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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"Woven Dodec," by Edmund Harriss (University of Arkansas, Fayetteville) Laser cut paper, 20 x 20 x 20 cm, 2014

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. 32 pieces of paper cut into two shapes connect and weave together to form a ball mixing the dodecahedron and icosahedron. Inspired by Quintron by Bathsheba Grossman. --- Edmund Harriss

"Ammann Cushion," by Maggi Harriss (Great Malvern, UK)Cotton cross-stitch, 38 x 38 x 5 cm, 2009

I am fascinated by mathematical patterns and enjoy using them to make something useful. Cushion with each tile shape for the Ammann-Beenker tiling in a different colour. --- Maggi Harriss

"Sword Dancing," by George Hart (Stony Brook University, Stony Brook, NY USAWood (dyed) and cable ties, 32 x 45 x 45 cm, 2015
Best textile, sculpture, or other medium, 2016 Mathematical Art Exhibition

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. This is a model for a large wood sculpture consisting of two congruent but mirror-image orbs of this design, each two meters in diameter. The sixty components of the design are "affine equivalent," meaning they can be stretched linearly to become congruent to each other. They lie in groups of three in twenty planes--the planes of a regular icosahedron which had been compressed by a factor of 1/2 along a five-fold axis. --- George Hart

"Waves - Offering to the Moon," by Veronika Irvine & Lenka Suchanek (University of Victoria, British Columbia, Canada) Stainless steel wire, shell, driftwood cedar frame, 40 x 36 x 9 cm, 2015

"Waves" was designed and created by lenka using a tessellation pattern generated algorythmically by Veronica. Bobbin lace, a 500-year-old art form, features delicate patterns formed by alternating braids. Lenka: "I had a beautiful frame made from old growth, driftwood red cedar and I needed a pattern that would look like the waves of the Pacific Ocean... The model is an incredible source of designs--every graph has so many variations for working the stitches and each combination results in a different pattern. I love the experimental nature of the work. --- Veronika Irvine & Lenka Suchanek

"45 Poppies," by Karl Kattchee (University of Wisconsin-La Crosse)Digital print, 18 x 31 cm, 2015
Best photograph, painting, or print, 2016 Mathematical Art Exhibition

This image is a classification of all closed paths, on a 6x6 grid, with the following properties: First, each path must proceed around the center of the grid and be orthogonal in the sense that every turn is 90 degrees. Also, the path must use each row and column exactly once. Finally, we require that each path be asymmetrical, and we do not distinguish between paths which differ by a rotation or flip. Each center square is colored black, and the shades of red are dictated by the winding number of each region. Acknowledgements: Craig Kaplan (Waterloo), for helpful notation and the coloring scheme idea, and artists Kate Hawkes and Misha Bolstad (UW-La Crosse) for the poppies idea. --- Karl Kattchee

"Catalan Connections: Level Four," by Margaret Kepner (Washington, DC)Archival inkjet print, 40 x 60 cm, 2015

I enjoy exploring the possibilities for conveying ideas in new ways, primarily visually. I have a background in mathematics, which provides me with a never-ending supply of subject matter. The Catalan numbers are a sequence of positive integers that provide answers to certain combinatorial questions. For example, in how many ways can a polygon with n+2 sides be cut into triangles? A hexagon (setting n=4) can be triangulated in fourteen different ways, so the 4th Catalan number is 14. Other types of problems also lead to the Catalan numbers: counting binary trees, balancing parentheses, finding paths through a grid, shaking hands in a circle, etc. This piece is composed of diagrams representing seven different problems; for each of these, the answer is the 4th Catalan number. The solution sets for the problems are displayed in diagonal bands. The columns indicate correspondences between elements in different solution sets. --- Margaret Kepner

"Hexahedron 2," by Dorothy McGuinness (Everett, WA)Watercolor paper, acrylic paint, polyester thread, 38 x 38 x 38 cm, 2014

I create forms art of diagonal twill and structures not normally found in the basketry world. My medium for this unique work is watercolor paper, which I've painted and cut into very narrow uniform strips to achieve the precision I seek. I am very much interested in the math and geometric constraints of the work. Using hundreds of strips of paper at a time, I explore new structural forms. An interpretation of a hexahedron using diagonal twill to form a woven sculptural basket. --- Dorothy McGuinness

"Ovoid Bead with three Hyperbolic Axes as a Lamp," by Gabriele Meyer (University of Wisconsin, Madison)Photograph, 2016

I like to crochet hyperbolic surfaces. This object started as a hollow ovoid, top and bottom missing. I then crocheted three vertical axes down the sides. These three axes are the basis for the hyperbolic crochet. The object is hung from the ceiling.
I then entered a light tube. The photograph was taken in the dark without flash. --- Gabriele Meyer

"Shield 1," by Kerry Mitchell (Phoenix, AZ)Digital print on aluminum panel, 40 x 40 cm, 2014

