The basic piece is obtained by cutting out part of one side of the shape, then rotating this part, with the centre of rotation on one of the vertices, and then adding this piece to one of the other sides. The angle of rotation is either 60º or 120º and the vertices forming the centre of rotation are not adjacent.

2.- Design your own tessellation using this second rule of construction.

3.- Use the rule to design pieces based on other polygons which are not hexagons.

4.- How much choice do you have as to how you fit the pieces together if you only cut out a shape from one of the sides?

The general principle governing the formation of Tessellations is the repetition of a basic element or piece which is translated in two different directions. Different types of tessellations are produced by adding more specific conditions of regularity and piecing together to this general principal.

5.- Work out how the reptile shapes you have should be pieced together by thinking about the way the pieces were made.

6.- Identify which of the shapes in the pattern is the basic motif which is then moved in two different directions in order to form the tessellation.

7.- Work out which other movements are necessary to form the tessellation.

8.- What happens if the angles of rotation are neither 60º nor 120º?

9.- Investigate different designs which are made form squares.

Due to the way the pieces are formed they cannot all be in the same position when we want to piece them together. Therefore, we have to refer to dynamic geometry in order to work out how the pieces should be joined together.

10.- Follow the instructions given in the first activity to make the reptile shapes out of paper. Investigate which different movements the shapes need to undergo in order to form the tessellation.

11.- Use the tessellation in the activity above to find other possible ways of translating the basic piece to form different patterns which, after the corresponding translations, can be used to tessellate the plane.

Once you have the basic pattern, which is made up of joining the first few pieces together, the rest of the tessellation is formed by moving this pattern in two different directions.

12.- Work out which two directions the pattern needs to be moved in to form the tessellation.

13.- Work out the corresponding vectors governing these translations.

14.- Compare the directions and vectors with the sides and dimensions of the original hexagon.

15.- Are there any axes of symmetry which can be used to obtain a piece of the tessellation which is a reflection of another fixed piece of the same tessellation?