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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 Hamid Naderi Yeganeh, "Heart" (November 2014)1475 viewsThis image contains a heart-like figure. It shows 601 line segments. For each i=1, 2, 3, .... , 601 the endpoints of the i-th line segment are: (sin(10π(i+699)/2000), cos(8π(i+699)/2000)) and (sin(12π(i+699)/2000), cos(10π(i+699)/2000)). I created this image by running my program. --- Hamid Naderi Yeganeh
 "Ten Triangular Prisms," by Magnus Wenninger (Saint John's Abbey, Collegeville, MN)1473 viewsPaper, 9" x 9" x 9", 2010 Robert Webb's Stella program is now the computer program I use for the construction of all the  polyhedron models I have recently been making. It is the program par excellence I now use for  the discovery of any new polyhedra, especially any I have never made before. The photo shows a model of Ten Triangular Prisms, recently made by me. I found the Stella version on  a web page called '75 Uniform Polyhedra' done by Roger Kaufman. It is #32 on this web page. The Stella  version gives me a 3D computer view in 10 colors and allows me to choose the size of the model and  thus also the size and shape of the net to be used for the construction of the model. However, I wanted  my model to be done using only 5 colors. This is where the artwork comes into play. The model now  shows each prism with its faces in one color of the five. Thus it becomes uniquely artistic in appearance. --- Magnus Wenninger (http://www.saintjohnsabbey.org/wenninger/)
 ashley.jpg1469 views"Ashley Knot," by by Rob Scharein (Centre for Experimental and Constructive Mathematics, Simon Fraser University, B.C., Canada) This example illustrates the SE rendering mode in KnotPlot, which visualizes the symmetric energy distribution. KnotPlot is a program to visualize and manipulate mathematical knots in three and four dimensions, and the website includes a wealth of resources and pictures. This picture is a direct screen capture from KnotPlot, rendered entirely in OpenGL, an environment for portable, interactive graphics applications. --- Rob Scharein
 symmetry3.jpg1466 views"Symmetry Energy Image III," by Rob Scharein (Centre for Experimental and Constructive Mathematics, Simon Fraser University, B.C., Canada) This example illustrates the SE rendering mode in KnotPlot, which visualizes the symmetric energy distribution. KnotPlot is a program to visualize and manipulate mathematical knots in three and four dimensions, and the website includes a wealth of resources and pictures. This picture is a direct screen capture from KnotPlot, rendered entirely in OpenGL, an environment for portable, interactive graphics applications. --- Rob Scharein
 "Imaginary Garden," by Anne M. Burns (Long Island University, NY)1450 views"Mathscapes" are created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector. See the Gallery of Mathscapes and find citations for my articles on modeling trees, plants and mountains, and on "blending and dithering" at http://myweb.cwpost.liu.edu/aburns/gallery/gallery.htm. --- Anne M. Burns (Long Island University, Brookville, NY)
 "Snowflake Model 13," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1445 viewsIn nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David Griffeath
 "Spirolateral_{6}^{1,3,4} ," by Mary Candace Williams. Quilt copyright 2005 Mary Candace Williams; design is copyright Robert Krawczyk; photograph by Robert Fathauer.1442 viewsThe challenge of doing a quilt that has only rotational symmetry and uncommon angles lies in a technique called partial seaming. The printed fabric has a lot of pursuit curves overlaide on top of one another and shows a lot of contrast with the black of the spirolateral. --- Mary Candace Williams
 Circle 11441 viewsComputers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.
 "Snowflake Model 3," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1437 viewsIn nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David Griffeath
 "Crane," by Zdenka Guadarrama, Rockhurst University, Kansas City, MO (2008)1422 viewsMobile--Gauze, papyrus, silver and wood, 10" x 10" x 15". "'Crane' represents the continuous dimensional transition from a point, represented by a silver sphere, to a line, a plane and finally a crane. This transition is depicted in parallel to the evolution of the creative process which starts with an idea, represented by the same silver sphere, and which through refinements and trials culminates in the bird as well. [My] projects consist in artistic explorations that happen in parallel to the teaching/learning of mathematics (measure theory or complex analysis, for example). I search to generate art using mathematics and art inspired in the mathematics that I share with my students in order to motivate them to learn more mathematics, to make some extra connections, and to create some art of their own." --- Zdenka Guadarrama, Assistant Professor of Mathematics, Department of Mathematics and Physics, Rockhurst University, Kansas City, MO
 "Scherk's First Surface," copyright Andrew Lipson. Made of Lego®1417 viewsThis is a nice example of a saddle point. The model shows (most of) one cell of a doubly-periodic Scherk surface. Actually Scherk discovered more than one minimal surface in 1835, but this one has the particularly simple parametrisation given by exp(z) = cos(x)/cos(y). This model shows the surface in the region |x|, |y| < p/2 - 0.01. As with most of my mathematical surfaces, I made use of some computer assistance. On my website you can find more pictures and an LDRAW .DAT file generated by my program for this sculpture. Beware--the .DAT file builds it out of 1x1 bricks. Actually constructing this out of larger bricks so that it holds together is a (non-trivial) exercise! Lego ® is a trademark of The Lego Group. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)
 "Flow 4," by Elizabeth Whiteley (2008)1413 viewsMuseum board and acrylic paint, 7.5" x 13.5" x 10.5". "'Flow 4' is created by the close proximity of two Golden Triangles (base angles of 72 degrees and vertex angle 36 degrees). The plane of each triangle is curved in opposing directions to create an aesthetically pleasing form. The sculpture changes our perception of a static and planar geometric shape and makes for a dynamic visual experience. The curves move the eye around the form and suggest multiple points of view." --- Elizabeth Whiteley, Studio artist, Washington, DC
 Circle 31406 viewsComputers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.
 Ulfah (Harmony)," by Nathan Voirol (2008)1399 viewsSilkscreen print on paper, 20" x 24". "Islamic star pattern based on a tessellation of a 54 pointed star surrounded by 9 and 18 pointed stars in a hexagonal repeat. My primary artistic interest is in designing repeatable patterns--I particularly enjoy creating geometric star and floral designs, which stem from my fascination with Islamic art." --- Nathan Voirol, CAD Drafter / Freelance Artist, Santa Barbara, CA
 "Julia's Loops," by Jennifer Ziebarth (California College of the Arts, Oakland, CA)1389 viewsDigital print, 16" x 13", 2009. This fractal image is based on a Julia set, visible in dark blue along the intersections of the loops. The loops, which all begin and end on the Julia set, also exhibit self-similarity, and hint at the existence of more small loops hidden behind the larger loops. "I have always been fascinated with repetition, abstraction, and the search for pattern, and this is what drew me to mathematics. As a mathematical artist, this love of repetition and detail has lead me to fractal art. As a mathematician teaching at an art college, some of my work is pedagogical in the sense of illustrating mathematical concepts in aesthetically pleasing ways; some of it is purely visual play." --- Jennifer Ziebarth (California College of the Arts, Oakland, CA)
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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