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.

 Hamid Naderi Yeganeh, "1,000 Line Segments (2)" (August 2014)This image shows 1,000 line segments. For each i=1,2,3,...,1000 the endpoints of the i-th line segment are: (-sin(4πi/1000), -cos(2πi/1000)) and ((-1/2)sin(8πi/1000), (-1/2)cos(4πi/1000)). I created this image by running my program on a Linux operating system. --- Hamid Naderi YeganehOct 01, 2014
 Hamid Naderi Yeganeh, "1,000 Line Segments (1)" (August 2014)This image shows 1,000 line segments. For each i=1,2,3,...,1000 the endpoints of the i-th line segment are: (-sin(2πi/1000), -cos(2πi/1000)) and ((-1/2)sin(8πi/1000), (-1/2)cos(12πi/1000)). I created this image by running my program on a Linux operating system. --- Hamid Naderi YeganehOct 01, 2014
 "Sierpinski Tringle at Brean Down," by Simon Beck (https://www.facebook.com/snowart8848/) Pattern made of footprints in the sand and a rake as a tool was made at Brean Down, UK. It's a race against the tide, it's nine hours of furious work. -- Simon Beck (https://www.facebook.com/snowart8848/Aug 19, 2014
 "Serpinski Triangle at Peisey-Vallandry," by Simon Beck (https://www.facebook.com/snowart8848/)Pattern made in snow with snowshoes at Peisey-Vallandry. --- --- Simon Beck (https://www.facebook.com/snowart8848/)Aug 08, 2014
 "Mandelbrot set in the sand, view two" by Simon Beck (https://www.facebook.com/snowart8848/) Pattern made of footprints in the sand and a rake as a tool at Brean Down in the U.K. It's a race against the tide, it's nine hours of furious work. Find a time lapse video of the creation at http://youtu.be/MVzGyAAtHiU --- Simon Beck (https://www.facebook.com/snowart8848/) Aug 08, 2014
 "Mandelbrot set in the sand," by Simon Beck (https://www.facebook.com/snowart8848/) Pattern made of footprints in the sand and a rake as a tool at Brean Down in the U.K. It's a race against the tide, it's nine hours of furious work. Find a time lapse video of the creation at http://youtu.be/MVzGyAAtHiU --- Simon Beck (https://www.facebook.com/snowart8848/) Aug 08, 2014
 "Hopf Knott, " by Peter Sittner (Kensington, MD)13.5" x 13.5" x 13.5", red oak, 2013 Hopf Knott is the offspring of two forms that have intrigued me for some time, the Hopf Link and the Borromean Rings. While the sculpture may appear to be a series of connected Hopf Links, it is actually two sets of Borromean Rings, an inner set and an outer set. Of course, the "rings" are stretched to make the crossings possible and shaped to resemble Hopf Links. The 6 outermost points correspond to the vertices of a regular octahedron. --- Peter Sittner May 05, 2014
 "Dance of Stars I, " by Reza Sarhangi (Towson University, Towson, MD)12" X 12" X 12", heavy paper, 2013 "I am interested in Persian geometric art and its historical methods of construction. " Dance of Stars I is one of the Kepler-Poinsot polyhedra, the Small Stellated Dodecahedron, with Schläfli Symbol (5/2, 5). It has been ornamented by the sâzeh modular tiles, that are used in a majority of tiling that conform to local fivefold symmetries. In an article that appeared in Science, the authors proposed the possibility of the use of a set of tiles, girih tiles, by the medieval craftsmen, for the composition of the underlying pattern. Ink outlines for these girih tiles appear in panel 28 of the Topkapi scroll. I used girih tiles and left the dashed outlines in the final tessellation. I also included rectilinear patterns that appear as an additional small-brick pattern in the decagonal Gunbad-i Kabud tomb tower in Maragha, Iran. --- Reza Sarhangi May 05, 2014
 "Longest Crease J, " by Sharol Nau (Northfield, MN)10" x 10" x 10", book sculpture, 2013 "A classical calculus problem, the so-called Paper Creasing Problem, deals with extremizing the length of the crease formed when a rectangular sheet of paper is altered by folding one corner, say the lower right-hand corner, to the opposite edge. The length of the crease depends on the distance the active corner is from the upper left-hand corner." This sculpture formed by folding individual pages is an example of the variety of the three-dimensional forms which can be obtained by incremental changes in the length of the crease from page to page and by extending the points that can be used. --- Sharol Nau (http://www.sharolnau.com) May 05, 2014
 "Jose, " by So Yoon Lym (Paterson, NJ)22" x 30", acrylic on paper, archival pigment print, 2009 "The Dreamtime is inspired by the Aboriginal stories and visions of creation. Each braided pattern, carried by the students, is a map of the ancient universe, a topographical palimpsest of the world in pattern: valleys, mountains, forests, oceans, rivers, streams. " The link to Dr. Ron Eglash's website on cornrow hair-braiding best details the mathematic content behind my hair and braid paintings. Professor Eglash's website called "Transformational geometry and iteration in cornrow hairstyles" outlines one aspect of his research in ethno-mathematics and cybernetics. Ethnomathematics "aims to study the diverse relationship between math and culture." More information: --- So Yoon (http://www.soyoonlym.com/works/dreamtime/) May 05, 2014
