TESSELLATIONS

=TILINGS WITH POLYGONS =

Tilings are constructed from tiles, by which we mean planar shapes with no holes or gaps in them. So, a tiling of the plane is a covering of the whole plane with tiles in such a way that there are no gaps or overlaps. A tiling of the plane is often called a tessellation.

Here are some examples of tessellations.

The study of tilings is a fascinating subject. However, here we can do no more than offer you an introduction to the subject. We will therefore restrict our study to those tilings with straight-edged tiles that is, polygonal tilings, such as those in figures B and C.
 * ==A [[image:tiling1.jpg width="206" height="204"]]== || ==B [[image:tile2.gif width="285" height="185"]]== || ==C [[image:regular.jpg width="249" height="173"]]== ||

B and C are examples of edge-to-edge tilings because each edge of each polygon is an edge of another polygon, and no two polygons meet along more than one edge.

Figure C is an example of a regular tiling. A regular tiling is an edge-to-edge tiling with congruent regular polygons.

In a tiling the edges meet at a vertex. Figure C is made up of regular hexagons and 3 hexagons meet at each vertex. Do you remember the interior angle of a hexagon? Well, it is 120 degrees. So the angles around any vertex in a tiling of regular hexagons sum to 360 degrees.

The only other polygons we can use to cover the plane are regular triangles and squares. In a tiling of regular triangles, 6 triangles meet at each vertex and 6 x 60 degrees = 360 degrees. In a tiling of squares, 4 squares meet at each vertex and 4 x 90 degrees = 360 degrees.

What about Figure B? It is made up of regular triangles and regular hexagons. It is not regular but it does cover the plane. Why, because the sum of the angles at each vertex is again 360 degrees, this time made up by 60 + 60 + 90 +90 degrees.

We would like you to investigate the different ways of tiling the plane using combinations of two or three different regular polygons. How many tessellations can you create from triangles and squares? Or from pentagons and triangles, for example?

You can use Geogebra to create your tessellation and then copy and paste it into the table. We would also like you to describe the angles around each vertex in your tessellation. Use the TAB key to add more rows to the table.
 * <span style="font-family: Verdana,Geneva,sans-serif; font-size: 120%;">TESSELLATION<span style="color: #ffffff; font-family: Verdana,Geneva,sans-serif; font-size: 120%;">................. || <span style="font-family: Verdana,Geneva,sans-serif; font-size: 120%;">POLYGONS AND ANGLES<span style="color: #ffffff; font-family: Verdana,Geneva,sans-serif; font-size: 120%;">....................  || <span style="font-family: Verdana,Geneva,sans-serif; font-size: 120%;">NAME<span style="color: #ffffff; font-family: Verdana,Geneva,sans-serif; font-size: 120%;">..........................  ||
 * [[image:Imagen1.jpg width="165" height="106"]] || SQUARES AND TRIANGLES.
 * 90 + 60 +60 + 90 + 60 || MAR ||

<span style="font-family: Verdana,Geneva,sans-serif; font-size: 110%;">And what about trying to create a tessellation using non regular convex polygons?


 * ==TESSELLATION== || ==POLYGONS AND ANGLES .... == || ==NAME ..................... == ||
 * [[image:irreg.jpg width="276" height="188"]] || Triangles
 * 20º + 50º + 110º || ANE ||
 * [[image:pent.png width="293" height="196"]] || Pentagons
 * 90º x 4
 * 140º + 140º + 80º || GEMMA ||