Recurrence in Lissajous Curves and the Visual Representation of Tuning Systems
Carlos A. Sierra
Abstract
Curves in the two-dimensional plane that describe the trajectory of two perpendicular oscillators are commonly know as Bowditch or Lissajous curves. Jules Antoine Lissajous popularized these curves with his ingenious experiments in which he projected a beam of light to two vibrating tuning forks set at right angles, and with a mirror attached to their tip (Lovering, 1880 ; Hales, 1945 ; Gallozzi and Strollo, 2023 ). The resulting light projected on a screen described the motion of the two forks vibrating at different frequencies but at fixed ratios, producing visually appealing shapes. Similar experiments were reproduced by others using different types of vibrating systems, commonly pendulums (Lovering, 1880 ). In 1877, Samuel C. Tisley introduced at a Royal Society Meeting the harmonograph, a device consisting of two pendulums moving perpendicularly, designed to reproduce Lissajous curves on a piece of paper. Writing in Foundations of Science, Gallozzi and Strollo ( 2023 ) give a comprehensive introduction to the history of Lissajous curves and the harmonograph.