To make a p4m tiling, begin with the tile shown to the left. The symmetry consists of reflections across the red line, 90 degree rotations about the yellow corners and 180 degree rotations about the purple corners.

Imagine how the p4 symmetries effect the tiling as a whole. If we reflect the entire tiling across any of the red lines, then the reflected tiling matches the original perfectly. Similarly when rotating 90 or 180 degrees about the yellow or red points, respectively, the rotated tiling will match the original. So in addition to having the four- and two-fold rotational symmetry that p4 tilings have, our tiling also has reflexive symmetry.

By looking at it, you can see p4m tilings are periodic, which is just the same as saying the tiling is symmetric under translations. If we rotate and reflect our original tile we can make the sqaure at the right. Then we can recreate the original p4m tiling by just translating this larger tile.

Kali denotes this symmetry by "*442". You can go to now to experiment with symmetry p4m.