Objective: To explore tessellations which combine different regular polygons.
Materials: For teaching students, hands-on manipulatives are recommended.
The only regular polygons which will tile by themselves are triangles, squares, and hexagons. If different types of regular polygons are used together, however, other types of polygons will tessellate. For simplicity, the possibilities will be restricted to tessellations in which all of the tile edges are the same length. We will consider tessellations known as edge-to-edge. This means that the full edges (or sides) of two tiles meet. A subset of all edge-to-edge tessellations is the set of eleven Archimedean tessellations. Three of these are the aforementioned triangle, square, and hexagon tessellations. The other eight are sometimes referred to as semi-regular or homogeneous tessellations. The Archimedean tessellations are edge-to-edge tessellations in which every vertex is of the same type. Examples are shown below of, respectively, a non-edge-to-edge tessellation, an edge-to-edge tessellation that is not Archimedean, and an edge-to-edge tessellation that is Archimedean.
The only other regular polygons which can be used in edge-to-edge tessellations are octagons and dodecagons (twelve-sided polygons). Try to construct examples of edge-to-edge tessellations using octagons and using dodecagons which are Archmedean.
Now try to construct the remaining five Archimedean tessellations.
Next, try to construct tessellations which are edge-to-edge but not Archimedean using octagons, and using dodecagons. You will find that there are none using octagons.
Materials provided courtesy of Dr. Robert Fathauer of www.tessellations.com