So far we have only used identical pieces which are placed so that they fit together exactly when constructing Escher's tessellations.

2.- Turn half of the pieces over (Careful! Make sure you don't turn them  upside down!) and try to fit the pieces together again as explained above. Can you do it now?

3.- What is the geometric relationship between a piece and the same piece which has been turned over?

4.- Look at the tessellation in the window. It is the basic outline of a piece which forms part of an engraving called Horseriders? Look at the Internet web pages listed in the introduction to this unit to see the finished engraving and compare it to what you can see in this window. 

After looking carefully at the tessellation above you will have noticed that the yellow and pink pieces are facing in opposite directions. Furthermore, they are not in line with each other. In other words the pink pieces are located either above or below the yellow pieces. This effect is obtained through reflection and translation.

6.- Copy one of the pink pieces from the tessellation onto squared paper. Choose different straight lines and carry out the corresponding reflection of the piece about these lines of symmetry. Is it possible to obtain the yellow piece in the same position as it is found in the tessellation?

7.- Choose which of the vertical lines of symmetry gives us the exact position of the reflection.

8.- What is the connection between the piece obtained through reflection and the yellow piece in the tessellation?

10.- Look carefully at the shape of the quadrilateral. If we cut out part of one of the sides on the left of the shape can we reflect it about a line of symmetry which is located further to the right than the straight line joining the two central vertices together? Do two lines of symmetry exist?

11.- The quadrilateral which is used is a symmetrical shape with the horizontal line joining the two vertices furthest to the left and the right together as its line of symmetry. Is this an important consideration when choosing the basic polygon? Repeat the exercise with other regular and non-regular polygons and look carefully at the results.

You could make dotted paper to help you construct Escher's tessellations.

7.- We have suggested two types of paper to help you depending on what the basic polygon is that you are going to use. Squared paper is good for drawing right-angled triangles and parallelograms easily. The second type of paper suggested here is good for drawing equilateral triangles and regular hexagons.

8.- Work out the measurements between the dots to make it easier to produce your own sheets.

9.- As well as these sheets of paper you will also need scissors, a ruler, a compass and sellotape to stick the pieces you cut out from the sides back onto the polygon as we have seen during the explanation of the rules.

10.- Now you just need to decorate the pieces you have designed in order to create human or animal-like pieces or whatever other designs your imagination comes up with.