  For the Newtonian 4-body problem in space we prove that any zero angular
momentum bounded solution suffers infinitely many coplanar instants, that is,
times at which all 4 bodies lie in the same plane. This result generalizes a
known result for collinear instants ("syzygies") in the zero angular momentum
planar 3-body problem, and extends to the $d+1$ body problem in $d$-space. The
proof, for $d=3$, starts by identifying the center-of-mass zero configuration
space with real $3 \times 3$ matrices, the coplanar configurations with
matrices whose determinant is zero, and the mass metric with the Frobenius
(standard Euclidean) norm. Let $S$ denote the signed distance from a matrix to
the hypersurface of matrices with determinant zero. The proof hinges on
establishing a harmonic oscillator type ODE for $S$ along solutions. Bounds on
inter-body distances then yield an explicit lower bound $\omega$ for the
frequency of this oscillator, guaranteeing a degeneration within every time
interval of length $\pi/\omega$. The non-negativity of the curvature of
oriented shape space (the quotient of configuration space by the rotation
group) plays a crucial role in the proof.
