Classifying two-dimensional shapes

A tessellation

One further way of classifying two-dimensional shapes is to distinguish between those that tessellate and those that do not. A shape is said to tessellate if it can be used to make a tiling pattern, or a tessellation. This means that the shape can be used over and over again to cover a flat surface, the shapes fitting together without any gaps. In practical terms we are asking whether the shape can be used as a tile to cover the kitchen floor (without worrying about what happens when we reach the edges).

Pupils can investigate which shapes tessellate and which do not, discovering, for example, that all triangles and all quadrilaterals do. They could use a plastic or card shape as a template, drawing round it in successive positions.

The commonest shapes used for tiling are, of course, squares and other rectangles, which fit together so neatly without any gaps. This is no doubt part of the reason why the rectangle is such a popular shape in a technological world. A remarkable fact is that any triangle tessellates. If the three angles of the triangle are called A, B and C, then it is instructive to identify the six angles that come together at a point where six triangles meet in the tessellation. Because the 3 angles in any triangle A, B and C add up to 180degrees, a straight angle, we find that they fit together at this point, neatly lying along straight lines. By repeating this arrangement in all directions the triangle can clearly be used to form a tessellation. It is also true that any quadrilateral tessellates. Because the four angles add up to 360 degrees, we can arrange for four quadrilaterals to meet at a point with the four different angles fitting together without any gaps. This pattern can then be continued indefinitely in all directions.

Interestingly, apart from the equilateral triangle and the square, the only other regular polygon that tessellates is the regular hexagon, as seen in the familiar honeycomb pattern.