Symmetry Reflects Us

A story of freedom and constraint, mathematician Frank Farris uses functions and symmetry to turn everyday scenes into stunning art.

Symmetry Reflects Us
View full image. Image by Frank Farris

Symmetry Reflects Us. It brings us “face-to-face with the grand structure of mathematics,” as Frank Farris puts it. Years ago Farris, an associate professor of mathematics and computer science at Santa Clara, bridged mathematical realms and photographic material with what he calls a “domain-coloring algorithm.” His resulting work weds waveforms to nature photography, and it has been exhibited in galleries and at universities across the country.

The book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns (Princeton University Press) collects scores of his images: beginning in the familiar plane of Euclidean geometry, employing complex equations to generate rosettes, friezes, and wallpaper patterns (any pattern with translational symmetry in two independent directions). To follow the mathematical threads requires understanding calculus. The big story unfolding is the beauty of the Earth.

Here are a few of his creations. See the gallery below for some of the source photos that he transformed.

Playing off the color wheel of a rhododendron blossom, this rosette function showcases both mirror and five-fold symmetry. 

Symmetry Reflects Us

A Sierra Nevada Lake transformed by octahedral symmetry. An octahedron, an eight-sided solid, shares the same symmetry group as a cube.

Sliced yellow peppers meet tetrahedral transformation. The four faces of a tetrahedron are equilateral triangles. Black, scalloped regions represent infinite values.

Twenty-sided figures gird the globe in “Icosahedral Lampflower.” Also at play: ray-tracing, a technique for simulating the path of light as it bounces through the world.

“They Arrive” as balls painted with polyhedral patterns and land by moonlight on Upper Twin Lake in the Sierra Nevada.

“Starry Night of Fire” captures a spiral of stars mirrored on a lake. The floating globe reflects the same pattern but in reverse.

A pumpkin pie surprisingly turns into carousel horses using a trick created by Farris’ wave approach.

“When the Butterfly Gates Open” shows color-reversing wallpaper symmetry on the gates and icosahedral symmetry on the floating globe. 

“Temple of the Peach” uses a pattern representing non-Euclidean geometry around the inside of a cylinder.