5 September 2026
The #SciArtSeptember day 5 prompt is ‘Orbit’.
It is well-known that the three-body problem does not have a general closed-form analytic solution. But there are many known periodic solutions of varying complexity. So I offer the following ‘artistic’ plot of twelve of these periodic solutions:
The solutions are from three papers by R. Broucke, by Broucke & D. Boggs, and by A. Chenciner & R. Montgomery. From left to right along each row, top row first, the solutions are Broucke’s solutions A1/1, A1/7, A2/3, A3/5, A4/5, R1/1;1 Broucke & Boggs’s solutions 1, 4, 30, 40, 122,2 and Chenciner & Montgomery’s solution.3
The solutions were plotted using a small Python program that generated Asymptote code specifying the motions of the three bodies as paths. This was imported into an Asymptote program that actually did the drawing of the paths with a fading effect. The resulting a PDF is available here. The PDF was converted to the SVG image on this page using Inkscape.
[The image was used as an illustration at the start of the main text of my book Form & Number: A History of Mathematical Beauty.]
R. Broucke ‘On Relative Periodic Solutions of the Planar General Three-Body Problem’. In: Celestial Mechanics. 12, no. 4 (December 1975), pp. 439–462. DOI: 10.1007/bf01595390 ↩
R. Broucke & D. Boggs ‘Periodic Orbits in the Planar General Three-Body Problem’. In: Celestial Mechanics. 11, no. 1 (February 1975), pp. 13–38. DOI: 10.1007/bf01228732 ↩
A. Chenciner & R. Montgomery ‘A remarkable periodic solution of the three-body problem in the case of equal masses’. In: Annals of Mathematics. 152, no. 3 (November 2000), pp. 881–901. DOI: 10.2307/2661357 ↩