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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"Toroweave," by David Bachman (Pitzer College, Claremont, CA)10 x 25 x 25 cm, 3D printed "sandstone" (gypsum powder + binder), 2016

These pieces were each created from two copies of a diamond tiling of a torus. The interior of each face of the tilings was removed, and the resulting webs were alternately offset in opposite directions to form a weave. Complementary colors are used to distinguish the two interlinked forms, which are completely disjoint. -- David Bachman

"A Fine Mesh We're In," by Dan Bach (Northern California, USA)35 x 45 cm, inkjet print on canvas, 2016

A central curve (not shown) has tangent, normal, and binormal directions at each point, making a local {T, N, B} frame. Using a trigonometric combination of the N and B vectors, we describe a toroidal mesh of curves with hues of green, yellow, and orange. Spheres of varying colors and sizes are placed along an equatorial helix and some try to escape their bonds. More information. --- Dan Bach

"Invertible Infinity," by Ellie Baker (Lexington, MA)45 x 45 cm, silk crepe de chine (custom printed via Spoonflower.com), 2016

This reversible infinity scarf is a specially constructed cloth torus such that its shape is invariant under inversion AND it folds flat into a six-layer equilateral triangle. Since the meridians and longitudes of a torus swap places under inversion, one might think the invariance property dictates construction from a square piece of fabric (with opposite edges sewn together). However, although inversion invariance can be achieved with a square construction, the equilateral triangle folding cannot. Can you figure out a possible shape for the flat fabric layout used? The fabric designs, both P6M wallpaper group patterns that I created with Richter-Gebert’s app iOrnament, are a clue, and permitted sewing the pattern to match at the seams.The mathematical ideas incorporated into the design of this scarf were developed in collaboration with Charles Wampler. Thank you to Carol Maglitta for modeling. More information. -- Ellie Baker

"Twelve Pi," by Sarah Berube (Diametric Arts, Shutesbury, MA)5 x 5 x 5 cm, 3D printed steel, 2016

My work currently revolves around the polyhedral symmetry groups. I pursue this form of art because I am strongly drawn to symmetry and the satisfying sense of beauty and perfection it evokes. My design process is an exploration of the properties of polyhedra. The design for this sculpture is comprised of twelve pi symbols arranged with tetrahedral symmetry. At four different locations, the right legs of three pi symbols spiral into each other, forming vortices in lieu of vertices. --- Sarah Berube

"Bicycle Wreck," by Fielding Brown (Westwood, MA)45 x 45 x 45 cm, wood & aluminum frame, silvered mylar strings, 2016

My sculptures are Lissajous figures in 3D. This sculpture is in two parts: an outer frame of laminated hardwood and aluminum, and an inner web of strings. The frame is defined by three, simultaneous, parametric equations, time as parameter. The strings are silvered, package-wrap string, space-defined by a commercial graphing calculator program. The web of strings guides the eye to see imaginary lines and surfaces. --- Fielding Brown

"Penrose Skates," by Douglas Burkholder (Lenoir-Rhyne University, Hickory, NC)50 x 50 cm, digital print, 2016

This artwork evolved from a search for beauty and patterns within Penrose’s non-periodic tiling of the plane with kites and darts. Half darts and half kites can be repeatedly subdivided into five smaller components. Start by labeling these five subcomponents A-E. Then, similar to creating the Sierpinski triangle, alternately subdivide and remove all the components with a certain label. After removing tile type A for three iterations we change to removing tile type B for five more iterations. Instead of painting the tiles remaining, pursuant curves are constructed on the regions removed. --- Douglas Burkholder

"Streamlines," by Anne Burns (professor emerita, Long Island University, Huntington, NY)48 x 32 cm, digital print, 2016

I began my studies as an art major, but switched to mathematics. When I went to my first conference on fractals I was hooked. Visualizing mathematical concepts allowed me to combine both of my interests. The streamlines of the vector field dx/dt = x^2 - y^2, dy/dt = 2xy (the real and imaginary parts of the complex function f(z) = z^2 ) are the directed paths along which the tangent vector is equal to (dx/dt, dy/dt). They are circles tangent to the real axis. the attached vectors are colored according to their slope. --- Anne Burns

"Map of Invariability I," by Conan Chadbourne (San Antonio, TX)60 x 60 cm, archival digital print, 2016

