Math ImageryThe 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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Home > Frank Farris :: Seeing Symmetry

Most viewed - Frank Farris :: Seeing Symmetry
"Tiffany Glass from a Mountain Gentian and its Negative," by Frank A. Farris, Santa Clara University, CA546 viewsInk jet on paper, 2012.

Using a composite photograph of a mountain gentian juxtaposed with its negative to produce an intense contrast of purple with the original green, I used just the right waves to make wallpaper with horizontal mirrors, vertical glides, and 2-fold rotational symmetry. In the notation of the International Union of Crystallographers, the symmetry group of this pattern is pmg, but if we allow color-swapping transformations as generalized symmetries, the larger group is cmm, so this pattern type is called cmm/pmg. The "Seeing Symmetry" virtual exhibition at includes more works and details. --- Frank A. Farris
"Turtles with Local Symmetry," Frank A. Farris, Santa Clara University, CA498 viewsInk jet on paper, 2012.

There is no mirror symmetry in this image, only 6-fold rotational symmetry. And yet our eye cannot help seeing symmetry in the turtle shapes. This "local symmetry" is the topic of an article, "Local symmetry in wallpaper," in preparation. --- Frank A. Farris
"A cmm Pattern from Peppers on a Cutting Board," by Frank A. Farris, Santa Clara University, CA484 viewsInk jet on paper.

The bright oranges in this cmm pattern come from a photograph of assorted chopped peppers, with collard greens and the glint of the knife as nice contrasts. --- Frank A. Farris
"A cm Pattern from a Minneapolis Skyline, (vertical format)," by Frank A. Farris, Santa Clara University, CA346 viewsInk jet on paper.

Fantastical samurai appeared when I used a photograph of the Minneapolis skyline on an autumn day in conjunction with wave functions adapted for cm patterns. --- Frank A. Farris
"A Strawberry Lemon Spiral," by Frank A. Farris, Santa Clara University, CA62 viewsInk jet on paper, 2015.

This image started life as a wallpaper pattern with symmetry group p2, based on a photograph of strawberries with a cut of lemon. After scaling it correctly, I applied the complex exponential map to wind it around the origin, creating a spiral of yellows, greens, and reds. I particularly enjoy how the lemon became a string of yellow wax beans, while the strawberries turned into peppermint candies. --- Frank Farris
"Octahedral Globe from a Window," by Frank A. Farris, Santa Clara University, CA59 viewsInk jet on paper, 2015.

This coloring of the sphere, based on a photograph of a stained-glass window by Hans Schepker, is invariant under the action of the octahedral group. I created it by mapping the sphere (and the group action) to the plane via stereographic projection and using known techniques for creating complex-valued functions invariant under groups that act on the plane. This image is part of a larger work, "Imaginary Planets." --- Frank Farris
"Alternating Wood Bugs," by Frank A. Farris, Santa Clara University, CA35 viewsInk jet on paper, 2015.

Had I used a source photograph whose colors reverse exactly when you turn it upside down, the image computed with these wallpaper waves would have exact color-reversing symmetry of type p4g/cmm. (See my book Creating Symmetry for explanation.) However, when you rotate my picture of a freshly cut pine stump, the colors only more-or-less reverse. This causes what I call "approximate color-reversing symmetry." The blond bugs marching northwest have the same outlines as the dark bugs marching northeast, but the details of the insides are quite different. --- Frank Farris
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