TESSELLATING THE PLANE
Geometry in the art of M.C. Escher
1.  TESSELLATING THE PLANE WITH TRIANGLES.

M. C. Escher spent a large part of his artistic career designing engravings which included tessellations of animal-shaped pieces. He got his inspiration to form these tiles from Moorish arabesques. These decorations were designed using polygons (mostly regular) and submitting them to different transformations in order to produce shapes which covered the plane uniformly, without leaving any gaps.
1.- Imagine that you want to cover a surface with tiles, for example the floor of a room or the walls of a bathroom. Think of examples of shapes which you could use to do so without leaving any gaps.

2.- If you thought of a square as a solution to the problem above you should see that triangles can also be used, as if we cut the square tiles along their diagonal we get two triangles. What sort of triangles are they?

The shape which is inside the dark blue square can be changed by moving the two control points found on two of its vertices.

The shape inside the light blue square can be moved onto the brown area by dragging the control point on its lower vertex on the left.
3.- Use the window to prove that the above hypothesis is true.

Each piece that you move has a letter. Around the window there are parameters like rotateA. You can rotate each piece by changing the corresponding parameter value, which is given in sexagesimal degrees.

4.-Generalise the answer you got for activity 3 so that it can be applied to any kind of triangle.

5.- Show how you can use any set of equal triangles of your choice to tessellate a strip of the plane between two parallel lines.

6.- Can you explain why the triangles tessellate the plane? Clue: Think about the relation that exists between the three angles of a triangle.

You can tessellate the plane with any kind of  triangle. 
2. TESSELLATING THE PLANE WITH QUADRILATERALS

You can see many examples of this kind of tessellation around you. Most tessellation is made up of quadrilaterals, especially squares and rectangles.

7.- Note that you can tessellate the plane in many different ways using squares. For example, by moving the squares in one row a bit further to the left or right so that the edges of the squares don't meet. However, in this unit we are not going to focus on this kind of tessellation. We are going to study tessellation where the edges of the shapes meet exactly.

8.- Investigate tessellation of the plane using parallelograms.

9.- Use the results obtained from the activity above to show that the plane can be tessellated by using parallelograms.

10.- Prove that the plane can be tessellated by using any type of quadrilateral, including those which are not convex (e.g. the head of an arrow).

We can tessellate the plane with any kind of quadrilateral, including those which are not convex.

11.-Tessellate the plane using quadrilaterals in the window and notice that when you join two quadrilaterals together you get a hexagon with parallel sides.

Not all hexagons will tessellate the plane, only regular hexagons will. In fact, the only regular polygons which can be used to tessellate the plane are equilateral triangles, squares and regular hexagons.
3. TESSELLATING THE PLANE WITH PENTAGONS

Up until now we expect any irregular polygon with any number of sides to be able to tessellate the plane. However, this is not true. Can you tessellate the plane using pentagons?

12.- Prove that in general we cannot tessellate the plane using pentagons.

13.- Try to tessellate using a pentagon which is shaped like a house. Does it work?

14.- Note that these kind of pentagons are made up of a rectangle and a triangle which cover strips of the plane.

This is a complicated area and even today we still don't know exactly which other types of pentagons can be used to tessellate the plane.

However, it has been proved that it is impossible to tessellate the plane with a convex polygon which has 7 or more sides.
Enrique Martínez Arcos
© Spanish Ministry of Education and Science. Year 2001