The Three-Body Problem and the Shape Sphere

The three-body problem defines dynamics on the space of triangles in the plane. The shape sphere is the moduli space of oriented similarity classes of planar triangles and lies inside shape space, a Euclidean three-space parameterizing oriented congruence classes of triangles. We derive and investigate the geometry and dynamics induced on these spaces by the three-body problem. We present two theorems concerning the three-body problem whose discovery was made through the shape space perspective.