Tantalising Tessellations

Everybody knows that squares tessellate.

What exactly is meant by this?

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.

The fact is …

only three regular shapes will tessellate

The equilateral triangle

The square

The regular hexagon

The regular hexagon

Extensions

Extension 1

Do all pentagons with one pair of parallel lines 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

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

Extension 3

Why does this method always work?