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.
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 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.
