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.

27-torus.jpg"The 27-Crossing Torus Knot T(9,4)," by Dror Bar-Natan (University of Toronto, Canada)

This is an example of a torus knot. A torus is a surface best described as a doughnut. A torus knot can be thought of as looping around and through the torus. The symbol T(9,4) means that the string making the knot loops through the hole of the torus 9 times, making 4 revolutions. This knot is drawn with TubePlot.

--- Dror Bar-NatanJul 24, 2006

8-torus.jpg"The 8-Crossing Torus Knot T(4,3)," by Dror Bar-Natan (University of Toronto, Canada)

This is an example of a torus knot. A torus is a surface best described as a doughnut. A torus knot can be thought of as looping around and through the torus. The symbol T(4,3) means that the string making the knot loops through the hole of the torus 4 times, making 3 revolutions. This knot is drawn with TubePlot.

--- Dror Bar-NatanJul 24, 2006

ashley.jpg"Ashley Knot," by 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 SchareinJul 24, 2006

symmetry2.jpg"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 SchareinJul 24, 2006

symmetry3.jpg"Symmetry Energy Image III," 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.