Understanding and Visualising the Shape Sphere

Summer research 2024: the three-body problem · Part 5 of 11

A weekly research log from my summer project at the University of Edinburgh, supervised by Prof. Jenni Smillie.

← Problem of Two Fixed Centres (Part I)  ·  Problem of Two Fixed Centres (Part II) →

This post introduces the shape sphere, a way to picture the motion of three bodies by keeping track only of the shape of the triangle they form.

Reducing the three-body problem

Reduction simplifies the three-body problem by removing degrees of freedom associated with symmetries.

Let the three bodies lie in the plane, identified with \(\mathbb{C}\). Before reduction, the configuration space is \(\mathbb{C}^3\), the positions of the three masses. By Galilean invariance we can fix the centre of mass at the origin, which reduces the configuration space to

\[X = \left\{ x = (x_1, x_2, x_3) \in \mathbb{C}^3 \;\middle|\; \sum_{i=1}^3 x_i = 0 \right\} \cong \mathbb{C}^2 .\]

Next come the rotations and reflections of the plane, which act by

\[R_\theta \cdot (x_1, x_2, x_3) = (e^{i\theta} x_1, e^{i\theta} x_2, e^{i\theta} x_3), \qquad S \cdot (x_1, x_2, x_3) = (\bar{x}_1, \bar{x}_2, \bar{x}_3).\]

Taking the quotient of \(X\) by the rotation group \(SO(2)\) gives a space homeomorphic to \(\mathbb{R}^3\), whose points are oriented congruence classes of triangles. Fixing the overall size as well, by setting the moment of inertia to \(I = 1\), leaves a 2-sphere whose points are oriented similarity classes of triangles: the shape sphere.

The maps

For equal masses, the Jacobi map \(\mathcal{J}: X \rightarrow \mathbb{C}^2\) is

\[\left(z_1, z_2\right)=\mathcal{J}\left(x_1, x_2, x_3\right)=\left(\frac{1}{\sqrt{2}}\left(x_3-x_2\right),\ \sqrt{\frac{2}{3}}\left(x_1-\frac{1}{2}\left(x_2+x_3\right)\right)\right),\]

and the Hopf map \(\mathcal{K}: \mathbb{C}^2 \rightarrow \mathbb{R} \times \mathbb{C}=\mathbb{R}^3\) is

\[\mathcal{K}\left(z_1, z_2\right)=\left(u_1,\ u_2+i u_3\right)=\left(\left|z_1\right|^2-\left|z_2\right|^2,\ 2 \bar{z}_1 z_2\right) .\]

One checks that

\[\left|\mathcal{K}\left(z_1, z_2\right)\right|^2=u_1^2+u_2^2+u_3^2=\left(\left|z_1\right|^2+\left|z_2\right|^2\right)^2=I^2,\]

where \(I = \lvert z_1\rvert^2 + \lvert z_2\rvert^2\) is the moment of inertia. So \(\mathcal{K} \circ \mathcal{J}\) sends the 3-sphere \(I = 1\) onto the unit 2-sphere in \(\mathbb{R}^3\), and rotating the triangle (multiplying \((z_1, z_2)\) by \(e^{i\theta}\)) does not change the image. This is the Hopf fibration. The details are in Chenciner and Montgomery’s paper, A remarkable periodic solution of the three-body problem in the case of equal masses.

Plotting in Python

Here is the figure-eight orbit traced on the shape sphere:

The same picture works for other periodic orbits; the Three-body gallery is a good place to explore them.

Progress

Working out how to plot the shape sphere, and reading the underlying theory, took several days. Long calculations can obscure the intuition, which is exactly what pictures like these help to restore. Next I return to the problem of two fixed centres, to classify the orbits of a body attracted by them.

Credits

This post is based on A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Mathematics 152 (2000), 881–901.

Changes on 25 October 2026

What was wrong. Nothing in the content.

What changed. The reduction is explained more clearly, the notation is consistent, and there is a link to Chenciner and Montgomery’s paper.