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.

Hilbert's Square-Filling Curve1836 views"Hilbert's Square-Filling Curve" by The
3DXM Consortium

In 1890 David Hilbert published a construction of a continuous curve whose image completely fills a square, which was a significant contribution to the understanding of continuity. Although it might be considered to be a pathological example, today, Hilbert's curve has become well-known for a very different reason---every computer science student learns about it because the algorithm has proved useful in image compression. See more fractal curves on the 3D-XplorMath Gallery.

--- adapted from "About Hilbert's Square Filling Curve" by Hermann Karcher

"36 circles in a dual tetrahedron pattern" by Bradford Hansen-Smith1816 viewsThe tetrahedron has four faces. The symbol of the circle is used as metaphor for nothing and for everything, and endless parts in-between. Folding circles appears to have little history: Somewhere in the history of origami lies the circle, unrecognized and discarded in favor of the square; Buckminster Fuller also folded the circle, with informational intent. Fuller is the inspiration for my own exploration into geometry and provided the seed for folding and joining circles-9" paper plates.

-- Bradford Hansen-Smith, Wholemovement

Hamid Naderi Yeganeh, "1,000 Line Segments (1)" (August 2014)1805 viewsThis 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 Yeganeh

"Figure eight knot," copyright Andrew Lipson. Made of Lego ®1795 viewsI think this is the most difficult single construction I have ever made out of Lego®. Those long sweeping curves, hanging unsupported in space... It's only when you get about 2/3 of the way up that you start to discover exactly which bits 1/3 of the way up aren't strong enough. And there are never enough 1x3 bricks... But I didn't cheat anywhere. The figure-eight knot has a nice tetrahedral skew-symmetry which the model illustrates quite well. 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)

Hamid Naderi Yeganeh, "A Bird in Flight" (November 2014)1795 viewsThis image is like a bird in flight. It shows 2000 line segments. For each i=1, 2, 3, ... , 2000 the endpoints of the i-th line segment are:
(3(sin(2πi/2000)^3), -cos(8πi/2000))
and
((3/2)(sin(2πi/2000)^3), (-1/2)cos(6πi/2000)).

I created this image by running my program. --- Hamid Naderi Yeganeh

"Black and Blue Ricochet Trio," by Gary R. Greenfield, University of Richmond, VA (2008)1764 viewsDigital print, 14" x 24". "Many of my computer generated algorithmic art works are based on visualizations that are inspired by mathematical models of physical and biological processes. These three side-by-side black and blue "ricochet compositions" were generated by placing particles on each of the sides of a 16-gon, assigning them starting angles, and then letting each move in a straight line until it encounters an existing line segment at which point it is reflected--the ricochet--and then paused so that the next particle may take its turn. Further, if a particle ricochets off its own path, then the area it has just enclosed is filled using the requisite black or blue drawing color that particles were alternately assigned." --- Gary R. Greenfield, Associate Professor of Mathematics and Computer Science, University of Richmond, Richmond, VA

"Snowflake Model 4," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1753 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

"Escher's 'Belvedere'," copyright Andrew Lipson. Made of Lego ®1736 viewsDaniel Shiu and I worked on this as a joint project. We discovered a few nasty surprises that Escher had hidden in the picture (other than the obvious one). And we had to get the camera position just right for the picture to come out OK. The domes on top, and the slightly protruding cell wall at the near end of the bottom level, were both interesting exercises in half-brick spacing, and many of those useful 1x2 plate offset bricks with the single stud on top were used. We took a small liberty with the guy in the red hat at the bottom of the picture. In Escher's original, he's holding an "impossible cube", but in our version he's holding an impossible Lego® square. Well, OK, not quite impossible if you've got a decent pair of pliers (ouch). See photos of the construction in progress . Lego® is a trademark of The Lego Group. On my website I post images of M.C. Escher's original works (C) Cordon Art, Baarn, the Netherlands on his website, used with permission, so that you may compare with the Lego® creations. All rights reserved. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)

