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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"Kharragan I," by Reza Sarhangi (Towson University, Towson, MD)16" 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)

"Seven Towers," by Radmila Sazdanovic (University of Pennsylvania, Philadelphia)16" 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/)

"Round Möbius Strip," by Henry Segerman (University of Melbourne, Australia)152mm 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/)

"Process Print 3 from Trefoil," by Nathan Selikoff (Orlando, FL)4" 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)

"Lawson's Minimum-Energy Klein Bottle," by Carlo Séquin (University of California, Berkeley)9" 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/

"Spring," by Jeff Suzuki and Jacqui Burke (Brooklyn, NY)24" 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/)

"101-smooth numbers," by Graeme Taylor (University of Bristol, UK)Print 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)

"Lisbon Oriente Station," by Bruce Torrence (Randolph-Macon College, Ashland, VA)Panoramic Photograph, 2011

I've been exploring recent developments in digital imagery which allow me to utilize mathematics and computer programming to solve visual problems. This is a projection made from a panorama of 13 photographs. The individual photos were shot from precisely the same point in space, and when stitched together they comprise the entire "viewable sphere" centered at that vantage point. That is, the panorama has complete coverage of the scene---360 degrees around, and 180 degrees from top to bottom. Stereographic projection was then applied to the spherical panorama, with the projection taken from the North Pole so that the point directly overhead becomes the point at infinity. This produces a lovely "little planet" effect, with the geometry of the roof structure framing the scene. The panorama was shot at Oriente Station in Lisbon, Portugal. --- Bruce Torrence (Randolph-Macon College, Ashland, VA, http://www.flickr.com/photos/thebrucemon/, http://faculty.rmc.edu/btorrenc/)

"Great Ball of Fire," by Eve Torrence (Randolph-Macon College, Ashland, VA)Craft Foam, 2010

I love the symmetric beauty of polyhedra and enjoy creating models to study. Through the process of building a model I am able to truly understand its form. I like to use color to help reveal the structure and patterns of an object. This sculpture is based on the third stellation of the dodecahedron. A stellation of a regular polyhedron is formed by extending the faces until they intersect and enclose a region of space. The faces of the dodecahedron will intersect three times as they are extended, forming the small stellated dodecahedron, the great dodecahdron, and the great stellated dodecahedron. Twelve identical pieces of craft foam were slotted at the edge of each stellation and then tightly woven. This open skeleton allows one to follow each face to view the intersections and the outline of the dodecahedron and the three stellations. Six colors of foam are used and parallel faces are the same color. Each of the five arms of each face intersects three others to form 20 colorful "flames" in an icosahedral arrangement. --- Eve Torrence (Randolph-Macon College, Ashland, VA)

"Hyperbolic Coasters," by Mikael Vejdemo-Johansson (University of St. Andrews, Scotland)Laser-etched glass, 14 items, 12cm diameter each, 2011

The advent of accessible automated tools opens up a number of new approaches to art: especially algorithmic and mathematical art works. The computational control allows us to write algorithms to generate concrete physical art; and their precision allows a higher resolution than what the eye can discern. These pieces highlighting and reifying different mathematical concepts, giving them physical presence and accessibility and turning abstract geometry into hands-on displays and objects. Among the most successful of the reified mathematics art-pieces I produced where these--hyperbolic disk tilings with the Poincare disk model were etched onto glass disks, producing a collection of reified hyperbolic geometries and symmetries. --- Mikael Vejdemo-Johansson (University of St. Andrews, Scotland, http://mikael.johanssons.org)