One of the defining characteristics of tessellations is that they have repeated patterns. Thus, let us place a restriction on the type of tessellations we are trying to find. We will consider only tessellations in which the same regular polygon arrangement is repeated at every vertex. The regular tessellations satisfy this condition, because at every vertex in the 3.3.3.3.3.3 tessellation, you can find six equilateral triangles; at every vertex in the 4.4.4.4 tessellation, you can find four squares; and at every vertex in the 6.6.6 tessellation, you can find three regular hexagons.

A logical process for testing whether an arrangement will tessellate is as follows:

1. Start with the initial arrangement.
2. Continue this arrangement at other vertices.
3. If a problem arises, stop, and the arrangment fails. Problems include gaps that cannot be filled and polygon arrangements that do not match the original one.
4. If it becomes clear what the overall pattern is, the arrangement succeeds.

Here are examples of polygon arrangements that cannot be made to tessellate:

Here is an example of a polygon arrangement that can be made to tessellate:

If all 18 arrangements of regular polygons are tested, it turns out that only 8 of them will tessellate.