My work is motivated by a fascination with the occurrence of mathematical and scientific imagery in traditional art forms, and the mystical, spiritual, or cosmological significance that is often attached to such imagery. The Klein Quartic is a genus-three Riemann surface which can be covered by a regular tessellation of 24 heptagons. In this image, the Klein quartic is projected into the Poincaré disk, and this heptagonal tessellation is given a regular 8-coloring. Each triplet of heptagons of any given color is fixed by a subgroup of order 21 of the full automorphism group of the surface. --- Conan Chadbourne

"Secret Hexagons," by Moira Chas (Stony Brook University, Stony Brook, NY)5 x 35 x 35 cm, crochet

This piece addresses the question: What is the maximum number of regions a surface can be divided into, so that each pair of regions share a length of their border (vertices don’t count)? In the torus, the maximum number is seven. --- Moira Chas

"Klein Bottle," by Jennifer Doyle (Lawrence Berkeley National Laboratory Berkeley, CA)23 x 23 x 20 cm, galvanized steel wire, 2015

My first experiments with mathematically-themed wire sculpture have been Klein bottles. The concept of producing a sculptural Klein bottle fascinates me, as the two concepts seem to be at odds: a sculpture is, by its nature, 3-dimensional, yet a Klein bottle is not; a sculpture, by its nature, has volume, yet a Klein bottle does not. The "classic" Klein bottle is considerably more "bottle-shaped" than my piece; I decided to shape my piece into a form more resembling two semi-toroids. --- Jennifer Doyle

"Fractal Monarchs," by Doug Dunham and John Shier (University of Minnesota, Duluth)Best photograph, painting, or print - 2017 Mathematical Art Exhibition

30 x 40 cm, color printer, 2016

This is a fractal pattern whose motifs are monarch butterflies. We modify our usual rule that motifs cannot overlap by allowing the antennas - but not the rest of the motif - to overlap another motif. Expanding on the area rule of the Goals statement, the area of the n-th motif is given by A/(zeta(c,N)(N+n)^c), where A is the area of the region, and zeta(c,N) is the Hurwitz zeta function, a generalization of the Riemann zeta function (for which N = 1; our algorithm starts with n = 0). For this pattern c = 1.26, N = 1.5, and 150 butterflies fill 72% of the bounding rectangle. --- Doug Dunham

"They Arrive," by Frank A. Farris (Santa Clara University San Jose, CA)20 x 25 cm, digital aluminum print on DuraPlaq mount, 2016

Glowing globes with three types of polyhedral symmetry drift over a moonlit mountain to land on the lake. Are they wafting from the Platonic world into ours? (The patterns on the globe were created with domain colorings of meromorphic functions invariant under the actions of the three chiral polyhedral groups. In the past, I always used rectangular photographs to paint spheres, resulting in images with singularities. New techniques allow the source photograph to live on the Riemann sphere, allowing poles to be painted just as if they were zeroes. Ray tracing and manipulation of the original daytime mountain photograph were done in Photoshop.) --- Frank A. Farris

"Hexa Form," by Robert Fathauer (Tessellations Company, Phoenix, AZ)20 x 20 x 20 cm, ceramics, 2016

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 abstract sculpture, which mixes biological and geometric forms, is based on a regular hexahedron (more commonly known as a cube). It retains most, but not all, of the symmetries of the cube. --- Robert Fathauer

"The Double-Knitting Groups," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City, MD)40 x 40 cm, silk/merino yarn, 2016

The Double-Knitting Groups exhibits all of the wallpaper symmetry types possible in double knitting. For a harmonious overall design, I grouped the nine structures into three pattern families: scrolls, hearts, and vines. Roughly speaking, the symmetry groups get more complex from top left to bottom right: the upper left pattern has only translations, the three adjacent to it have only translations and one other type of plane symmetry (clockwise: reflections, rotations, and glide reflections), and the remaining patterns have at least three symmetry types. --- Susan Goldstine

"Champy," by George Hart (Stony Brook University, Stony Brook, NY)32 x 32 x 32 cm, laser-cut wood, stained, 2016

Thirty components suggestive of "sea monsters" dance around each other, only touching at the hands and mouths. The arrangement of the thirty identical planar parts comes from the face planes of a rhombic triacontahedron, which provides a mathematical foundation for the structure. There are six parts in each of five colors, arranged with a five-color pattern based on the compound of five cubes. The order of the five colors of heads is different around each five-sided opening---all the even permutations. This was a prototype model for a larger (4-foot diameter) version of this design, installed at the Burr and Burton Academy in Vermont. The name "Champy" comes from the legend of a reputed lake monster said to be living in Lake Champlain. --- George Hart