Polyomino Tiling

A polyomino tiling is a tiling of the plane by specified types of polyominoes. Tiling by polyominoes has been investigated since at least the late 1950s, particularly by S. Golomb (Wolfram 2002, p. 943).

Interestingly, the Fibonacci number gives the number of ways for dominoes to cover a checkerboard.

Each monomino, domino, triomino, tetromino, pentomino, and hexomino tiles the plane without requiring flipping. In addition, each heptomino with the exception of the four illustrated above can tile the plane, also without flipping (Schroeppel 1972).

Recently, sets of polyominoes that force non-periodic patterns have been found. The set illustrated at left above was announced by Roger Penrose in 1994, and the slightly smaller set illustrated at right below was found by Matthew Cook (Wolfram 2002, p. 943).

Both of these sets yield nested patterns, as illustrated above for Cook's tiles (Wolfram 2002, p. 943).

Consider now those collections of all -ominoes which form a rectangle. The polyominoes of orders and form only a square and rectangle, respectively. The two polyominoes of order cannot form a rectangle, nor can the five polyominoes of order or the 35 polyominoes of order (Beeler 1972). There are several rectangles formed by the 12 polyominoes of order , as summarized in the following table (Fletcher 1965, Beeler 1972).

size	solutions
	2
	368
	1010
	2339
two	2
with hole	65