Configurations, and parallelograms associated to centers of mass
  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. 
F. R. Cohen, Y. Kamiyama