Frank A. Farris

Follow the spiral from the center, where the story begins with a circle, then a torus. The complicated curve appears to have 5-fold rotational symmetry, but in fact does not; the illusion is created by an envelope of the high-frequency curve, a phenomenon we call "phantom symmetry." The phenomenon is explained by interpreting the high-frequency curve as a mapping of a torus into the plane, where it acquires singularities. Puffing up the torus and cutting across the singular set produces the bowls shown. In the last frame, the bowls remain hanging in the air as their components fall apart. The shapes are quite difficult to understand; a jumble of pieces helps depict some of the eccentric details.

Rachel Quinlan's innovative paper about the six subtypes of cmm wallpaper patterns for the Bridges 2024 conference inspired me to apply Fourier methods to create this kind of symmetry. Subtypes are based on extra symmetry in the frieze strips bounded by neighboring parallel glide and mirror axes in the pattern. In a typical cmm pattern, these have only vertical mirror symmetry, and this is type 1. This image helps us understand the subtypes by isolating those strips, where the symmetry can be revealed.