Tantalising Tessellations Everybody knows that squares tessellate. What exactly is meant by this? Welcome to ‘Tantalising Tessellations’ : This investigation is introduced as part of the Maths Extension Programme for Years 4,5 & 6 in Chigwell School. ©DDimoline, Chigwell School, 2001

A tessellation originally was the result of covering an area with tesserae – the small square blocks used by the Romans to make mosaics.

Nowadays the word tessellation is used to represent any ‘tiling’ of a plane surface by a regular pattern of one or more congruent, non-overlapping shapes

This is a tessellation

This is not a tessellation Can you spot the odd tiles out?

This is not a tessellation Can you spot the odd tiles out?

Can you show that a tessellation can be made from any parallelogram?

If you can think of parallelograms forming strips, you can also see that they fit together very easily.

Can you show that a tessellation can be made from any triangle?

If you think of it, a parallelogram can be made from two triangles. Finding other patterns could be more worthwhile.

How would you show that no tessellation is possible from a regular pentagon?

The 108o corners cannot be fitted together to form 360o

The 108o corners cannot be fitted together to form 360o

It stands to reason then that if the corners do not add up to 360o the shapes will not tessellate.

only three regular shapes will tessellate The fact is … only three regular shapes will tessellate

The equilateral triangle

The square

The regular hexagon

The regular hexagon

Extensions

Do all pentagons with one pair of parallel lines tessellate? Extension 1 Do all pentagons with one pair of parallel lines tessellate?

Are there other pentagons which tessellate? Extension 2 Are there other pentagons which tessellate?

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2

Extension 2 The second quadrilateral was obtained by rotating the first through 180o about O, the midpoint of a side.

Extension 2

Keep doing this and you will soon see a tessellation of quadrilaterals Extension 2 Keep doing this and you will soon see a tessellation of quadrilaterals

Extension 3 Try the method used in Extension 2 for re-entrant quadrilaterals such as …

Extension 3

Extension 3

Why does this method always work? Extension 3 Why does this method always work?