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.

"Seven Sided Seven Color Torus," by Faye E. Goldman (Ardmore, PA)Strips of polypropylene ribbon, 2014
This toroid shape is made from over 3200 strips of ribbon. I love the fact that there needs to be as many heptagons making the negative curvature in the center as there are pentagons around the outside. It is the fourth torus I've made and the most interesting. When I decided to create a seven sided torus, it was obvious that it needed to have seven colors to show the seven color map problem on a torus. --- Faye E. Goldman (http://www.FayeGoldman.com) Apr 06, 2015

"Trifurcation," by Robert Fathauer (Tessellations, Phoenix, AZ)Ceramics, 2014
This sculpture is a fractal tree carried through five generations. With each iteration, the number of branches is tripled. The scaling factor from one generation to the next is the inverse of the square root of 3, approximately 0.577. As more and more branches are added, the top surface begins to display the classical fractal known as the Sierpinski triangle. More information: http://mathartfun.com/shopsite_sc/store/html/Art/Trifurcation.html. --- Robert Fathauer (http://robertfathauer.com) Apr 06, 2015

"Hope in Base 8," by Sally Eyring (Watertown, MA)Woven cotton, 2013
Weaving technology is closely related to the computer industry - Hollerith cards were a direct inspiration from dobby looms. In this piece the word HOPE is translated into a weaving pattern using an 8 shaft loom. Using ANSII codes - A is represented as 101, B as 102, etc. up to Z represented as 132. First 100 was subtracted from each code to create a workable weave structure. Next, 1 was added to each code (using base 10) because weaving software programs number the shafts from 1 to 8. That resulted in representing A by 12, B by 13, etc. with Z represented as 43. Thus, the word "HOPE" is represented by 8 threads. Rotating the set of numbers by 1, 7 times, created a twill weave with a repeat of 64 threads, producing "HOPE", woven in base 8. The colors depict the colors of a sunrise; the red and orange raising out of the black of night. --- Sally Eyring (http://sallyglassdreams.blogspot.com/) Apr 06, 2015

"Constructing the Inner Apollonian," by Jeffrey Stewart Ely (Lewis and Clark College, Portland, OR)Digital print on archival paper, 2014
Unlike pictures of two-dimensional Apollonian gaskets, most renderings of the three-dimensional analogue, Apollonian sphere packing, tend to be disappointing because they do not reveal the interior structure the way that their two-dimensional
cousins do. This image tries to reveal the inner structure in several ways. First, some of the larger spheres that obstruct the view have been removed. The negative spaces caused by their removal are plain to 'see'. Second, the observer has been located in one of these negative spaces, affording a more intimate view. Finally, the process has been deliberately left incomplete, giving a sense of both the coarser and finer stages of the construction. --- Jeffrey Stewart Ely Apr 06, 2015

"The Lost Art of Cyclides Islands Weavers," by Francesco De Comité (University of Lille, France)Digital print on cardboard, 2014
Dupin cyclides are the images of tori by sphere inversion. Since sphere inversion preserves circles, the set of Villarceau circles one can draw on a torus is transformed in a set of circles on the cyclide. The game is then to find nice images illustrating this fact, together with some story 'à la Raymond Roussel' to reinforce the magic. 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. Things become yet more interesting, when you can transform your two-dimensional dream objects in real three dimensional sculptures. You can then handle your creations, and look at them from an infinity of view angles. --- Francesco De Comité ([url=http://www.lifl.fr/~decomite]http://www.lifl.fr/~decomite]/url]) Apr 06, 2015

"19 x 31," by Larry Crone (associate professor emeritus, American University, Alum Bank, PA)Print, 2014
In addition to its beauty, this image is of interest because the underlying quadratic rational function has an attracting fixed point cycle of order 19, and another of order 31. Just as a mountain presents many targets to a photographer, this function can be viewed from different perspectives, and it was hard to decide which one to use. The windows program Gplot, the camera which took this picture, is available for free download at http://www.american.edu/cas/faculty/lcrone.cfm. --- Larry Crone (http://www.american.edu/cas/faculty/lcrone.cfm) Apr 06, 2015

"Intrinsic Transformation III," by Conan Chadbourne (San Antonio, TX)Archival inkjet print, 2014
This work is part of a series of visual meditations on the structure of the alternating group on 5 elements, also known as the icosahedral group. This image explores the structure of the icosahedral group through a particular presentation by two generators. The group's elements, which appear as yellow disks in this image, are arranged at the vertices of a rhombicosidodecahedron, shown here in stereographic projection, while the group's generators, of orders 3 and 5, correspond to the regions between the disks, colored green and blue, respectively. The image is composed of multiple hand-drawn images which are digitally composited and output as an archival digital print. --- Conan Chadbourne (http://www.conanchadbourne.com) Apr 06, 2015

"Basilica," by Anne Burns (professor emerita, Long Island University, Brookville, NY)Digital print, 2014
The Julia Sets of z^n + c are familiar objects to mathematicians. In particular c=-1 yields the well-known "Basilica". Adding a term d/z(z^2-1) introduces three poles: z=0, z=-1, z=1. The orbits of initial points near the poles rapidly diverge to ∞; for very "small" (real, positive) d, amazingly, the boundary of the set of points whose orbit escapes (the Julia Set) contains an infinite number of tiny decorations resembling the decorations on the original "Basilica". --- Anne Burns Apr 06, 2015

"Three (2k+2, 2k) links," by sarah-marie belcastro (Hadley, MA)Knitted hand-dyed wool, 2013
A (p,q) torus link traverses the meridian cycle of a torus p times and the longitudinal cycle q times; when p and q are coprime, the result is a knot, and when not (ha!) the result is a gcd(p,q)-component link with each component a (p/gcd(p,q), p/gcd(p,q)) torus knot. Here we have (in increasing order of complexity) a (4,2) torus link, a (6,4) torus link, and an (8,6) torus link. Each is knitted so that both the knotting and the linking are intrinsic to the construction (rather than induced afterwards via grafting). They were made as proof-of-concept for the methodology for knitting torus knots and links that the artist introduced at the 2014 JMM. --- sarah-marie belcastro (http://www.toroidalsnark.net) Apr 06, 2015