Absolutely Flat Idempotents
  A real $n$-by-$n$ idempotent matrix $A$ with all entries having the same absolute value is called {\it absolutely flat}. We consider the possible ranks of such matrices and herein characterize the triples: size, constant, and rank for which such a matrix exists. Possible inequivalent examples of such matrices are also discussed. 
Jonathan M. Groves, Yonatan Harel, Christopher J. Hillar, Charles R.   Johnson, Patrick X. Rault