Conway criterion


In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a fast way to identify many prototiles that tile the plane; it consists of the following requirements: The tile must be a closed topological disk with six consecutive points A, B, C, D, E, and F on the boundary such that:
Any prototile satisfying Conway's criterion admits a periodic tiling of the plane—and does so using only translation and 180-degree rotations. The Conway criterion is a sufficient condition to prove that a prototile tiles the plane but not a necessary one; there are tiles that fail the criterion and still tile the plane.

Examples

In its simplest form the criterion states that any hexagon whose opposite sides are parallel and congruent will tessellate the plane by translation. But when some of the points coincide, the criterion can apply to other polygons and even to shapes with curved perimeters.
The Conway criterion is sufficient, but not necessary, for a shape to tile the plane. For each polyomino up to order 8 that can tile the plane at all, either the polyomino satisfies the Conway criterion or else two copies of the polyomino can be combined to form a polyform patch that satisfies the criterion. The same is true of every tiling nonomino, except for the two tiling nonominoes on the right.