Whatever the size of a pattern you find, you cannot tile the plane with translations (just translations) using that pattern: you need to rotate it. The definiti

Whatever the size of a pattern you find, you cannot tile the plane with translations (just translations) using that pattern: you need to rotate it. The definition is just that. So, there are "patterns" (as a matter of fact, the elementary tile is repeated infinitely on the tiling) but you cannot fill the plane with mere translations of one. See: https://personal.math.ubc.ca/~cass/courses/m308-02b/projects/schweber/tilings.html https://personal.math.ubc.ca/~cass/courses/m308-02b/projects...