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.

"Science/Art," by Erik Demaine (MIT, Cambridge, MA) and Martin Demaine (MIT, Cambridge, MA)931 views22" (tall) x 28" (wide), framed poster, elephant hide paper, 2011

The crease pattern (top) folds into both SCIENCE and ART (bottom, not to scale). More precisely, the rectangular paper sheet folds into the 3D structure of the word SCIENCE, while the gray inking in the sheet (top) forms the inked ART in the background (bottom). The message is that science and art can exist on a common plane, as two different perspectives of the same object. The crease pattern was designed using an algorithm by Demaine, Demaine, and Ku (2010), which describes how to efficiently fold any orthogonal "maze" (including word outlines like SCIENCE) from a rectangle of paper. Red lines fold one way and blue lines fold the other way. --- Erik Demaine (Massachusetts Institute of Technology, Cambridge, MA, http://erikdemaine.org/art/scienceart/)

"Nueve y 220-B," by Juan G. Escudero (Universidad de Oviedo, Spain)926 views50cm x 26 cm, Digital Print, 2011

A possible way to remove the gap between the worlds of sciences and humanities, is the search for interconnections between mathematics and physics with the sound and visual arts. This work is based on a family of algebraic surfaces with many nodal singularities. They have been introduced recently, by means of a kind of duality in the basic geometric constructions corresponding to the generation of substitution tilings ("A construction of algebraic surfaces with many real nodes". http://arxiv.org/abs/1107.3401). Here the surface is a nonic with 220 real nodes. In general, the surfaces have degrees divisible by three and cyclic symmetry. They appear as mirror pairs not necessarily topologically inequivalent (see the sextic with 59 real nodes in arXiv:1107.3401). --- Juan G. Escudero (Universidad de Oviedo, Spain)

"Conical panoramic view of the George Eastman House grounds," by Andrew Davidhazy (Rochester Institute of Technology, NY)861 viewsPhotograph, circa 1990

My area of interest is the application of mathematical concepts in technical applications of photography. Be it quantification of phenomena or the design and use of photography to visualize physical and mathematical concepts. A camera that rotated a circular piece of film past a radial slot acting as a shutter exposed the film for more than two rotations of the camera and thus recorded two plus views of the House grounds each covering a sector of about 120 degrees or so designed so that the 360 degree view of the grounds would produce a sector that could be cut and formed into a conical lampshade. Sometimes this photo is confused with those that a fisheye lens might make but the fisheye lens could only make a single image of the House per frame. Here there are two. --- Andrew Davidhazy (Rochester Institute of Technology, NY, http://people.rit.edu/andpph/)

"Pythagorean Tree," by Larry Riddle (Agnes Scott College, Decatur, GA)729 views16" X 20", Digital print, 2011

The traditional Pythagorean Tree is constructed by starting with a square and constructing two smaller squares such that the corners of the squares coincide pairwise (thus enclosing a right triangle), then iterating the construction on each of the two smaller squares. When viewed as an iterated function system, however, one can start the iteration with any initial set. For this image I began with a common picture of Pythagoras as the initial set. The trunk of the tree was constructed using 10 iterations of a deterministic algorithm based on an iterated function system with three functions - the identity function, a scaling and rotation by 45 degrees, and a scaling and rotation by 45 degrees with a reflection. This gives a reflective symmetry for the trunk. The leaves consist of 500,000 points plotted using a random chaos game algorithm and colored based on a "color stealing" algorithm for iterated function systems described by Michael Barnsley in a 2003 paper. To give the leaves a realistic shading, the colors were from a digital photograph of a field of green and yellow grass. -- Larry Riddle (Agnes Scott College, Decatur, GA, http://ecademy.agnesscott.edu/~lriddle)

"Round Möbius Strip," by Henry Segerman (University of Melbourne, Australia)723 views152mm x 62mm x 109mm, PA 2200 Plastic, Selective-Laser-Sintered, 2011

My mathematical research is in 3-dimensional geometry and topology, and concepts from those areas often appear in my work. The usual version of a Möbius strip has as its single boundary curve an unknotted loop. This unknotted loop can be deformed into a round circle, with the strip deformed along with it. This shows a particularly symmetric result. The boundary of the strip is the circle in the middle, and the surface "goes through infinity", meaning that the grid pattern should continue arbitrarily far outwards. To save on costs, I have removed the grid lines that would require an infinite amount of plastic to print. --- Henry Segerman (University of Melbourne, Australia, http://www.ms.unimelb.edu.au/~segerman/)

"Fractal Tessellation of Spirals," by Robert Fathauer (Tesselations, Phoenix, AZ)708 views16" x 16", Archival inkjet print, 2011

This artwork is based on a fractal tessellation of kite-shaped tiles I discovered several years ago. Grouping of the kite-shaped tiles into spirals allowed a fractal tessellation to be created in which two colors were sufficient to ensure that no two adjacent tiles have the same color. All of the spirals in the print have the same shape (more precisely, they are all similar in the Euclidean plane). --- Robert Fathauer (Tessellations, Phoenix, AZ, http://www.robertfathauer.com)

"Pleated Multi-sliced Cone," by Thomas Hull (Western New England University, Springfield, MA), Robert Lang (Robert J. Lang Origami) and Ray Schamp (Ray's Origami)704 views16" x 16" x 5", elephant hide paper, 2011
Second Place Award, 2012 Mathematical Art Exhibition

