  For a manifold with an affine connection, we prove formulas which
infinitesimally quantify the gap in a certain naturally defined open geodesic
quadrilateral associated to a pair of tangent vectors $u$, $v$ at a point of
the manifold. We show that the 1st order infinitesimal obstruction to the
quadrilateral to close is always zero, the 2nd order infinitesimal obstruction
to the quadrilateral to close is $-T(u,v)$ where $T$ is the torsion tensor of
the connection, and if $T = 0$ then the 3rd order infinitesimal obstruction to
the quadrilateral to close is $(1/2)R(u,v)(u+v)$ in terms of the curvature
tensor of the connection. Consequently, the torsion of the connection, and if
the torsion is identically zero then also the curvature of the connection, can
be recovered uniquely from knowing all the quadrilateral gaps. In particular,
this answers a question of Rajaram Nityananda about the quadrilateral gaps on a
curved Riemannian surface. The angles of $3\pi/4$ and $-\pi/4$ radians feature
prominently in the answer, along with the value of the Gaussian curvature. This
article is essentially self-contained, and written in an expository style.
