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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"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)

"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)

"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)

"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)

"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)

"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)

"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)

"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)

"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)

"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)

"Parabola-C for curve," by Sharol Nau (Northfield, MN)9" 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)

"Laplacian Growth #1," by Nervous System generative designers8" x 8" x 8", Selectively Laser Sintered Nylon, 2011

We designers at Nervous System are attracted to complex and unconventional geometries. Our inspirations are grounded in the natural forms and corresponding processes which construct the world around us. Laplacian Growth #1 is an instance of growth using a model of 3D isotropic dendritic solidification. The form is grown in a simulation based on crystal solidification in a supercooled environment. This piece is part of a series exploring the concept of laplacian growth. Laplacian growth involves a structure which expands at a rate proportional to the gradient of a laplacian field. Under the right circumstances, this leads to instabilities causing intricate, fractal branching structure to emerge. This type of growth can be seen in a myriad of systems, including crystal growth, dielectric breakdown, corals, Hele-Shaw cells, and random matrix theory. This series of works aims to examine the space of structure generated by these systems. --- Nervous System generative designers (http://n-e-r-v-o-u-s.com)

"SPHERE," by Dominique Ribault (Paris, France)60cm x 60cm, Digital Print (Hahnemuhle Canvas Goya)

Eleph-Zero and its clones are tessellations of the plane made with the crystallographic group P3. With this work I wanted also to illustrate links between Algebra and Topology. Eleph-Zero walks on two spirals from the south to the north. --- Dominique Ribault, Artist, Paris, France

"Pythagorean Tree," by Larry Riddle (Agnes Scott College, Decatur, GA)16" 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)

"Snail Shell," by Ian Sammis (Holy Names University, Oakland, CA)20" 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)