While working on Exploring Tessellations: A Journey through Heesch Types And Beyond, I discovered that if a square template is shaped with identical center-point rotation edges, there are exactly four distinct shapes possible. Two of those shapes can be in two orientations, one in four orientations, and the last in eight orientations, for a total of 16 orientations.
If the shapes are connected not edge to edge but vertex to vertex, they form a checkboard-like pattern of tiles and voids. The voids have four edges that are formed by the four adjacent tiles and thus they too must have the same four shapes in 16 orientations that the tiles have. By arranging the tiles in different ways, a huge number of different patterns become possible. I tried to find some place on the web that explored these patterns, but could not. However, it is unclear what the best search terms would be.
I have taken this insight and made a series of eleven typefaces that are now available on MyFonts.com. A person using these typefaces can, by simply typing in letters from A to P or a to p, explore possible patterns that these four shapes can generate. Each typeface has two different designs on it. Below are a couple examples.
I was able to use these patterns to illustrate 11 of the 17 wallpaper groups of symmetry.
They are available at www.myfonts.com/fonts/ingrimayne/fab-fours/.