 "Untitled, " by Jack Love (graduate student, George Mason University, Fairfax, VA)Spherical, 18" in diameter, medium-density fiberboard, 2013 "The Platonic solids have been the inspiration for the pieces I have created thus far. My work explores the structure of these objects and their relationships to one another, and attempts to express this structure in a way that is aesthetically appealing. " Take either the icosahedron or dodecahedron and center it at the origin. Project its vertices outward from the origin onto the surface of a sphere surrounding it, giving a collection of points on a sphere. Draw a great circle through two points if they are images of two adjacent vertices in the original polytope. Each of these great circles is partitioned into arcs by its intersection with the other great circles thus produced. The arcs come in three lengths and are projections of the edges of, respectively, an icosahedron, a dodecahedron, and a third polytope whose facets are rhombic. This model exhibits this construction. The convex arcs correspond to the icosa, the concave to the dodeca, and the straight to the rhombic. --- Jack Love May 05, 2014
 "Magic Square 8 Study: A Breeze over Gwalior," by Margaret Kepner (Washington, DC)20" x 20", archival inkjet print, 2013 "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. " This work is based on a magic square of order 8, expressed in a visual format similar to a traditional quilt pattern. The magic square, known as the Gwalior Square, is an 8x8 array of numbers from 0 to 63, such that every row and column adds up to 252, the 'magic constant.' The two main diagonals, as well as every broken diagonal, also sum to 252. The numbers in the square are represented in base 2 and base 4. Nested squares serve as the number places in the base systems, and suggest the Log Cabin quilt structure. For each of the 64 squares, half is shown in base 2 and the other half in base 4. The squares are oriented to create the 'pinwheel' quilt pattern. This pattern groups together 2x2 arrays of 4 numbers, all of which sum to 126. --- Margaret Kepner (http://mekvisysuals.yolasite.com) May 05, 2014
 "Evolution of the Twin Tornados, " by Hartmut F. W. Höft (professor emeritus, Eastern Michigan University, Ann Arbor, MI)7.5" x 9" (10.5" x 12" framed), archival photographic digital paper, 2013 "I like to experiment with simple planar geometric figures. Starting with one outline I move it, rotate it, and change its proportions. " In this picture 12 images are arranged in a spiral. A primordial sea in the center starts the process. One parameter of the spine function increases, then stays constant, while the other increases and then decreases again. As the sequence progresses along the spiral the spine function increases in complexity so that the images billow out like clouds and then start separating into two funnels as the second parameter decreases again. Each image on the spiral is composed of 289 ellipses with a closed spine curve though it does not appear to be closed since the axes of the elipses decrease to zero at the bottom. The graphics were rendered in Mathematica 8 and printed on archival, photographic digital paper. --- Hartmut F. W. Höft (http://people.emich.edu/hhoft/) May 05, 2014
 "View of the platycosm -a2, decorated with AMS, " by Charles Gunn (Technisches Universität Berlin, Germany)18" x 18", print of computer image, 2013 "I enjoy exploring two- and three-dimensional tessellations -- euclidean, spherical and hyperbolic, especially the "insider's" view of such spaces. I have been experimenting recently with alternative perspective rendering based on a spherical canvas surrounding the viewer. " This is an insider's view of the platycosm -a2. ("Platycosm" is John Conway's term for a compact euclidean 3-manifold.) This platycosm is generated by a translation, a glide reflection and a screw motion with order-2 rotational part, in three mutually perpendicular directions. The fundamental domain is a cube, whose edges are rendered via the textured beams. The "geometry" consists of the initials AMS, in honor of one of the hosting orgranizations of the art show. The image is rendered conformally from the viewable sphere of an insider positioned in the scene. I sometimes call this "six-point perspective" rendering since one may see vanishing points not just in the x,y, and z directions but also in the -x, -y, and -z directions. --- Charles Gunn (http://page.math.tu-berlin.de/~gunn) May 05, 2014
 "Blue Torus, " by Faye E. Goldman (Ardmore, PA)10" x 10" x 2.5", strips of ribbon, 2011 Honorable Mention 2014 Mathematical Art Exhibition "I was drawn to modular origami by its structure and mathematical properties. More recently I found the Snapology technique by H. Strobl, which allows great creativity with very few rules using only strips of material… It has provided insights into mathematical ideas. " This toroid shape is made from over 2400 strips of ribbon. It was the first non-convex shape I've made. 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. --- Faye Goldman (http://www.FayeGoldman.com) May 05, 2014
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 Art & Music, MathArchives Geometry in Art & Architecture, by Paul Calter (Dartmouth College) Harmony and Proportion, by John Boyd-Brent International Society of the Arts, Mathematics and Architecture Journal of Mathematics and the Arts Mathematics and Art, the April 2003 Feature Column by Joe Malkevitch Maths and Art: the whistlestop tour, by Lewis Dartnell Mathematics and Art, (The theme for Mathematics Awareness Month in 2003) MoSAIC - Mathematics of Science, Art, Industry, Culture Viewpoints: Mathematics and Art, by Annalisa Crannell (Franklin & Marshall College) and Marc Frantz (Indiana University) Visual Insight, blog by John Baez