I think a more interesting pattern with four common vertices is one that uses a G1G1G2G2 arrangement. Of the 64 possible tiles with the identical centro-symmetric sides, only eight shapes will form patterns that are true tessellations. Two are shown below.
In the figure below all eight are shown. Six tiles of each of the eight surround two voids of the same shape. Can you identify them?
Fitting together other shapes results in voids that are different than the surrounding tiles.Delightful Designs: A Coloring Book of Magical Patterns if I ever revise it.