Conway criterion

In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will 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:

• the boundary part from A to B is congruent to the boundary part from E to D by a translation T where T(A) = E and T(B) = D.
• each of the boundary parts BC, CD, EF, and FA is centrosymmetric—that is, each one is congruent to itself when rotated by 180-degrees around its midpoint.
• some of the six points may coincide but at least three of them must be distinct.

Any prototile satisfying Conway's criterion admits a periodic tiling of the plane—and does so using only 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.

Every Conway tile is foldable into either an isotetrahedron or a rectangle dihedron and conversely, every net of an isotetrahedron or rectangle dihedron is a Conway tile.