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.

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).

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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.

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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.

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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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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

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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.

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

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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.

These images illustrate a variety of kinds of symmetrical figures; most were produced for "The Symmetries of Things," written with John H. Conway and Heidi Burgiel (A.K. Peters, 2008), using a variety of proprietary software tools.

I have been interested in geometry, pattern, and mathematical illustration of one form or another since I was a child. Abstraction is the basis of the power of mathematics, but too often we forget that mathematics is also a descriptive language, with meaning anchored in intuitive experience of the world around us. How many students emerge from, say, an undergraduate linear algebra course for math majors, knowing full well proofs of the existence and characterization of eigenspaces and eigenvalues, having no simple, clear idea of what such objects might look like?

Though I am as seduced by abstraction as any research mathematician, I am drawn to mathematics I can see and touch; mathematical illustration, carried out in a graphically rigorous manner, is a natural extension of my work as a mathematician--and indeed may be more fundamental to me.

As a sculptor of constructive geometric forms, my work deals with patterns and relationships derived from classical ideals of balance and symmetry. Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic. I use a variety of media, including paper, wood, plastic, metal, and assemblages of common household objects.

"Fractal Art: Beauty and Mathematics" was an exhibit at the 2006 International Congress of Mathematicians in Madrid, Spain. The works were selected through the International Contest of Fractal Art ICM2006, which Benoit Mandelbrot, "the father of fractal geometry," chaired. (Read more about Mandelbrot.) The 25 works featured in the exhibit and in this album represent current fractal art as well as all the different techniques used in it.

"How wonderful that one object should be at the same time so easy to draw and so complex, involving as it does one of the most challenging and frustrating open conjectures in mathematics. And in addition--as shown by this beautiful Exhibit--it is sweet and friendly to every citizen's eye. How wonderful that the organizers imagined and implemented this Exhibit. Between a Congress defining the cutting edge of mathematics and all the citizens of the world, it does not erect the wall of an Ivory Tower (or Fortress) but an extraordinarily mutually beneficial bridge." --- Benoit Mandelbrot

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Some years ago, my colleague Arjeh Cohen asked me if I could visualize a Seifert surface. I quickly became fascinated with these strange and difficult surfaces. Seifert surfaces are orientable surfaces, bounded by a knot or link. Their shapes are real-world and not abstract, but also strange and difficult to comprehend. Therefore, I developed a tool called SeifertView to define and display these fascinating mathematical objects.

"Symmetry Energy Image II," by Rob Scharein (Centre for Experimental and Constructive Mathematics, Simon Fraser University, B.C., Canada)

This example illustrates the SE rendering mode in KnotPlot, which visualizes the symmetric energy distribution. KnotPlot is a program to visualize and manipulate mathematical knots in three and four dimensions, and the website includes a wealth of resources and pictures. This picture is a direct screen capture from KnotPlot, rendered entirely in OpenGL, an environment for portable, interactive graphics applications.

--- Rob Scharein

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The symbol of the circle is used as metaphor for nothing and for everything, and endless parts in-between. Folding circles appears to have little history: Somewhere in the history of origami lies the circle, unrecognized and discarded in favor of the square; Buckminster Fuller also folded the circle, with informational intent. Fuller is the inspiration for my own exploration into geometry and provided the seed for folding and joining circles—9" paper plates.

This striking object is an example of a surface in 3-space whose intrinsic geometry is the hyperbolic geometry of Bolyai and Lobachevsky. Such surfaces are in one-to-one correspondence with the solutions of a certain non-linear wave-equation (the so-called Sine-Gordon Equation, or SGE) that also arises in high-energy physics. SGE is an equation of soliton type and the Breather surface corresponds to a time-periodic 2-soliton solution. See more pseudospherical surfaces on the 3D-XplorMath Gallery.

--- Richard Palais (Univ. of California at Irvine, Irvine, CA)

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This is a version of the Ow-Hull "Five Intersecting Tetrahedra." The visually stunning object should be a familiar sight to those who frequent the landscapes of M.C. Escher or like to thumb through geometry textbooks. Read about the object and how it is constructed on the Origami Gallery.

--- Thomas Hull. Photograph by Nancy Rose Marshall.

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People have long been fascinated with repeated patterns that display a rich collection of symmetries. The discovery of hyperbolic geometries in the nineteenth century revealed a far greater wealth of patterns, some popularized by Dutch artist M. C. Escher in his Circle Limit series of works. The cover illustration on this issue of the Notices portrays a pattern which is symmetric under a group generated by two Möbius transformations. These are not distance-preserving, but they do preserve angles between curves and they map circles to circles. See Double Cusp Group by David J. Wright in Notices of the American Mathematical Society (December 2004, p. 1322).

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