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 ands, 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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"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 Mar 09, 2016

"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 Mar 09, 2016

"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 Mar 09, 2016

"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 Mar 09, 2016

"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 Mar 09, 2016

"Fibonacci Downpour," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City, MD)Merino 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 Mar 09, 2016

"Brown and Green Egg -163," by Faye Goldman (Ardmore, PA)Strips 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 Mar 09, 2016

"Dragony Curve," by Robert Fathauer (Tessellations Company)Ceramics, 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 Mar 09, 2016

"An Iris Spiral," by Frank A. Farris (Santa Clara University, San Jose, CA)Aluminum 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 Mar 09, 2016

"A Fractal Circle Pattern on the {3,12} Polyhedron," by Doug Dunham (University of Minnesota - Duluth, MN)Printed 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 Mar 09, 2016

"A Steiner Chain Trapped Inside Two Sets of Villarceau Circles," by Francesco De Comité (Univeristy of Lille, France)Digital 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 Mar 09, 2016

"Poincaré’s 'Pas de deux'," by Jean Constant (Hermay, Santa Fe, NM)Mathematics 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 ConstantMar 09, 2016

"Intrinsic Regularity," by Conan Chadbourne (San Antonio, TX)Archival 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 Mar 09, 2016

"Tangent Discs I," by Anne Burns (professor emerita, Long Island University Brookville, NY)Digital 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 Mar 09, 2016

"A Radin-Conway Pinwheel Lace Sampler," by Douglas G. Burkholder (Lenoir-Rhyne University Hickory, NC)Digital 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. BurkholderMar 09, 2016