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Motivated by a question of R.\ Nandakumar, we show that the Euclidean planecan be dissected into mutually incongruent convex quadrangles of the same areaand the same perimeter. As a byproduct we obtain vertex-to-vertex dissectionsof the plane by mutually incongruent triangles of unit area that arearbitrarily close to the periodic vertex-to-vertex tiling by equilateraltriangles.

Incongruent equipartitions of the plane into quadrangles of equal perimeter