Would you like to make your own repeating patterns, a la Escher? Grab some scissors and make some tessellating tiles!

- Square of paper, maybe 10 cm across
- Some more paper
- Good pair of scissors
- Sticky tape

- Use the scissors to cut a shape out of one side of the square.
- Put the pieces you cut off on the opposite side of the square. Make sure you match the straight side of the square with the straight side of the piece you cut off (don’t spin the pieces).
- Use a small piece of sticky tape to stick the small piece to the square.
- You can repeat this process of cutting off a piece and sticking it on the opposite side as many times as you like. Make sure you always stick it at the same spot on the opposite side, without sliding it further along the side.
- When you are finished you should have a shape that tessellates. Take a new sheet of paper and trace the outline of your shape onto it. Then trace it again and again, trying to cover the sheet of paper without leaving any gaps.

The shapes you have made cover a flat surface with no gaps. Any pattern that does this is called a tiling. This tiling is a very special type of tiling called a tessellation.

A tessellation is a tiling that repeats. If you take your tiling, and you move it across just the right amount, it matches exactly on top of itself. Mathematicians are very interested in tilings and tessellations, because they occur a lot in the world (see ‘Applications’ below).

Mathematicians are especially interested in tessellating with regular shapes – those shapes where all the sides are equal and all the angles are the same. There are only three regular shapes that tessellate – the square, the equilateral triangle, and the regular hexagon. All other regular shapes, like the regular pentagon and regular octagon, do not tessellate on their own. However, if you use more than one shape there are a whole lot more tessellations that you can make. For instance, you can make a tessellation with squares and regular octagons used together. Sometimes you see this tessellation on bathroom tiles.

Tessellations are used all the time to design things like bricks and bathroom tiles. More complicated maths is used to design interlocking shapes that can cover a surface. These can be used for things like retaining walls and roofs.

Tessellations are used for more than just tiling. Every tessellation is a repeating pattern. Even more importantly, every repeating pattern is based on a tessellation. A lot of things in the world have repeating patterns on them, and understanding the maths going on behind the pattern makes it a lot easier to design them. Tessellations are used on everything from carpets to wrapping paper.

Manufacturing companies also use tessellations. Often they want to cut as many pieces out of a sheet of metal as possible. If the pieces tessellate, then they can minimise the gaps, and that means they get more pieces. Trying to avoid gaps in this way is a very famous problem, known as a packing problem. Mathematicians have been trying to solve these types of problems for hundreds of years.

Check out a new tessellation discovered recently by mathematicians

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