Not just for squares

23rd February 2001 at 00:00
Jenny Houssart presents original ways to use MCEscher's extraordinary works to teach about shape and tessellation.

Is it a bird? Is it a triangle? It is hard to look at the work of graphic artist MC Escher without asking a host of questions. He specialised in pictures that make you look again. Triangles become birds, hexagons become bees and shapes in the form of lizards fit together perfectly. Perhaps the most compelling question is "How did he do it?" There have been many attempts to analyse the mathematical principles behind Escher's work. Also of interest, however, is a wider view of his approach to the study of shape.

His early work in particular was based on observations and drawings of shapes he made while travelling around in town and countryside. He was fascinated by nature and architecture. As every primary teacher knows, both are rich starting points for a study of shape. He looked at the shapes of apparently insignificant natural objects, such as leaves, shells and stones. He compared the shapes of birds, lizards and fish to simpler mathematical shapes. He sketched the three-dimensional shapes making up buildings from unusual angles.

Escher also loved looking at things in different and unusual ways; his drawings were sometimes framed in arches or portholes, or reflected in mirrors or spheres. He was fascinated by patterns, such as those found on floor or wall coverings. These were an important influence on his later work, when he developed an interest in "regular divisions of the plane" - Jfitting shapes together, called tessellation. He specialised in using increasingly complex shapes, many of them looking like birds, fish or other creatures. It is this work for which he is probably best known and which has particularly fascinated mathematicians.

There is a striking similarity between Escher's approach to the study of shape and the approach frequently used with young children by teachers. The good news is that it is perfectly compatible with the detail of the numeracy framework and the spirit of the national curriculum. Reception children, for example, might go on a "shape walk", look for patterns in floor coverings, or make models of buildings using wooden shapes. Children in Years 1, 2 and 3 may combine simple shapes to create pictures or to make more complex shapes. These techniques can be used to form Escher-type shapes, reminiscent of birds, animals or objects. Older children become more aware of the symmetry in shapes and the corresponding regularities in patterns. Escher's tessellated animal designs reveal examples of tessellating shapes by means of reflection, translation and rotation.

While adapting and tessellating shapes, children are becoming aware of their propertis. In attempting to emulate Escher's unusual designs, they are encouraged to use a wider range of shapes, including the irregular, the concave and those with curved edges. Such work should prove challenging and motivating, and children will soon realise that tessellation is not just for squares.


Cut and stick method You need - card, scissors, sticky tape.

Start with a square cut from card.

Draw a shape on one side of the square.

Cut the shape out, move to the opposite side of the card, stick on.

Use the shape as a template for making tessellated design.

Decorate the shape to transform it into something recognisable.

For a more complicated design, this process can be repeated so that all four sides of the square are altered.

Draw a shape on one sid e of the square.

Cut the shape out, move to the opposite side of the card, stick on.

Draw another shape on one of the remaining straight sides of the square.

Cut this shape out and stick it on the opposite side.

Challenge Use the cut-and-stick method starting with one of the shapes below. Rhombus, Parallelogram, Rectangle Using grid sheets You need: paper with squares or isometric paper (with equilateral triangles).

Try using three or more squares.

Try to find more complex shapes made from squares.

Try using the isometric paper.


Can you alter your shape so that it contains some curves or details but can still be tessellated?

LOOKING AT TESSELLATIONS In all tessellations, shapes are repeated in a systematic way. Escher's work provides good examples of different ways of doing this.

You need: copies of Escher tessellations, tracing paper.

Translation In some tessellations, the shapes are simply repeated by translation - duplicating them horizontally andor vertically.

The shape is always the same way up and facing the same direction. To show that a tessellation uses translation, trace one of the shapes, then move the tracing horizontally or vertically across the tessellation. It should fit over the next shape.

Glide reflection Escher's designs sometimes use glide reflection, whereby shapes are reflected and then moved, with adjacent rows of shapes facing opposite directions. To test for this, trace one of the shapes, flip the tracing over and fit it over an adjacent shape (see the horsemen in Regular Division of the Plane III, right).

Rotation Many of Escher's designs use rotation. Several shapes meet at one place, the centre of rotation (see Circle Limit III, above). To test this, trace round a shape and pin the tracing to the centre of rotation; then rotate the tracing to fit over an adjacent shape.

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