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 120 o.  So if you put more than two of them together you will have over 360 o 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 488 or 4·8 2.

 What two shapes make this tessellation?  The same polygons appear in the same order at each vertex: triangle, dodecagon, dodecagon.  It is a 12 2 ·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.