  The purpose of this article is to
  \begin{enumerate}
  \item define $M(t,k)$ the $t$-fold center of mass arrangement for $k$ points
in the plane,
  \item give elementary properties of $M(t,k)$ and
  \item 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.
  \end{enumerate}
  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.
