# Frank A Farris

Professor of Mathematics and Computer Science
Santa Clara University
Santa Clara, California, USA

After publishing Creating Symmetry: The Artful Mathematics of Wallpaper Patterns with Princeton University Press, I continue to find new ways to apply the same circle of ideas from analysis (symmetry, PDEs, and complex variables). The basic technique uses photographs in domain colorings of complex-valued functions that are invariant under various group actions. My Imaginary Landscapes series constructs surrealistic views of virtual objects painted with symmetric patterns, seen against natural or constructed backdrops. Ray-tracing techniques create reflections and shadows that might make us ask whether this is mathematics or some previously unseen part of the world.

Art Fesitival at Filoli
51 x 61 cm
Digital print on aluminum
2017

Bold flower sculptures decorated with tulip rosettes seem to tower over a swimming pool, where a sphere and a saddle surface float. The saddles are decorated with a rose and iris mandala. The sphere carries a pattern with icosahedral symmetry, made from a photo of this same pool, taken at a time when the orange tulips in the foreground were blooming more vigorously. The large flower shapes are harmonic functions with 6-fold symmetry, constructed in Maple; decorating them with rosettes is a separate step. After the shapes are defined and decorated, ray-tracing creates reflections on the water. An interesting technical challenge: how do the reflections in the pool stay behind the orange tree, which is technically in the background?

A Gooseberry/Fibonacci Spiral
51 x 51 cm
Digital print on aluminum
2017

A twist on John Edmark's spirals, this pattern winds a walllpaper pattern of type p31m around the plane with the complex exponential map to create a Fibonacci spiral. The mathematical underpinnings involve a Fibonacci-like sequence of Eisenstein integers, which then determine a lattice of frequency vectors for wallpaper waves that will land correctly in the winding. The pattern is selected by "tuning" the waves: adjusting frequencies and amplitudes to find a beautiful pattern. The Western (or Sierra) Gooseberry tastes about like the eastern one, which is translucent and green, but ripens to a deep red and is covered in thorns, which make it quite inconvenient to pick. The delicious jelly is a longtime family tradition.