Tessellations with Regular Polygons

Many regular polygons or combinations of regular polygons appear in nature and architecture.

Floor Designs Honeycomb

These model how regular polygons cover a surface. This is called tessellating the surface or a tessellation.

You already know that squares and regular hexagons create a monohedral tessellation.

Because a regular hexagon can be divided into six equilateral triangles we can logically conclude that equilateral triangles also create a monohedral tessellation.

For shapes to fill the plane without gaps or overlaps around a point, their angles measures must be a factor of 360. Why is this?

Regular polygons with more than six sides have angles that are more than 120o.

So if you put more than two of them together you will have over 360o so the sides will overlap. Is this what happened with the heptagon?

So the only regular polygons that create a monohedral tessellation are equilateral triangles, squares, and regular hexagons.

A tessellation with congruent regular polygons is called a regular tessellation.

Tessellations can have more than one shape.

You can use an octagon and two squares to tessellate the surface.

Notice that you can put your pencil on any vertex and that the point is surrounded by one square and two octagons.

We call this tessellation a numerical name or vertex arrangement of 4•8•8 or 4·82.

What two shapes make this tessellation? The same polygons appear in the same

order at each vertex: triangle, dodecagon, dodecagon.

It is a 122·3 vertex arrangement

What polygons make this tessellation? The same polygons appear in the same

order at each vertex: square, hexagon, dodecagon.

What would the vertex arrangement be? 4·6·12

There are eight different semiregular tessellations. You have seen three of them.

You will investigate to find the other five tessellations.

The remaining five only use triangles, squares, and hexagons.