Conway criterion

The Conway criterion is an important idea in the mathematical study of patterns that repeat without gaps, called tessellations. It was named after the English mathematician John Horton Conway, who created this rule to help determine when a single shape, known as a prototile, can cover an entire flat surface.

According to the Conway criterion, a shape must meet certain conditions. It needs to be a closed shape with six points along its edge. These points must line up in a special way: part of the edge from one point to the next must match another part of the edge when moved straight, as if by a translation. Also, certain middle sections of the edge must look the same when flipped around their middle points, a property called being centrosymmetric.

If a shape meets the Conway criterion, it can be used to create a repeating pattern across the whole plane, using only 180-degree flips. While this rule guarantees that a shape can tile the plane, there are also shapes that can tile the plane without meeting all of Conway’s conditions. Interestingly, every shape that fits the Conway rule can also be folded into special three-dimensional shapes called an isotetrahedron or a rectangle dihedron.

History

The Conway criterion helps us know if a shape can cover a flat surface completely without gaps. The artist M.C. Escher used ideas similar to this rule in the 1920s, even before it was officially named. Later, in 1963, the mathematician Heinrich Heesch described five types of shapes that follow this rule.

Mathematician John Horton Conway developed the Conway criterion after reading a 1975 article by Martin Gardner in Scientific American. Gardner’s article talked about shapes that can tile a plane, and Conway used this to study which of the 108 heptominoes can do the same.

Examples

The Conway criterion helps us understand which shapes can fit together to cover a flat surface completely, like a puzzle. For example, any hexagon with two opposite sides that are parallel and the same length will tile the plane. This is also true for parallelograms, which are four-sided shapes where opposite sides are equal and parallel.

The Conway criterion works well with polyforms, which are shapes made by connecting squares edge-to-edge. Almost all polyominoes (shapes made of squares) up to size 7 meet the Conway criterion, or two of them together can form a group that does. This pattern continues for most shapes up to size 9, with just a few exceptions.