kerry Mitchell (Phoenix, AZ)Digital print onto aluminum panel, 2014
Best photograph, painting, or print
2015 Mathematical Art Exhibition
Underlying this image is a non-periodic Penrose tiling, using the kite and dart tiles. Each tile is rendered using pursuit curves. To accommodate the concave dart tile, it was split into two triangular halves. Each half was filled with three pursuit curves, while the kite tiles have four. --- Kerry Mitchell (http://kerrymitchellart.com) Apr 06, 2015

"Irregular hyperbolic disk as lamp shade," by Gabriele Meyer (University of Wisconsin, Madison)Photograph, 2013
I like to crochet hyperbolic surfaces. More recently I am interested in irregular hyperbolic shapes and how they interact with light. This is an image of a crocheted hyperbolic surface used as a lamp shade. The object itself is about 1 yard in diameter. On one side it has a more negative curvature than on the other, an irregularity, which makes it appear more interesting. The surface is created with white yarn, so that nothing detracts from the shape. --- Gabriele Meyer (http://www.math.wisc.edu/~meyer/airsculpt/hyperbolic2.html) Apr 06, 2015

"Partitions Study: On the Grid," by Margaret Kepner (Washington, DC)Archival Inkjet Print, 2014
A multiplicative partition of a number is an expression consisting of integer factors that produce the number when multiplied together. An unordered multiplicative partition is usually called a factorization. This work presents each of the factorizations of the integers from 1 to 28 in a symbolic representation based on subdividing a square. For example, "7 x 3" is a factorization of 21. It is represented by a square divided into a grid of 7 rows and 3 columns – see the symbol in the lower-left corner. The uniform grids corresponding to square numbers are highlighted in red. This piece is formatted so it can be cut in a spiral fashion and folded to create a 64-page accordion book of factorization diagrams. --- Margaret Kepner (http://mekvisysuals.yolasite.com)

Apr 06, 2015

"Winter," by Veronika Irvine (University of Victoria, British Columbia, Canada)White cotton, DMC Cebelia No 20, 2014
Periodic bobbin lace patterns can be described by a mathematical model. Key elements of the model are 1) a toroidal embedding of a directed graph describing the movement of pairs of threads and 2) a function that maps each vertex of the digraph to a braid word. Using an intelligent combinatorial search, over 100,000 patterns matching the fundamental properties of this model were found. Of these, three closely related patterns were chosen (see inset). The three patterns can be transformed into one another by a small number of changes. The submitted piece was designed to show a gradual transition from one pattern to the next resembling the transformation from perfect, large snowflakes to the slanted, driving snow of a blizzard. When I first learned to make bobbin lace, some 20 years ago, I was struck by its mathematical nature. The patterns are diagrams, not a linear set of instructions. The order in which braids are worked and most of the decisions about how threads should move so that they arrive in the correct position as needed, are left up to the lacemaker. Over the past 4 years, I have been developing a mathematical model for bobbin lace and discovering the joy of designing my own pieces. More information: http://arxiv.org/abs/1406.1532. --- Veronika Irvine (http://web.uvic.ca/~vmi/) Apr 06, 2015

"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) Apr 06, 2015

"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) Apr 06, 2015

"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) Apr 06, 2015

"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 Apr 06, 2015

"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) Apr 06, 2015

"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) Apr 06, 2015