The steady state packing fraction of a tapped granular bed is studied for different grain shapes via a discrete element method. Grains are monosized regular polygons, from triangles to icosagons. Comparisons with disc packings show that the steady state packing fraction as a function of the tapping intensity presents the same general trends in polygon packings. However, better packing fractions are obtained, as expected, for shapes that can tessellate the plane (triangles, squares and hexagons). In addition, we find a sharp transition for packings of polygons with more than 13 vertices signaled by a discontinuity in the packing fraction at a particular tapping intensity. Density fluctuations for most shapes are consistent with recent experimental findings in disc packing; however, a peculiar behavior is found for triangles and squares.