# Tiling the plane

A shape *tiles the plane* if it is possible to cover an infinite grid with copies of it such that no holes are left. These tilings, or *tessellations*, have been well known for millenia, being featured in places such as Islamic architecture to the works of M.C. Escher.

Given their structure, it is a natural question to ask: which polyominoes tile the plane?

## The Translation Criterion

In the most restricted sense, we can ask if a polyomino tiles the plane by *translation* only--that is, by sliding copies of itself without rotating or flipping. Whether a polyomino can do this is determined by the *translation criterion*:

A polyomino satisfies the Translation Criterion if its border can be divided into six segments A, B, C, D, E, F such that the pairs A-D, B-E, and C-F are translations of each other.

As an example, consider the following polyomino:

This polyomino tiles the plane because blue, green, and yellow segments are translations of each other. This translation is realized as shown here:

A polyomino tiles the plane by translation if and only if it satisfies the translation criterion.

## The Conway Criterion

The translation criterion covers some tilings but not all. We may ask which polyominoes tile the plane if we are allowed to rotate or reflect copies in addition to sliding. One powerful tool at our disposal is the *Conway Criterion*:

A polyomino satisfies the Conway criterion if its border can be divided into six segments A, B, C, D, E, F such that:

- A and D are translations of each other
- B, C, E, and F are symmetric with respect to rotation about a center point

As an example, consider the following polyomino:

This polyomino satisfies the criterion since the grey segments are translations of each other, and each of the other segments can be rotated about their center to yield the same segment. The tiling for this example is as such:

Note that, unlike the translation criterion, the Conway criterion is *sufficient* but not *necessary*. Some polyominoes do not satisfy it but still tile the plane. However, it is still very powerful, and all tileable polyominoes up to size 8 satisfy the criterion, or you can glue two copies of it to form a shape that satisfies the criterion, in this example:

With the translation and Conway criteria, we can find all the polyominoes that tile the plane for up to size 8. For more information on tilings for larger polyominoes, see the references below.

## Reference

- "Planar tilings by polyominoes, polyhexes, and polyiamonds" by Glenn C. Rhoads.
- "Polyomino, polyhex and polyiamond tiling" by Joseph Myers.