For some time, the first lead has had a rich toolbox for creating objects that, when strobed correctly, exhibit a magical “blooming” quality. The mathematics involves Fibonacci spirals, which are patterns with a special property: when rotated through the golden angle, the scale of the pattern increases just slightly. When a succession of rotated images is shown (perhaps by strobing), the effect is an almost continuous enlargement, as if the pattern is blooming. In 2017, the second lead began adapting his method of wallpaper functions (as described in Creating Symmetry) to create bitmap versions of these Fibonacci spirals. He also used the wallpaper function concept with Maple to create some shapes that bloom under strobing.
The second lead has determined that the key to creating wallpaper functions with Fibonacci spiral symmetry lies in a family of Fibonacci sequences with complex initial data. We have a complete description of how to wind patterns based on square and hexagonal lattices around in a spiral using the complex exponentiation map. Yet to be investigated are the details for other lattices.
We propose to put our two approaches together. One under-exploited idea is to develop shapes for strobing that appear not just to bloom, but to morph as they expand. Current examples show shapes remaining the same while appearing to grow in size (or perhaps rotate, as in the case of “Tumbling Cubes”) as they expand outward. New techniques should allow other transformations of the shapes, perhaps even an organic appearance of flower buds that open into flowers.
- Frank A. Farris, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, Princeton, 2015.