  The purpose of this article is to
  1. define M(t,k) the t-fold center of mass arrangement for k points in the
plane,
  2. give elementary properties of M(t,k) and
  3. give consequences concerning the space M(2,k) of k distinct points in the
plane, no four of which are the vertices of a parallelogram.
  The main result proven in this article is that the classical unordered
configuration of k points in the plane is not a retract up to homotopy of the
space of k unordered distinct points in the plane, no four of which are the
vertices of a parallelogram. The proof below is homotopy theoretic without an
explicit computation of the homology of these spaces.
  In addition, a second, speculative part of this article arises from the
failure of these methods in the case of odd primes p. This failure gives rise
to a candidate for the localization at odd primes p of the double loop space of
an odd sphere obtained from the p-fold center of mass arrangement. Potential
consequences are listed.
