The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. We can group these intersections into roughly three categories: Origami mathematics, which includes the mathematics that describes the underlying laws of origami; Computational origami, which comprises algorithms and theory devoted to the solution of origami problems by mathematical means; Origami technology, which is the application of origami (and folding in general) to the solution of problems arising in engineering, industrial design, and technology in general. One genre blends into another. Origami math defines the "ground rules" for computational origami's goal of solving origami design problems (and quantifying their difficulty). The results of computational origami, in turn, can be (and have been) pressed into service to solve technological problems ranging from consumer products to the space program. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality website includes galleries of my designs, crease patterns, schedule of my lectures, appearances and exhibitions, commissioned works, and more on the science of origami.

--- Robert J. Lang

17 files, last one added on May 22, 2013
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The 2013 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in San Diego, CA. Here on Mathematical Imagery is a selection of the works in various media. Mathematical Art Exhibition Awards were given: "Bended Circle Limit III," by Vladimir Bulatov was awarded Best photograph, painting, or print; "Inlaid Wooden Boxes of Makoto Nakamura's Tessellations," by Kevin Lee was awarded Best textile, sculpture, or other medium; and "Tessellation Evolution," a beaded necklace by Susan Goldstine received Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

27 files, last one added on May 16, 2013
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Since high school I have been fascinated by geometry. I enjoyed constructing the more complicated Platonic solids with ruler and compasses, as well as reading about the 4th dimension. While at Bell Labs in Murray Hill, I was introduced to the field of Computer Graphics, and later developed the Berkeley UniGrafix rendering system, so that I could depict objects more complex than I could build. Since then, the focus of my work has been on computer-aided design (CAD) tools -- for engineers, architects, and artists. When creating abstract sculptures I see myself as a composer in the realm of pure geometry. The artistic achievement then lies in finding a procedural formulation that can reflect the inherent symmetries and constructive elegance that seems to lie beneath many sculptural master pieces as well as at the foundations of the physical laws of our universe.

--- Carlo Séquin

16 files, last one added on Mar 18, 2013
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I'm a math teacher, illustrator, and dad. Having begun entertaining my children with weekly pancakes earlier this year, I'm always looking for new themes; in this album you'll find some of the fractal pancakes I cooked up one morning. On my blog, www.10minutemath.com, you can find more about fractals and other topics that interest me as a teacher. --- Nathan Shields (www.10minutemath.com)

4 files, last one added on Jun 14, 2012
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I create geometric patterns in the snow, walking along the frozen lakes of Savoie, France in snowshoes. On average the works take about 10 hours to really do it properly, some are a little unfinished, if my feet get cold or hurt too much. The setting out is done using handheld orienteering compass and distance determination using pace counting or measuring tape. Curves are either judged or arcs of circles are made using a clothesline attached to an anchor at the centre. Designs are chosen from the world of geometry. The Koch curve and Sierpinski triangle in this album are among my favorites. The works are very large (the size of several soccer fields), and many of the mathematical patterns appear 3D, especially when viewed from above. --- Simon Beck

5 files, last one added on May 21, 2012
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The 2012 Mathematical Art Exhibition, held at the Joint Mathematical Meetings held in Boston, MA, was the largest exhibition to date. Here on Mathematical Imagery is a selection of the works in various media. Mathematical Art Exhibition Awards were given: First Place to Sylvie Donmoyer for "Still Life with Magic Square"; Second Place to Thomas Hull, Robert Lang, and Ray Schamp for "Pleated Multi-sliced Cone"; and Third Place to Carlo H. Séquin for "Lawson's Minimum-Energy Klein Bottle. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

40 files, last one added on May 14, 2012
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All my life I have vacillated between mathematics and art and find I am happiest when doing both. It's generally acknowledged that math and music are closely related in human developmental processes. I guess it boils down to basic communication. Math and music are languages, and art is a visual means of communication. My medium of choice is glass, and my favorite designs are mathematical and usually the ones that nature presents in both anatomical and botanical spheres of life. --- Erica Rollings Glass Works (www.ericarollings.net)

4 files, last one added on Apr 03, 2012
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I love to make mathematical Lego® sculptures. They aren't constructed entirely without computer assistance; usually I write some C code to generate whatever the shape is and figure out which cells in a grid made up of 1x1x1 Lego® bricks should be filled in. The code outputs this as an LDraw .DAT file, separated into construction steps adding one complete layer of the structure in each step. Then I use MLCad to view the .DAT file. I play around with the parameters and repeat until I have something that looks nice and which will probably be able to balance. But that's the easy part. Now I have to try to construct it out of actual Lego® bricks so that it actually holds together. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)

5 files, last one added on Feb 06, 2012
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The Mathematical Art Exhibition held at the 2011 Joint Mathematics Meetings in New Orleans, included juried works in various media. Robert Fathauer was the curator of the exhibition, and the exhibition website was prepared by Anne Burns. The 2011 Mathematical Art Exhibition Awards were awarded: First Place to Margaret Kepner, for her work, "Magic Square 25 Study"; Second Place to Carlo Sequin for "Torus Knot (5,3)"; and Third Place to Anne Burns for "Circles on Orthogonal Circles". The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

57 files, last one added on Mar 10, 2011
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I am a mathematician and artist, fascinated by patterns, both theoretical and visual and their communication. My research is based on substitution tilings, tilings with a scaling symmetry like the Penrose Tiling. I write about ideas on maths, art and communication on my blog, Maxwell's Demon, at www.maxwelldemon.com. --- Edmund Harriss (University of Leicester).

