Tiling By Regular Polygons - Combinations of Regular Polygons That Can Meet at A Vertex

Combinations of Regular Polygons That Can Meet At A Vertex

The internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular -gon has internal angle degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex. Only eleven of these can occur in a uniform tiling of regular polygons. In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd.

With 3 polygons at a vertex:

• 3.7.42 (cannot appear in any tiling of regular polygons)
• 3.8.24 (cannot appear in any tiling of regular polygons)
• 3.9.18 (cannot appear in any tiling of regular polygons)
• 3.10.15 (cannot appear in any tiling of regular polygons)
• 3.122 - semi-regular, truncated hexagonal tiling
• 4.5.20 (cannot appear in any tiling of regular polygons)
• 4.6.12 - semi-regular, truncated trihexagonal tiling
• 4.82 - semi-regular, truncated square tiling
• 52.10 (cannot appear in any tiling of regular polygons)
• 63 - regular, hexagonal tiling

Below are diagrams of such vertices:

• 3.7.42

• 3.8.24

• 3.9.18

• 3.10.15

• 3.12.12

• 4.5.20

• 4.6.12

• 4.8.8

• 5.5.10

• 6.6.6

With 4 polygons at a vertex:

• 32.4.12 - not uniform, has two different types of vertices 32.4.12 and 36
• 3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4
• 32.62 - not uniform, occurs in two patterns with vertices 32.62/36 and 32.62/3.6.3.6.
• 3.6.3.6 - semi-regular, trihexagonal tiling
• 44 - regular, square tiling
• 3.42.6 - not uniform, has vertices 3.42.6 and 3.6.3.6.
• 3.4.6.4 - semi-regular, rhombitrihexagonal tiling

Below are diagrams of such vertices:

• 3.3.4.12

• 3.4.3.12

• 3.3.6.6

• 3.6.3.6

• 4.4.4.4

• 3.4.4.6

• 3.4.6.4

With 5 polygons at a vertex:

• 34.6 - semi-regular, Snub hexagonal tiling, comes in two entianomorphic forms. The vertex figures of the two entianomorphs are the same, but the resulting tilings are different.
• 33.42 - semi-regular, Elongated triangular tiling
• 32.4.3.4 - semi-regular, Snub square tiling

Below are diagrams of such vertices:

• 3.3.3.3.6

• 3.3.3.4.4

• 3.3.4.3.4

With 6 polygons at a vertex:

• 36 - regular, Triangular tiling

Below is a diagram of such a vertex:

• 3.3.3.3.3.3