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.

"Broken Dishes, Mended Edges," by Margaret Kepner (Washington, DC)6" x 16", Archival Inkjet Print, 2011

The traditional quilt pattern “Broken Dishes” and certain edge-matching puzzles share a common visual element – a square subdivided along its main diagonals to form 4 right triangles. This work presents 4 puzzle solutions using this visual element in a format suggesting Broken Dishes quilts. Edge-matching puzzles based on the square were introduced by MacMahon in the 1920s. One challenge was to arrange a set of 24 three-colored squares (all the possibilities) in a rectangle with same colors matching on the edges and a single color appearing around the border. If this is generalized to four colors, the complete set of puzzle pieces jumps to 70. These can be arranged in a 7x10 rectangle, providing a nice quilt proportion. This set of four designs is based on different matching “rules” ranging from strict matching to random placement, while maintaining the border requirement. To produce richer colors, each design is overlaid with a translucent scrim of the next design in the sequence. --- Margaret Kepner (Artist, Washington, DC)May 14, 2012

"Triaconthedron sphere," by Richard Kallweit (New Haven, CT)12" 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)May 14, 2012

"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)16" 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)May 14, 2012

"Color Wheel with a Twist," by Diane Herrmann (University of Chicago, IL)12" diameter, needlepoint canvas, 2011

"Color Wheel with Twist" is more than just a stitched version of the artist’s color wheel, and is also more than the mathematician’s non-orientable manifold. The colored leaves flow all around the band in their natural order on the color wheel; yet this mysterious shape has only one side. I wanted to capture both color theory and geometry in this piece. --- Diane Herrmann (University of Chicago, IL)May 14, 2012

"Four Sierpinskis," by George Hart (Museum of Mathematics, New York, NY)3" 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)May 14, 2012

"Tetradic Knot," by Mehrdad Garousi (Hamadan, Iran)20" 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)May 14, 2012

"Four Right Angles: Ascent (left) and Cantilever (right)," by Nat Friedman (Albany, NY)17" L x 11" H x 7 D", Steel, 2010

A sculpture is defined as a form in a position relative to a fixed horizontal plane (base, ground). To hypersee an outdoor sculpture, one walks around it to see overall views and close up detail views from different viewpoints as well as in different light conditions at different times. If two sculptures consist of the same form in different positions, then the sculptures are said to be congruent. Congruent sculptures can look so completely different that one does not realize the sculptures are congruent. A hypersculpture is a group of congruent sculptures, and a more complete presentation of the sculptural possibilities of a form. In order to hypersee a form, one presents it as a hypersculture. This hypersculpture Four Right Angles consists of two vertical and three horizontal congruent sculptures and is discussed in an article of the same title in the Spring, 2011 issue of Hyperseeing, www.isama.org/hyperseeing/ . The two vertical sculptures Ascent and Cantilever are shown here. The form consists of four identical angle iron sections welded together. Each section is 5" x 5" x 6 ½" and ½" thick. --- Nat Friedman (Professor Emeritus, University at Albany, NY, Founder and Director of ISAMA, http://www.isama.org)May 14, 2012

"Beaded Star Weaves: Five Bracelets," by Gwen Fisher (beAd Infinitum, Sunnyvale, CA)Sizes vary from 1.5" to 2.5" wide by 5.5" to 8" long, Seed bead weaving, 2011

I weave beads to appeal to people's affinity for organization in design. I use mathematics, including geometry, symmetry, and topology, as an inspiration for the structure of my creations. In this series, I explore how tilings of the plane can be interpreted as beaded angle weaves. Tilings of the plane, especially periodic tilings, can be used as the basis for flat bead weaving patterns called angle weaves. The “star tilings” used to design these five bracelets are generated from the three regular tilings of the plane and two other Laves tilings. I converted each star tiling into a star weave by placing beads on the vertices and edges of the tiling and weaving them together with a needle and thread. Because all of the vertices in a regular tiling are similar, all of the stars are similar in the three regular star weaves (i.e., Kepler’s Star, Archimedes’ Star, and David’s Star). The other two star weaves (i.e., Night Sky and Snow Star) include stars of two types, reflecting the two types of vertices in their respective Laves tilings. --- Gwen Fisher (beAd Infinitum, Sunnyvale, CA, http://www.beadinfinitum.com)May 14, 2012

"Hybrid 101," by Michael Field (University of Houston, TX)24" x 24" (framed), Archival inkjet print, 2011

Hybrid 101 is a representation of an invariant measure for a dynamical system on a 2-torus with deterministic and random components. Deterministic dynamics is given by the product of two identical circle maps with topological degree 2 ('doubling maps') together with a random component which is a place dependent iterated function system: probabilities and direction and size of jumps depend on the position on the torus. Hybrid dynamics combining deterministic dynamics with an iterated function system was first studied mathematically by Kobre and Young in the context of extended dynamical systems on the line. In Hybrid 101, dynamics is defined by doubly 1-periodic maps on the plane and we reduce mod the integer lattice to obtain dynamics on a torus. We lift the measure back to the plane to obtain a repeating pattern. Appearances can be deceptive: the only symmetries of the repeating pattern are translations (the pattern is of type p1) and all the lines are straight. --- Michael Field (University of Houston, TX, http://www.math.uh.edu/~mike)May 14, 2012

"Fractal Tessellation of Spirals," by Robert Fathauer (Tesselations, Phoenix, AZ)16" 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)May 14, 2012

"Nueve y 220-B," by Juan G. Escudero (Universidad de Oviedo, Spain)50cm 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)May 14, 2012

"Butterflies 6-4," by Doug Dunham (University of Minnesota Duluth, MN)11" x 11", Color printer, 2009

This is a hyperbolic pattern of butterflies, six of which meet at left front wing tips and four of which meet at their right rear wings. The pattern is inspired by M.C. Escher's Euclidean image Regular Division Drawing Number 70, and is colored similarly. Disregarding color, the symmetry group of this pattern is generated by 6-fold and 4-fold rotations about the respective meeting points of the wings, and is 642 in orbifold notation (or [4,6]+ in Coxeter notation). This pattern exhibits perfect color symmetry and its color group is S3, the symmetric group on three objects. --- Doug Dunham (University of Minnesota Duluth, MN, http://www.d.umn.edu/~ddunham/)May 14, 2012

"Still Life with Magic Square," by Sylvie Donmoyer (Saumur, France)20" x 26", Oil paint on canvas, 2011
First Place Award, 2012 Mathematical Art Exhibition

It all arose from a sense of wonder when seeing the formal beauty of mysterious objects called polyhedra. Since then, I have joyfully played with geometric shapes and it led me to explore the possible representation of Geometry in classical painting. From Durer's magic square to strange cubes, painted by the precise brush of a would-be 17th century Dutch artist. --- Sylvie Donmoyer (Saumur, France, http://www.illustration-scientifique.fr/index-A.html)May 14, 2012

"Science/Art," by Erik Demaine (MIT, Cambridge, MA) and Martin Demaine (MIT, Cambridge, MA)22" (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/)May 14, 2012

"Sierpinski Cliffs," by Francesco De Comité (University of Sciences and Technology, Lille, France)50cm x 50cm, Digital print, 2011

Seeking ways to illustrate mathematical concepts and constructions is an endless game. Jumping from one idea to another, mixing techniques and computer code, and then waiting for the image to appear on my screen, leads often to surprising results. Playing around with Apollonian gaskets, recursivity and circle inversion can give rise to landscapes no one has seen before. --- Francesco De Comité (University of Sciences and Technology, Lille, France, http://www.lifl.fr/~decomite)May 14, 2012