**Safety: This activity uses a craft knife. Younger mathematicians should ask an adult to help.**

**You will need**

- Thin sheet of cardboard
- Craft knife
- Pencil
- Ruler
- String
- Pin
- Eraser
- Cutting mat

**What to do**

- Carefully stick the pin through the middle of your sheet of cardboard to make a tiny hole.
- Mark points that are 2 cm, 4 cm, 6 cm, 8 cm and 10 cm from the hole.
- Put the eraser underneath the hole in the cardboard, and stick the pin back through the hole and into the eraser.
- Cut a 30 cm length of string and tie one end around the pencil.
- Pin the other end of the string so the pencil lines up with the 10 cm mark. Keeping the string tight, rotate the pencil around the pin to draw a circle.
- Use the same process to draw circles at the other marks.
- Turn the cardboard over, then mark 3 cm, 5 cm, 7 cm and 9 cm from the hole. Using the method above, draw circles using each of these points.
- Get your craft knife and lightly run the blade along each circle you have drawn – this is called ‘scoring’. You don’t want to cut through the cardboard, you just want to make it easier to fold along the lines.
- Turn the cardboard over and score the circles on this side too.
- Cut out the 10 cm circle and discard the rest of the cardboard.
- Hold the cardboard circle in both hands, with the 9 cm circle on the far side. Bend the middle of the card toward you. Along the 9 cm circle, fold the cardboard so the edge comes towards you.
- Turn the circle over, and use the same technique to fold along the 8 cm circle. The fold should be in the opposite direction to the previous fold.
- Fold the remaining circles, remembering to flip the cardboard over after each fold.

**What’s happening?**

If you’ve ever made paper planes or folded origami, you might have noticed the creases tend to be in straight lines. When you’re making these things, the paper usually ends up flat at the end of each step, except in the final few stages when the wings fold out, or the box pops up.

A fold must be straight to lie flat. To understand why, imagine folding along a curve. To lie flat, the paper on the inside of the curve will have to match exactly to the paper on the outside of the curve. You would need to stretch the paper on the inside of the curve until it curved the other way! This is why the shape made in this activity pops up as soon as it is folded.

**Applications**

You can make a lot of different shapes using curved folds in a flat surface. Mathematicians call these shapes ‘developable surfaces’ and they can be beautiful, and also very useful. When manufacturers want to make an object, they often start with a flat sheet of a material, such as steel. It is relatively easy to put curved folds into steel, so designers working with this material try to make as many parts as possible from developable surfaces.