About 5-6 years ago, I sat down with graph paper, pencil, and ruler to design a new quilt block. I had three requirements: it had to have fourfold rotation; it had to extend past the confines of a 4-patch block; and, all the patches had to be identical.
First, I drew a 2x2 grid, and divided the sides into thirds with little tick marks:
Next, I drew two lines in one of the squares. Since this block would have fourfold rotation, I drew the same two lines in the remaining three squares, rotating the lines 90 degrees in each. The block now had four arms, ending at the outer edges of the block:
Next, I drew four more squares next to the truncated arms and drew my original two lines in each (remember, each patch has to be identical):
I drew another pair of diagonal lines, leading from the original pair of lines to an adjacent side of the square (and added the same lines to my original four squares). The block was still incomplete, because the arms still ended at the outer edges of the block:
Finally, I drew four more squares next to the truncated arms and drew all the lines I'd drawn so far in each of them:
Looking at my progress so far, there was only one place left for me to draw a line. At this point, there was a line beginning or ending at each of the tick marks I had drawn originally.
All the squares were identical; all the lines connected with the lines in the adjacent squares; and, look at the shapes!! I added some more squares and drew the same lines in them.
My design tile was born (though I didn't realize it at the time) . . .