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.

"Fibonacci Downpour," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City, MD)819 viewsMerino yarn, cotton thread, embroidery hoop, 21 x 26 x 26 cm, 2015

For me, the most exciting part of mathematics is communicating it to others. I am especially interested in models that make mathematical concepts tactile or visual. In Fibonacci Downpour, the vertical stitch lines branch and form drops following a physical version of the Fibonacci recursion. The number of drops and branchings in each row are consecutive Fibonacci numbers. As the Fibonacci numbers are asymptotically exponential, the fabric falls into a more or less pseudospherical form. --- Susan Goldstine

From "Serenity to Monkey-Mind and Back (Two Twisted Tessellated Transforming Tori)," by Ellie Baker (Lexington, MA) 724 viewsPrinted polyester crepe de chine, bead crochet (glass beads and thread), 70 x 50 cm, 2015

This infinity scarf and bead crochet necklace are twin tori. The fabric design is (an elongated version of) the infinitely repeating planar pattern that a tiny explorer could map by charting the surface of the necklace in all directions (the universal cover of the beaded rope). The two colors, identical tessellated wave motifs, gradually transform from "calm" to "busy." The pattern at each step has an increasing "busyness" quotient (a measure of how much the individual beads in a fundamental tile differ in color from neighboring beads). The scarf, sewn from a parallelogram to create a mobius-like twisted torus, has a small hole in one seam so that it can be turned inside out to explore the puzzling behavior of torus inversions. --- Ellie Baker

"A Steiner Chain Trapped Inside Two Sets of Villarceau Circles," by Francesco De Comité (Univeristy of Lille, France)722 viewsDigital print on cardboard, 60 x 80 cm, 2015

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. Here, ring cyclides are images of tori under sphere inversion. If certain conditions are fulfilled, a torus can contain a set of tangents spheres. Since the tangency property is preserved by inversion, this set of tangent spheres find its place inside the cyclide. --- Francesco De Comité

"Poincaré’s 'Pas de deux'," by Jean Constant (Hermay, Santa Fe, NM)697 viewsMathematics and mathematical visualizations are meaningful at many scientific and technological levels. They are also an endless source of inspiration for artists. The following artworks are part of the 12-30 project – one mathematical image a day for one year, 12 mathematical visualization software, January 1st, 2015 – December 31, 2015. See the 365 images portfolio and a printed compilation of the work will be available in the coming months at Hermay.org. --- Jean Constant

"Brown and Green Egg -163," by Faye Goldman (Ardmore, PA)677 viewsStrips of polypropylene ribbon, 13 x 10 x 10 cm, 2013

Loosely defined, a 'Buckyball' is a polyhedron made of pentagons and hexagons with every vertex of degree three (three edges meeting). Buckyballs must have exactly twelve pentagons. I enjoy creating Buckyballs and their duals. I discovered that if you rearranged the twelve pentagons in a semi-regular pattern you could get interesting shapes. Thus began my series of eggs. --- Faye Goldman

"Intrinsic Regularity," by Conan Chadbourne (San Antonio, TX)673 viewsArchival inkjet print, 60 x 60 cm, 2015

This image presents a visualization of the Steiner triple system S(2,3,7). This system, which is combinatorially equivalent to the Fano plane, consists of seven three-element subsets (or blocks) drawn from a seven element set such that any pair of elements occur in exactly one block, and any two blocks have exactly one element in common. --- Conan Chadbourne

"A Fractal Circle Pattern on the {3,12} Polyhedron," by Doug Dunham (University of Minnesota - Duluth, MN)670 viewsPrinted cardboard, 50 x 30 x 30 cm, 2015

The goal of my art is to create aesthetically pleasing repeating hyperbolic patterns. One way to do this is to place patterns on (connected) triply periodic polyhedra in Euclidean 3-space. This polyhedron is constructed by placing regular octahedra on all the faces of another such octahedron, so there are 12 equilateral triangles about each vertex. Each of the triangular faces has been 90% filled by a fractal pattern of circles provided by John Shier. The polyhedron consists of red and blue "diamond lattice" polyhedra and purple octahedra that connect the red and blue polyhedra. Each of the red and blue polyhedra consists of octahedral "hubs" connected by octahedral "struts", each hub having 4 struts projecting from alternate faces. The red and blue polyhedra are in dual position with respect to each other - they form interlocking cages. Each purple connector has a red and a blue octahedron on opposite faces. --- Doug Dunham

"Rainbow Brunnian Link Cowl," by sarah-marie belcastro (MathILy, Holyoke, MA)658 viewsKnitted wool (various sources) and printed photographs, 30 x 30 x 7 cm, 2015

The central property of the Borromean rings--that removing any component unlinks the remaining components, which collectively form the unlink--generalizes to the class of Brunnian links. The Rainbow Brunnian Link Cowl has seven components rather than the three components of the Borromean rings. All linking is intrinsic, rather than introduced post-construction via grafting. The Rainbow Brunnian Link Cowl is also a garment that can be worn two different ways, which are pictured alongside that cowl in the exhibit. --- sarah-marie belcastro