6 files, last one added on Feb 09, 2010
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The Mathematical Art Exhibition held at the 2010 Joint Mathematics Meetings in San Francisco, included works in various media by 64 artists. Robert Fathauer was the curator of the exhibition, and the exhibition website was prepared by Anne Burns. The exhibition was juried by Fathauer and Burns, along with Nat Friedman and Reza Sarhangi. The 2010 Mathematical Art Exhibition Prizes were awarded. Four judges, selected by the American Mathematical Society and the Mathematical Association of America, made the following awards: First Prize to Robert Bosch, for his work, "Embrace"; Second Prize to Harry Benke for "The Vase"; and Third Prize to Richard Werner for "Meditations". The Prize "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

63 files, last one added on May 10, 2010
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The Mathematical Art Exhibition held at the 2009 Joint Mathematics Meetings in Washington, DC, included 49 works in various media by 36 artists. Robert Fathauer was the curator of the exhibition, and the exhibition website was prepared by Anne Burns. The exhibition was juried by Fathauer and Burns, along with Nat Friedman and Reza Sarhangi. The inaugural Mathematical Art Exhibition Prizes were awarded. Four judges, selected by the American Mathematical Society and the Mathematical Association of America, made the following awards: First Prize to Goran Konjevod, for his origami work, "Wave (32), 2006;" Second Prize to Carlo Séquin, for his sculpture, "Figure-8 Knot, 2007;" and Third Prize to Robert Fathauer, for "Twice Iterated Knot No. 1, 2008." The Prize "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

49 files, last one added on Apr 28, 2009
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Dr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold.

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Weavers of beads use a needle and thread to sew beads together to make decorative objects including jewelry, wall hangings, sculptures, and baskets. Some bead weave designers weave beads into composite clusters, usually with at least one large hole, called beaded beads. Mathematically, many beaded beads can be viewed as polyhedra, with each bead (or, more precisely, the hole through the middle of each bead, which provides its orientation) corresponding to an edge of the polyhedron. Different weaving patterns will bring different numbers of these "edges" together to form the vertices of the polyhedron. So it is very natural to use various polyhedra as the inspiration for beaded bead designs. Mathematics, including geometry, symmetry, and topology, is an inspiration for the structure of these woven bead creations. Across cultures and continents, humans show a natural affinity towards the aesthetic of pattern and order, and this art form appeals to this aesthetic in a tactile, tangible form. --- Gwen L. Fisher, Ph.D., California Polytechnic State
University, San Luis Obispo, and beAd Infinitum (www.beadinfinitum.com)

9 files, last one added on Aug 26, 2011
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Mathematicians David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis) have built a model that generates detailed 3D images of all types of nature's snowflakes.

13 files, last one added on May 11, 2009
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An aspect of my art work that I particularly enjoy is that I write the software for all the programs I use and build the computers that run the software. In this sense, I like to feel that theory (mathematics), art (outcome), software (algorithms) and engineering (hardware) are integrated and interdependent and that no part survives without the others.

--- Mike Field

9 files, last one added on Aug 26, 2011
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Computers make it possible for me to "see" the beauty of mathematics. The artworks in the gallery of "Mathscapes" were created using a variety of mathematical formulas.

--- Anne M. Burns

17 files, last one added on Jul 23, 2010
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Mathematics plays a very particular role in the quest for knowledge. Whether mathematicians are involved in invention or discovery, the tools that they develop have constituted the very basis of science for more than 2000 years. Mathematics, which has been considered for too long as a mere language in which to formulate the laws of nature, is now recognized as a creative thought process that can be used to discover new entities and phenomena.

Yet scientific knowledge is undoubtedly not the only way of comprehending the infinite wealth of phenomena in our universe. Art, the quest for beauty and the indefinable, is another way forward, a means of progress that is parallel to the means provided by science, and we surmise that still more possibilities exist, probably more than we could ever imagine.

---Jean-Francois Colonna, Centre de Mathematiques Appliquees, Ecole Polytechnique, www.lactamme.polytechnique.fr

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I'm always fascinated by things that are symmetric. In symmetrical things we see beauty and thereby mathematics. It is therefore a quest and adventure for an inquisitive mind to go deeper and deeper to study the building blocks of things that are symmetric. The images that I produce show the ubiquity of mathematics and how mathematics is a key to understand the nature that we live in, in particular, and how it unlocks the secrets of our universe in general.

The usual elementary functions and their compositions can generate sophisticated graphs which are shown. The structures (or patterns) are superimpositions of polar surfaces resulted from several compositions of tilts and turns on the coordinate axes in three dimensions. When they are viewed from a different turn and tilt they generate a totally different, fascinating structure.

The structures are a few from my collection, which numbers more than 70. For those who are interested to see them, check my website and click on the gallery button.

---Dejenie A. Lakew, Virginia State University

6 files, last one added on Feb 28, 2008
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I love experimenting in the fuzzy overlap between art, mathematics, and programming. The computer is my canvas, and this is algorithmic artwork--a partnership mediated not by the brush or pencil but by the shared language of software. Seeking to extract and visualize the beauty that I glimpse beneath the surface of equations, I create custom interactive programs and use them to explore algorithms, and ultimately to generate artwork.

In the world of chaotic dynamical systems, minute changes in initial conditions produce radically different results. The interface of my software gives me hooks into the algorithms and allows me to exert some control. But there is always tension - between the computer and me, between simplicity and complexity, and between problem solving and spontaneity.

Art and mathematics, the right brain and the left, are inextricably linked in this work. My art depends on mathematics, yet simultaneously illuminates and unravels its beauty. I am an explorer who uncovers something extraordinary, bringing into view that which was always there to be discovered.

---Nathan Selikoff

6 files, last one added on Jul 23, 2010
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