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By introducing simple topological constraints and applying a binarydecomposition method, we show the existence of a set of prograde double-doubleorbits for any rotation angle $\theta \in (0, \pi/7]$ in the equal-massfour-body problem.  A new geometric argument is introduced to show that for any $\theta \in (0,\pi/2)$, the action of the minimizer corresponding to the progradedouble-double orbit is strictly greater than the action of the minimizercorresponding to the retrograde double-double orbit. This geometric argumentcan also be applied to study orbits in the planar three-body problem, such asthe retrograde orbits, the prograde orbits, the Schubart orbit and theH\'{e}non orbit.

Existence of prograde double-double orbits in the equal-mass four-body problem