  Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane
can be dissected into mutually incongruent convex quadrangles of the same area
and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections
of the plane by mutually incongruent triangles of unit area that are
arbitrarily close to the periodic vertex-to-vertex tiling by equilateral
triangles.
