The nine criteria for isohedral tiles.

Sometimes it is useful to be able to tell if a shape will tile isohedrally or not. This can be done by testing the shape to see if it fits into any of nine criteria which deserve to be better known. These are given in Schattschneider's book 'Visions of Symmetry' at the back in bold in the chart of the Heesch types. (These match the shapes given by Grunbaum and Shepherd in 'Tilings and Patterns' as IH numbers 4,5,2,1,3,6,7,21, and 28.) Note that certain edges may 'collapse' to zero.

I will use similarly marked arrows to represent congruent curves. Thanks to Paul Church and Craig Kaplan for pointing out the need for the final rules for criteria 5 and 6.

Criterion 1: The Conway criterion.

Rules:
1. Sides a and d will be translates of each other.
2. Sides b,c,e,f will be centrosymmetric.

Criterion 2:

Rules:
1. Sides a and d will be translates of each other.
2. Sides b and c will be centrosymmetric.
3. Sides e and f will glide reflect to each other.

Criterion 3:

Rules:
1. Sides a and d will be translates of each other.
2. Sides b and c will glide reflect to each other.
3. Sides e and f will glide reflect to each other.

Criterion 4:

Rules:
1. Sides a and d will be translates of each other.
2. Sides b and e will be translates of each other.
3. Sides c and f will be translates of each other.

Criterion 5:

Rules:
1. Sides a and d will be translates of each other.
2. Sides b and f will glide reflect to each other.
3. Sides c and e will glide reflect to each other.
4. The glide reflections above will be parallel.

Criterion 6:

Rules:
1. Sides a and d will glide reflect to each other.
2. Sides b and f will glide reflect to each other.
3. Sides c and e will be centrosymmetric.
4. The glide reflections above will be perpendicular.

Criterion 7:

Rules:
1. Side c will rotate 120 degrees around D to side d.
2. Side e will rotate 120 degrees around F to side f.
3. Side a will rotate 120 degrees around B to side b.

Criterion 8:

Rules:
1. Side c will rotate 120 degrees around D to side d.
2. Side e will be centrosymmetric.
3. Side a will rotate 60 degrees around B to side b.

Criterion 9:

Rules:
1. Side b will rotate 90 degrees around C to side c.
2. Side a will be centrosymmetric.
3. Side d will rotate 90 degrees around E to side e.

In practical use, we can combine some of these checks. All of the criteria 1-5 involve a translation. (side a to side d.) These are convenient to check together. In addition, criteria 7 and 8 both have 120 degree rotations. These could be checked together. For the artists out there, try using criterion #3. Escher used it for many of his tilings as it allows all the horses (or whatever) to remain upright.