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The figure-eight orbit found by C. Moore, R. Montgomery and A. Chenciner is depicted in blue, following coordinates published by C. Simó. The coordinate axes are marked with a scale that can be thought of as essentially arbitrary, and the plane is thought of as that of complex numbers. The three equal masses that travel under Newtonian gravitation are, arbitrarily again, numbered 0, 1 and 2. They form a triangle whose edges are given in green. The discrete Fourier transform, i.e. harmonic analysis on the group of integers modulo 3, applied to the vertices of the triangle, gives three new points also numbered 0, 1 and 2.
The 0'th harmonic value is fixed, by the conventional choice of initial
conditions at the origin. The edges of the triangle formed by the
Fourier transform values are given in light blue. As the Fourier transform point 1 evolves
with time it traces a curve given in red, and similarly the point 2 traces out a purple curve. The acceleration and force
vectors, as calculated approximately from the numerical values used in
the plots, and scaled by a factor of about a twentieth, are labelled
respectively a and f, at the ends of scarlet which are marked with grey squares.
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