Andy Liu defines seven levels of ability to tile strips of cells with a specified polyomino:
This is a strict hierarchy. Copies of a rectangle can form a bent strip. Two bent strips can form a branched strip, assuming that we do not require the arms to have equal width. Two branched strips can form crossed strips, again assuming that the arms' widths need not be equal.
The tiling of an arm of a crossed strip must eventually repeat a configuration of tiles, so that a periodic tiling is possible, and so an infinite straight strip can be tiled. Infinite straight strips can tile the whole plane.
Other classes can be added to this classification, not necessarily preserving hierarchy. Solomon Golomb, in his book Polyominoes, second edition, includes Quadrant and Strip, Quadrant, Half Plane, and Self (rep-tiling), but not Branched Strip or Crossed Strips.
It is not always easy to determine the class of a polyomino. Some of the classifications below are uncertain. If you find a higher-class tiling for a polyomino, please write!
For Class 1 tilings—rectifications—I show only the polyomino, not the tiling of the rectangle. You can find minimal known tilings at Mike Reid's Rectifiable Polyomino Page.
For Class 6 tilings—tilings of the whole plane—I show only the polyomino, not the tiling. For the tilings, see this page at Joseph Myers's site.
No polyominoes of order 6 or less belong to Class 7. All can tile the plane.
Last revised 2016-10-03.