This animation shows Marchal's celebrated family of choreographies bifurcating from the Lagrange equilateral triangle. The family, named \(P_{12}\), is characterized by

• \(2\pi\)-periodic solutions, where the three bodies are spaced out by a time shift of \(2\pi/3\),
• in a rotating frame of frequency \(\Omega\) (which is the continuation parameter)
• and having 12 symmetries per period

At the 1999 Evanston (Illinois, USA) conference honoring Saari's 60^th birthday, Marchal conjectured that \(P_{12}\) terminates at the figure eight choreography discovered numerically by Moore in 1993 and proven to exist by Chenciner and Montgomery in 2000.

After 25 years, this conjecture is now a theorem.

More information on \(N\)-body choreographies can be found in Montgomery's scholarpedia article, and more animations on Calleja's website!