Janna Levin, the host of the podcast “The Joy of Why,” recently had a conversation with mathematician Natalie Priebe Frank about the fascinating world of tiling patterns. They discussed the history of aperiodic monotiles, starting from the discovery of Wang tiles and the quest to find a single tile that can fill a plane without repeating.
The conversation touched on the significance of aperiodic tilings, such as the Penrose tilings, and their connection to quasicrystals. Quasicrystals, ordered materials that lack a consistent repeating structure, have been found to exhibit diffraction patterns similar to Penrose tilings, leading to breakthroughs in material science.
Natalie shared her insights on hierarchical tilings, self-similar structures that can be used to model physical solids and explore the limits of disorder in ordered systems. She also discussed the topological aspects of tilings, where the topology of a tiling space can reveal information about convergence rates and spectral analysis.
The conversation concluded with a discussion on ongoing research challenges in the field, including the search for aperiodic monotiles in higher dimensions and the exploration of undecidable tiling problems. Natalie also shared her personal experience with designing mathematically inspired tile floors in her home and her colleagues’ creative tiling projects.
Overall, the conversation highlighted the beauty and complexity of tiling patterns, showcasing the interdisciplinary nature of mathematics and its connections to art, physics, and material science. The exploration of tilings continues to inspire new research directions and push the boundaries of mathematical knowledge.
