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.

"Four Sierpinskis," by George Hart (Museum of Mathematics, New York, NY)647 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)

"Color Wheel with a Twist," by Diane Herrmann (University of Chicago, IL)631 views12" 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)

"Four Right Angles: Ascent (left) and Cantilever (right)," by Nat Friedman (Albany, NY)630 views17" 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)

"Laplacian Growth #1," by Nervous System generative designers629 views8" 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)

"Spring," by Jeff Suzuki and Jacqui Burke (Brooklyn, NY)627 views24" x 36", quilt, 2011

Our quilts are based on "Rule 30" (in Wolfram's classification of elementary cellular automata), applied to a cylindrical phase space. "Winter" is the basic rule 30 to produce a two-color pattern. The successive patterns combine the history of two ("Spring"), three ("Summer), or four ("Fall") generations to produce a palette of four, eight, or sixteen colors. In this quilt, "Spring", the colors are determined by the history of a cell at times t = 2k and 2k + 1, treated as a two-bit number between 0 and 3. --- Jeff Suzuki and Jacqui Burke (Brooklyn College, NY, https://sites.google.com/site/jeffsuzukiproject/)

"Beaded Star Weaves: Five Bracelets," by Gwen Fisher (beAd Infinitum, Sunnyvale, CA)624 viewsSizes 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)

"101-smooth numbers," by Graeme Taylor (University of Bristol, UK)604 viewsPrint from digital, 2011

'The smoothness spiral' is an interactive applet (see http://maths.straylight.co.uk/archives/453) that plots the first 10,000 integers on an Archimedean spiral. Each point has a brightness depending on its number-theoretic smoothness (its largest prime divisor), controlled by a user-selected threshold. Curves of smooth numbers emerge, whilst large primes are conspicuous by their absence, causing 'missing' curves. This print from 'the smoothness spiral'; the threshold is set to show values which are at most 101-smooth, with brightness proportional to smoothness. --- Graeme Taylor (University of Bristol, UK, http://straylight.co.uk)

"Process Print 3 from Trefoil," by Nathan Selikoff (Orlando, FL)579 views4" x 6", Archival Pigment Print, 2011

I love to experiment in the fuzzy overlap between art, mathematics, and programming. Seeking to extract and visualize the beauty that I glimpse beneath the surface of equations and systems, I create custom interactive programs and use them to explore algorithms, and ultimately to generate artwork. When I prepare an image from my Aesthetic Explorations series of strange attractors for print, the first step is rendering a very high resolution, high quality 16-bit grayscale image from my custom software. While these images are destined to spend some time in Photoshop in a process of recoloring and enhancement, I find that they are very beautiful in and of themselves. The nature of algorithmic artwork (and fractal phenomena in nature in general) is that there is captivating detail at all scales. This is a crop from "Trefoil". --- Nathan Selikoff (Artist, Orlando, FL, http://nathanselikoff.com)

"Hybrid 101," by Michael Field (University of Houston, TX)561 views24" 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)

"Broken Dishes, Mended Edges," by Margaret Kepner (Washington, DC)533 views6" 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)