Frank A. Farris
I aspire to connect what I perceive to be our two principal sources of beauty: mathematics and the natural world. My method combines photographs with complex-valued functions in the plane to create images with various types of symmetry. When we consider that the Riemann sphere is just another kind of plane, this opens the door to polyhedral symmetry as well. The basic methods are explained in detail in my book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, published in 2015 by Princeton University Press, but new discoveries are always in progress.
Glowing globes with three types of polyhedral symmetry drift over a moonlit mountain to land on the lake. Are they wafting from the Platonic world into ours? (The patterns on the globe were created with domain colorings of meromorphic functions invariant under the actions of the three chiral polyhedral groups. In the past, I always used rectangular photographs to paint spheres, resulting in images with singularities. New techniques allow the source photograph to live on the Riemann sphere, allowing poles to be painted just as if they were zeroes. Ray tracing and manipulation of the original daytime mountain photograph were done in Photoshop.)
Mandalas are a timeless invitation for the eye to wander through repeating shape. The beautiful window I commissioned from glass artist Hans Schepker has appeared in many of my works. Here it is the source photograph for the domain coloring of a rational function with five-fold symmetry. You may be able to count that it has twenty poles and twenty zeroes and hence approaches a constant limit at infinity. I colored the lower half of the Riemann sphere (representing the range of the function) with the disk of the window and the upper half with its negative, with a short white/black band in between. Thus, zeroes of the function are colored with the creamy center of the window, while poles are colored a dark opposite.