"Tangent Discs I," by Anne Burns (professor emerita, Long Island University Brookville, NY)642 viewsDigital print, 30 x 30 cm, 2015

I am interested in the connections between mathematics, art and nature, especially the concept of evolution. Thus my mathematics interests are dynamical systems, differential equations and any area that deals with states that evolve with time. This image is an iterated Function System consisting of a group of (six) Mobius Transformations acts on six discs, five of which are tangent to the unit circle, to its two neighboring discs and to a sixth disc centered at the origin. --- Anne Burns

"Dragony Curve," by Robert Fathauer (Tessellations Company)639 viewsCeramics, 60 x 45 x 3 cm, 2014

I'm endlessly fascinated by certain aspects of our world, including symmetry, chaos, and infinity. Mathematics allows me to explore these topics in distinctive artworks that I feel are an intriguing blend of complexity and beauty. This sculpture is based on a particular stage in the development of a fractal curve known as the ternary dragon. This ceramic piece has been mounted on a board, with standoffs, partly to make it easier to handle without breaking. The resulting construct could be viewed as either a two-dimensional or three-dimensional artwork, which echoes the manner in which fractal curves can be considered as one-dimensional (a line), two-dimensional (a plane-filling object), or something in between. --- Robert Fathauer

"A Radin-Conway Pinwheel Lace Sampler," by Douglas G. Burkholder (Lenoir-Rhyne University Hickory, NC)636 viewsDigital Print, 50 x 50 cm, 2015

This artwork evolved from a search for beauty and patterns within Conway and Radin’s non-periodic Pinwheel Tiling of the plane by 1x2 right triangles. The Pinwheel tiling can be created by repeatedly subdividing every triangle into five smaller triangles. This lace resulted from alternately subdividing triangles and removing triangles. Triangles are removed based upon their location in the next larger triangle. First, on the macro level, the five distinctive removal rules are applied one to each row. This removal rule is especially easy to see in the bottom row. These same five rules are then applied, on the micro level, to the columns. The remaining triangles form a sampling of twenty-five styles of lace generated by the Pinwheel tiling. --- Douglas G. Burkholder

"An Iris Spiral," by Frank A. Farris (Santa Clara University, San Jose, CA)631 viewsAluminum print, 51 x 61 cm, 2015

My artistic impulse is to let the beauty of the real world shine into the realm of mathematical patterns. My method combines photographs with complex-valued functions in the plane to create images with all possible types of symmetry. I photographed the irises and used complex wave functions to turn the image into a pattern with four-fold rotational symmetry. Then I applied a complex exponential mapping to wind the wallpaper around the complex plane, choosing just the right scaling to make the pattern match, while also creating five-fold symmetry. I bleached an outer ring to bring focus to the center of the spiral and to allow the original photograph of the iris to stand out. Details about wallpaper waves appear in my book, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns. --- Frank A. Farris

"Hyperbolic Afghan {3, 7}," by Heidi Burgiel (Bridgewater State University, Bridgewater, MA)607 viewsKnitPicks Shine Sport yarn: 60% cotton 40% modal, 7 x 44 x 44 cm, 2015

"Hyperbolic Afghan {3, 7}" illustrates a tiling of the hyperbolic plane by triangles, 3 at a vertex, in crocheted cotton. Adapting techniques developed by Joshua and Lana Holden, the piece is not assembled from flat triangles but instead approximates constant curvature over its entire surface. Its coloration, inspired by William Thurston's rendition of the heptagon tiling underlying the Klein quartic, suggests the identifications required to construct that surface as a quotient of the hyperbolic plane. --- Heidi Burgiel

"Dragon Curve Double Knit Scarf," by Rachelle Bouchat (Indiana University of Pennsylvania, Indiana, PA)601 viewsMerino Wool Yarn, 137 x 18 cm, 2015

This double knit scarf brings together the recursive construction of a fractal, the dragon fractal, as well as the recursive construction of an integer sequence, the Fibonacci sequence. The main panels of the scarf are based on a pattern developed from the eleventh iteration of the dragon fractal. Moreover, the striping pattern in between the main panels is illustrative of the Fibonacci sequence with color changes after 1 row, after another 1 row, after 2 rows, after 3 rows, after 5 rows, and with another color change after 8 rows. As this is a double knit pattern, the back side of the scarf is shown in the reverse color pattern. --- Rachelle Bouchat

"The Jordan Curve Theorem," by Robert Bosch (Oberlin College Oberlin, OH)541 viewsLasercut woods, 15 x 45 cm, 2015

The Jordan Curve Theorem states that when a simple closed curve is drawn in the plane, it will cut the plane into two regions: the part lies inside the curve (here, the slightly darker-colored inset piece of wood), and the part that lies outside it (here, the slightly brighter and thicker frame). --- Robert Bosch