My work is composed primarily of computer generated, mathematically-inspired, abstract images. I draw from the areas of geometry, fractals and numerical analysis, and combine them with image processing technology. This image was created using a dynamic version of the Chaos Game algorithm. The Chaos Game is a simple example illustrating chaotic motion and strange attractors. It is typically implemented using three anchor points, which become the vertices of a Sierpinski triangle. The Dynamic Chaos Game allows the anchor points to move each iteration. Here, the image comprises seven panels. In each, the three anchor points slid along line segments, with different speeds. The pixels were then colored according how frequently that point was visited during the iteration. --- Kerry Mitchell

"OSU Triptych No. 2," by Robert OrndorffPaper and acrylic, 20 x 46 cm, 2015
Honorable Mention, 2016 Mathematical Art Exhibition

My paper folding art is math plus paper. Folded paper is simply tangible math. This is a permanent manifestation of an ephemeral artwork, namely, one solution for a specific one-straight-cut problem. Such problems are usually stated as follows: How must one fold a paper rectangle into a flat figure such that one straight cut through all of the layers will produce a given planar straight-line graph? Here the problem has been solved with paper and then represented in acrylic. To a significant degree the work relies on transmitted and reflected light, and so it never looks the same twice. The figure (the letters "OSU") has been divided into three frames. The crease patterns for the left and right letters are pedestrian but that for the central letter is sublime. --- Robert Orndorff

"Walnut Star," by David Reimann (Albion College, Albion, MI)Walnut veneer and brass fasteners, 38 x 38 x 38 cm, 2015

I am interested in creating patterns that convey messages at multiple levels and scales using a wide variety of mathematical elements and media. This form is based on the small rhombicosidodecahedron, an Archimedean solid with 120 edges. The underlying polyhedral edges have been replaced by 4.75 cm squares made from laser-cut paper-backed walnut veneer and connected at their corners with brass split-pin fasteners. The 62 faces (squares, hexagons, and pentagons) and 60 vertices of the underlying polyhedron are transformed into open negative space. The expansion of linear edges into squares results in a sphere-like shape with 20 knobs. --- David Reimann

"Levy Dragon Outside Tapestry," by Larry Riddle (Agnes Scott College, Decatur, GA)Back stitch embroidery on 18 point canvas, 31 x 31 cm, 2013

I have been working with needle crafts since graduate school. I have also been interested in fractals and fractal geometry for more than 20 years. The Levy Dragon Outside Tapestry consists of four copies of the Levy Dragon built from the edges of a square. The iteration steps are repeated on each of the four sides of the square with the initial triangle motif placed outside the square. This back stitch design shows the twelfth iteration for this outside construction.The outside tapestry was done in two shades of blue to better show each of the four copies of the Levy dragon. --- Larry Riddle

"Kokabi Stars," by Reza Sarhangi (Towson University, Towson, MD)Tile, 50 x 50 cm, 2015

I am interested in Persian geometric art and its historical methods of construction. Kokabi Star (the great pentagram) can be constructed using the lines of the 10/3 star polygon. Patterning this star can be achieved using different approaches. Some of the presented stars in this artwork have been made based on the actual tiling on existing buildings. Some others have been constructed based on old treatises and scrolls. Some of the patterns have been created using the traditional compass-straightedge process. Modularity is another approach in this regard. Moreover, the two decorated quasicrystal patterns of Star and Sun (the only two quasicrystal patterns with global five-fold rotational symmetry) and their striking relationships with Kokabi Star have been presented. Is this relationship a theorem? --- Reza Sarhangi

"Stereographic projection (grid)," by Henry Segerman (Oklahoma State University, Stillwater)3D printed nylon plastic, lamp, 10 x 9 x 9 cm, 2014

The light rays from the lamp are partly blocked by the shrinking design on the sphere; the resulting shadow is a regular tiling of the plane by squares. This illustrates how stereographic projection transforms the sphere, minus the north pole, into the plane. Note how shapes are slightly distorted near the south pole, and dramatically distorted near the north pole. --- Henry Segerman

"Snap-together Super-Bottle of Genus 4/σ," by Carlo Séquin (University of California, Berkeley)ABS plastic, printed on an FDM machine, 16 x 20 x 14 cm, 2015

Stimulated by the LEGO-Knot project, I aimed to design a set of modular parts that permits to compose not only various handle-bodies, but also single-sided surfaces of higher genus. The modular parts employed in my sculptures are tubular 3-way junctions, where one of the tubular stubs exposes the opposite side of the surface shown by the other two stubs. Depending on how the parts are connected, the resulting compositions remains orientable or becomes single-sided; in the latter case they correspond to sums of multiple Klein bottles; which I call "Super-Bottles." The two identical parts of which the sculpture is composed can be put together in three different ways. In two cases the resulting surface is single-sided (σ = 1) and in the third case it is double-sided (σ = 2). The genus of the resulting surface is 4/σ. The configuration shown is a non-orientable surface of genus 4, corresponding to the connected sum of two Klein Bottles, with two punctures. The insets show the two individual parts, and an assembly of them resulting in a 2-hole torus of genus 2 (with two punctures). --- Carlo Séquin