Conway criterion

[1] The Conway criterion is a sufficient condition to prove that a prototile tiles the plane but not a necessary one.

In 1963 the German mathematician Heinrich Heesch described the five types of tiles that satisfy the criterion.

[5] Conway was likely inspired by Martin Gardner's July 1975 column in Scientific American that discussed which convex polygons can tile the plane.

[6] In August 1975, Gardner revealed that Conway had discovered his criterion while trying to find an efficient way to determine which of the 108 heptominoes tile the plane.

[7] In its simplest form, the criterion simply states that any hexagon with a pair of opposite sides that are parallel and congruent will tessellate the plane.

Prototile Octagon satisfying the Conway criterion. Sections AB and ED are shown in red, and the remaining segments are shown in color with a dot on the point of centrosymmetry.
A tessellation of the above prototile meeting the Conway criterion.
Example tessellation based on a Type 1 hexagonal tile.
A tiling nonomino that does not satisfy the Conway criterion.
The four heptominoes incapable of tiling the plane, including the one heptomino with a hole.