Regular tessellations

A tessellation is created when one or more shapes are used to cover a plane completely, with no gaps or overlaps. An alternative name for a tessellation is a tiling. A tiled floor is a real-life example of a tessellation.

A regular tessellation is a special case where the plane is covered by shapes that:

• are all regular polygons of the same shape.
• are all the same size.
• use edge-to-edge tiling (described below).

There are only three shapes that can form regular tessellations. They are:

• Equilateral triangles.
• Squares.
• Regular hexagons.

We will look at these cases and also learn why no other regular tessellations are possible.

Here is a video covering this topic:

Equilateral triangles

Here is a regular tessellation made up of equilateral triangles:

Triangles of any type will tessellate, but if the triangles are not equilateral, we can't call it a regular tessellation.

In the scheme above, every edge of any triangle is joined to a complete edge of a different triangle. There are no edges that join two or more other edges. This is edge-to-edge tiling, mentioned earlier.

The example below is not edge-to-edge tiling, so again it cannot be called a regular tessellation:

Here, notice that the horizontal bands of triangles are shifted relative to each other, so the edges of some triangles do not line up. This means that in some cases, an edge of one triangle meets two edges of other triangles:

Here, the base of the red triangle meets the edges of two different blue triangles.

Squares

Here is a regular tessellation made up of squares:

Rectangles, or any other quadrilaterals, will tessellate, but if the quadrilaterals are not squares, it isn't a regular tessellation.

Regular hexagons

Here is a regular tessellation made up of regular hexagons:

Why can no other polygons form a regular tessellation?

The only regular polygons that tessellate are those with 3, 4 or 6 sides.

To understand why this is, we will look at a couple of regular shapes that don't tessellate. First, regular pentagons:

If we place three regular pentagons (5 sides) together, it leaves a small angle. The angle is too small for another pentagon to fit, so it is impossible to fill the plane without the shapes either overlapping or leaving a gap.

If we try the same thing with regular heptagons (7 sides), we see a slightly different issue:

After joining two heptagons, the remaining angle is too small to add an extra heptagon. If we try this with a regular octagon (8 sides) or any shape with more sides, this will also fail for the same reason.

For regular tessellation, the vertices of the shapes must meet at a point, which means that the internal angles of all the shapes that meet at a vertex must add up to 360°. This table shows how this works for the three shapes:

Shape	Interior angle	Number of vertices that meet
Equilateral triangle	60°	6
Square	90°	4
Regular pentagon	108°	3.3333
Regular hexagon	120°	3
More sides	>120°	<3

For a tessellating square, 4 vertices meet, and for a tessellating hexagon, 3 vertices meet.

So for a pentagon, the number of vertices would have to be greater than 3 but less than 4, so it can't be an integer. So regular pentagons can't tessellate.

For a shape with more than 6 sides, the number of vertices would have to be less than 3, which is impossible. So regular polygons with more than 6 sides can't tessellate.

See also

• Line symmetry
• Rotational symmetry