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.

"Equivalent," by Robert Bosch and Derek Bosch (Oberlin College, OH)6" x 6" x 2", Nylon (selective laser sintering), 2011

The mathematician in me is fascinated with the various roles that constraints play in optimization problems: sometimes they make problems much harder to solve; other times, much easier. Equivalent is a three-piece, 3D-printed sculpture that consists of three topologically equivalent variations of the Borromean rings. In the Borromean rings, no two of the three rings are linked, so if any one of them is destroyed, the remaining two rings will come apart. For the photograph, we positioned Equivalent on a piece of Lenox china (a wedding gift). --- Robert Bosch (Oberlin College, OH, http://www.dominoartwork.com)May 14, 2012

"Möbius Hanging Gardens," by Tatiana Bonch-Osmolovskaya (Sydney, Australia)1276 x 1800 pixels, computer graphics, 2011

I use 2D and 3D computer graphics as well as photographs made by myself or my friends, to show the beauty of these objects, thus uniting the intellectual wonder of perceiving a mathematical concept with the aesthetical pleasure of viewing a beautiful image. Hanging gardens of Babylon were built in the desert as a wonder of land amelioration and engineering. In our era humanity continues to perform such wonders, e.g. in desert Australia. While the flowers on my picture, which have grown on the Möbius strip over the Australian plain, were placed there by computer graphics, it is the hard work of those who make the Red Continent green that invokes our admiration. Photographs of Australian views and the Florida festival in Canberra were used in this image. --- Tatiana Bonch-Osmolovskaya (Sydney, Australia, http://antipodes.org.au)May 14, 2012

"K_7 embedded on a torus," by sarah-marie belcastro (Hadley, MA)11" x 11" x 4.5", Knitted cotton (Reynolds Saucy), 2010

I am a mathematician who knits as well as a knitter who does mathematics. It has always seemed natural to me to combine mathematics and knitting, and it is inevitable that sometimes the results will be artistic rather than functional. The Heawood bound shows that K_7 is the largest complete graph that can embed on the torus. This is an embedding of K_7 on the torus with all vertices centered on the largest longitude. It is the second knitted instantiation of this embedding; this version is larger and has a larger face-to-edge proportion than the first, which was exhibited at Gathering for Gardner 7 in 2006. --- sarah-marie belcastro (Hadley, MA, http://www.toroidalsnark.net)May 14, 2012

"Magneto-2," by Reza Ali (Palo Alto, CA)18" by 24" print, 2011

This image is a snap-shot from a real-time interactive particle simulation using Lorentz's Law to define each particle's movements. The color palette, perspective, magnetic field placement, and rendering style were designed by the artist. Physics and mathematics define the piece's motion and overall pattern formation. --- Reza Ali (Palo Alto, CA, http://www.syedrezaali.com/)May 14, 2012

"Moresque No.1," by Erica RollingsStained Glass. From the Grammar of Ornament series.

The Escher-esque quality of the shields appealed to me, especially as they leave a negative-space dodecagon in the center. I kept the colors in symphony with each other to add to the complexity of the basic design. --- Erica Rollings Glass Works (www.ericarollings.net) Apr 03, 2012

"Middle Ages No. 5," by Erica Rollings Stained Glass. From the Grammar of Ornament series.

I loved the combination of knots and leaves, with the letters providing a hint of heraldry. One can easily picture a knight's shield, and hence the background color of war. I chose the yellow and purple glass for their red references. This piece is constructed of dichroic glass which appears one color when backlit (as in the photo) and an entirely different color when light is reflected off the surface instead. --- Erica Rollings Glass Works (www.ericarollings.net) Apr 03, 2012

"Coelenterates," by Erica RollingsStained Glass.

Coelenterates--otherwise known to us laypeople as Jellyfish. If you've ever had the misfortune to encounter one while swimming then you know they're notoriously hard to see in water. To find it, search for hexagons. Its body is represented by regular hexagons, while its tail, the nasty part, is represented by elongated hexagons. --- Erica Rollings Glass Works (www.ericarollings.net) Apr 03, 2012

"Byzantine No.1," by Erica Rollings Stained Glass. From the Grammar of Ornament series.

This design appealed because of the interconnection of always-pleasing circles and squares. I wanted the geometric qualities to be represented in warm earthy colors to create an aesthetic tension and provided more tension by opposing with a cool background. --- Erica Rollings Glass Works (www.ericarollings.net) Apr 03, 2012

"Scherk's First Surface," copyright Andrew Lipson. Made of Lego®This 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)Feb 06, 2012

"Escher's 'Relativity'," copyright Andrew Lipson. Made of Lego ®Daniel Shiu and I worked on this as a joint project. There are no camera tricks, but the picture has to be taken from exactly the right place, and that was a challenge in itself. Unlike many of Escher's other "impossible" pictures (like "Ascending and Descending"), there is actually no optical illusion involved here. Gravity seems to be working in three different directions simultaneously, but the picture shows a perfectly self-consistent physical scene. So modelling it should certainly be feasible. But while Escher's picture has three different "up"s, Lego® isn't quite so flexible. 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)Feb 06, 2012

"Figure eight knot," copyright Andrew Lipson. Made of Lego ®I 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)Feb 06, 2012

"Escher's 'Belvedere'," copyright Andrew Lipson. Made of Lego ®Daniel 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)Feb 06, 2012

"Escher's 'Ascending and Descending'," copyright Andrew Lipson. Made of Lego ®Daniel Shiu and I worked on this as a joint project. There are no camera tricks, but the picture has to be taken from exactly the right place, and the final photograph was slightly distorted to emphasize the perspective effect. I'm especially pleased with the way the roof in the top left of the picture came out. 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)Feb 06, 2012

"RedCenter," (detail) by Mike Field (University of Houston)"RedCenter" is a section of a planar repeating "two-color" pattern of type pmm' (or pmm/pm in Coxeter notation). The underlying repeating pattern has reflection symmetries and two-fold rotational symmetries as well as translation symmetries and, less obviously, glide reflection symmetries. Roughly speaking, half the symmetries preserve colors and half interchange colors. (The 46 two-color repeating patterns of the plane were originally classified by H. J. Woods of the Textile Physics Laboratory, University of Leeds, in 1935-36.) The pattern was generated using a determinsitic torus map and the coloring reflects the density of two invariant measures on the torus. The name "RedCenter" is suggested by Uluru (Ayers Rock) in Central Australia. --- Mike FieldAug 26, 2011

"Floating Pentangle Construction," by Bente Simonsen (Landeryd, Sweden)Digital print, 20" x 24", 2010

Impossible pentangle construction, 2D and 3D mix-illusion. --- Bente Simonsen (http://geometric-impossibilities.blogspot.se) Mar 10, 2011