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.

---Chaim Goodman-Strauss, University of Arkansas

9 files, last one added on Jan 31, 2008
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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.

--- George W. Hart (www.georgehart.com)

6 files, last one added on Oct 19, 2010
Album viewed 980 times

"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

25 files, last one added on Mar 07, 2007
Album viewed 1593 times

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.

--- Jarke J. van Wijk

3 files, last one added on Mar 13, 2007
Album viewed 893 times

Straight, a professor of painting at the University of Delaware, uses prime numbers and geometric shapes to create visually compelling paintings.

3 files, last one added on Feb 01, 2007
Album viewed 432 times

I am a mathematical artist trying to give mathematical concepts a visual counterpart that everyone can enjoy.

-- Mary Candace Williams

6 files, last one added on Jun 20, 2008
Album viewed 582 times

"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

5 files, last one added on Jul 24, 2006
Album viewed 401 times

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.

--- Bradford Hansen-Smith (Wholemovement)

4 files, last one added on Jul 24, 2006
Album viewed 240 times

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)

4 files, last one added on Jul 12, 2006
Album viewed 449 times

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.

4 files, last one added on Mar 28, 2006
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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).

4 files, last one added on Jul 12, 2006
Album viewed 307 times