Imagine a long paper cone that is pleated with alternating mountain and valley creases so that its cross-section is star-shaped. Now slice the cone with a plane and imagine reflecting the top part of the cone through this plane. The result is exactly what one would get if we folded the pleated cone along creases made by the intersecting plane. Doing this repeatedly can result in interesting shapes, including the origami version presented here. This work is a collaboration. The concept and crease pattern for this work was devised and modeled in Mathematica by origami artist Robert Lang (http://www.langorigami.com/). The crease pattern was then printed onto elephant hide paper by artist Ray Schamp (http://fold.oclock.am/). The paper was then folded along the crease pattern by mathematician and origami artist Thomas Hull (http://mars.wne.edu/~thull). Part of the charm of paper folding is its capacity for simple, elegant beauty as well as stunning complexity, all within the same set of constraints. This mirrors the appeal of mathematics quite well. Geometric origami, which is where most of my artwork lives, strives to express in physical form the inherent beauty of mathematical concepts in geometry, algebra, and combinatorics. The constraints that origami provides (only folding, no cutting, and either one sheet of paper or further constraints if more than one sheet is allowed) challenges the artist in a way similar to being challenged be a mathematical problem. --- Thomas Hull (Western New England University, Springfield, MA, http://mars.wne.edu/~thull)

"Kharragan I," by Reza Sarhangi (Towson University, Towson, MD)680 views16" X 20", Digital print, 2011

I am interested in Persian geometric art and its historical methods of construction, which I explore using the computer software Geometer's Sketchpad. I then create digital artworks from these geometric constructions primarily using the computer software PaintShopPro. Kharragan is an artwork based on a design on one of the 11th century twin tomb towers in Kharraqan, western Iran. The artwork demonstrates two different approaches that are assumed to have been utilized centuries ago to create the layout of the pattern, which is at the center of the artwork. From left to right, the artwork exhibits the construction of the design based on a compass and straightedge. From right to left, we see another approach, the Modularity method, to construct the same design using cutting and pasting of tiles in two colors. --- Reza Sarhangi (Towson University, Towson, MD)

"Lawson's Minimum-Energy Klein Bottle," by Carlo Séquin (University of California, Berkeley)663 views9" x 6" x 4.5", FDM model, 2011
Third Place Award, 2012 Mathematical Art Exhibition

My professional work in computer graphics and geometric design has also provided a bridge to the world of art. This is a gridded model of a Klein bottle (Euler characteristic 0, genus 2) with the minimal possible total surface bending energy. This energy is calculated as the surface integral over mean curvature squared. --- Carlo Séquin (University of California, Berkeley, CA, http://www.cs.berkeley.edu/~sequin/

"Seven Towers," by Radmila Sazdanovic (University of Pennsylvania, Philadelphia)654 views16" x 16", Digital Print, 2011

This tessellation of the hyperbolic plane is inspired by Japanese pagodas but realized in classical black, red and white color scheme, emphasizing local 7-fold symmetry. --- Radmila Sazdanovic (University of Pennsylvania, Philadelphia, http://www.math.upenn.edu/~radmilas/)

"Snail Shell," by Ian Sammis (Holy Names University, Oakland, CA)636 views20" square, Digital Print on metal, 2011

I am particularly interested in creating visualizations of data and of mathematical structures, and more broadly in the creation of art directly from code. It has long been observed that a logarithmic spiral describes a snail shell quite well. I created this image as part of a series of pieces based upon logarithmic spirals. --- Ian Sammis (Holy Names University, Oakland, CA, http://www.hnu.edu/~isammis)

"Parabola-C for curve," by Sharol Nau (Northfield, MN)630 views9" x 6" x 12", folded book, 2011

Folded Book Sculpture. The collection of folds forms an envelope to the parabola. The abounding waves emanate as the book is opened and spread out. --- Sharol Nau (Northfield, MN)

"Triaconthedron sphere," by Richard Kallweit (New Haven, CT)614 views12" x 12" x 12", printed paper, 2011

My artworks are based on investigations into mathematical form concerning the arrangements of units in space. This is a model of a triacontehedron using minimal surface planes with an infinite regression pattern. --- Richard Kallweit (New Haven, CT, http://www.richardkallweit.com)

"Tetradic Knot," by Mehrdad Garousi (Hamadan, Iran)613 views20" x 20", Digital Art Print, 2010

I am interested in all types of mathematical arts which are generated in computers; from 2D and 3D fractals to 3D mathematical sculptures and knots. Every now and then I encounter a new imagery software working on the basis of mathematical algorithms, I try to examine its capacities in creating works containing acceptable amounts of aesthetics. This time I have used Surfer, a mathematical imagery software which creates and displays surfaces constructed according to zero sets of polynomial equations. (x^2+y^2+z^2-(0.5+2*a)^2)^2-(3.0*((0.5+2*a)^2)-1.0)/(3.0-((0.5+2*a)^2))*(1-z-sqrt(3)*x)*(1-z+sqrt(3)*x)*(1+z+sqrt(3)*y)*(1+z-sqrt(3)*y)=0 a= 0.15. It should be paid attention that opening my equations in the software might not have the same result in your viewer. Differences are because of zoom, color and/or position issues which are not contained in the equations. --- Mehrdad Garousi (Hamadan, Iran, http://mehrdadart.deviantart.com)

"Four Sierpinskis," by George Hart (Museum of Mathematics, New York, NY)610 views3" x 3" x 3", Nylon (selective laser sintering), 2011

Four Sierpinski triangles interweave in three dimensions, each linked with, but not touching, the other three. The twelve outer vertices are positioned as the vertices of an Archimedean cuboctahedron and the black support frame is the projection of this cuboctahedron to the circumsphere. These are fifth-level Sierpinski triangles, i.e., there are five different sizes of triangular holes. The strut diameters were made to vary with the depth of recursion, giving a visual and tactile sense of this depth. This hand-painted maquette is intended as a model for a possible large outdoor sculpture. --- George Hart (Museum of Mathematics, New York, NY, http://georgehart.com)