"Aristolochia Grandiflora," by S. Louise Gould, Central Connecticut State University, New Britain (2008)1733 viewsInkjet print on treated silk, quilted and sparsely beaded to emphasize symmetries, 20" x 21.5". "My artwork usually connects textiles or paper with mathematical, specifically geometric ideas. 'Aristolochia Grandiflora' is a floral fractal. When I first saw the plant at Frederik Meijer Gardens in Grand Rapids in full bloom in May, it seemed a natural subject for exploring the seventeen wallpaper patterns in the plane. Starting with a photograph that I had taken in the garden, I sampled sections of the plant image and used KaleidoMania to generate samples of each of the seventeen wallpaper patterns. These were printed on 8.5 by 11 inch treated silk pages and folded, cut, pieced, quilted and beaded to create mathematical art to wear." --- S. Louise Gould, Associate Professor, Department of Mathematical Sciences, Central Connecticut State University, New Britain, CT

"Artistic View of the Klein Bottle," by Jean-Francois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)1727 viewsIn mathematics, the Klein Bottle is a non-orientable surface, i.e. a surface with no distinct "inner" or "outer" sides. Other related non-orientable objects include the Mobius strip and the real projective plane. Whereas a Mobius strip is a two-dimensional object with one side and one edge, a Klein bottle is a three-dimensional object with one side and no edges.

27-torus.jpg1725 views"The 27-Crossing Torus Knot T(9,4)," by Dror Bar-Natan (University of Toronto, Canada)

This is an example of a torus knot. A torus is a surface best described as a doughnut. A torus knot can be thought of as looping around and through the torus. The symbol T(9,4) means that the string making the knot loops through the hole of the torus 9 times, making 4 revolutions. This knot is drawn with TubePlot.

--- Dror Bar-Natan

"Clover-52," by Jean-Francois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)1723 viewsThis image shows the lack of associativity for addition and multiplication inside a computer. In order to be able to obtain the exact same results over the years for a certain computation, I did include the definition of some "devices" in my own programming language, which allow the definition of the precise order of the arithmetic operations: +, -, *, and / (by the way, parentheses won't do that, for example, X=A+(B+C) does not mean T=B+C then X=A+T).

This opens the door to something very powerful: The possibility to dynamically redefine the arithmetic used when launching a program. This picture and "Clover-51" are the results of the combination of eight elementary pictures: 3-clover, 4-clover, ... ,10-clover with substitutions like (A+B) --> MAX (A,B), (A*B) --> (A+B).

"Recursive Construction for Sliding Disks," Adrian Dumitrescu, University of Wisconsin, Milwaukee (2008)1712 viewsDigital print, 11" x 5". "Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk slides in the plane without intersecting any other disk, so that its center moves along an arbitrary (open) continuous curve. One can easily show that 2n moves always suffice, while the above construction shows pairs of configurations that require 2n-o(n) moves for this task, for every sufficiently large n. Disks in the start configuration are white, and disks in the target configuration are shaded. " --- Adrian Dumitrescu, University of Wisconsin, Milwaukee

"Recursive Figure-8 Knot" by Carlo Sequin, University of California, Berkeley1703 viewsIn some depictions of a mathematical knot, some of the meshes formed between the criss-crossing strands resemble the overall outline shape of the whole knot. It is then possible to fit a reduced copy of the knot into every one of these meshes and reconnect the strands so as to obtain again a mathematical knot consisting of a single closed strand. Then this process can be continued recursively resulting in a self-similar pattern. This general process was applied to the 4-crossing Figure-8 knot. But rather than performing this process in a drawing plane as outlined above, subsequent generations of reduced knot instances were placed in planes that are roughly perpendicular to one another, resulting in a truly 3-dimensional sculpture. --- Carlo Sequin

"Twice Iterated Knot No. 1," by Robert Fathauer, Tessellations Company (2008)1697 viewsThird Prize, 2009 Mathematical Art Exhibition. Digital print, 19" x 12". Fathauer makes limited-edition prints inspired by tiling, fractals, and knots. He employs mathematics in his art to express his fascination with certain aspects of our world, such as symmetry, complexity, chaos, and infinity.

"The starting point for this iterated knot is a nine-crossing knot that has been carefully arranged to allow seamless iteration. Four regions of this starting knot are replaced with a scaled-down copy of the full starting knot, incorporated in such a way that the iterated knot is still unicursal. These same four regions are then replaced with a scaled-down copy of the iterated knot, resulting in a complex knot possessing self similarity." --- Robert Fathauer, Small business owner, puzzle designer and artist, Tessellations Company, Phoenix, AZ