This activity requires very careful folding. If it doesn’t work first time, try it again, and it might work better. If you keep getting stuck, download the crease diagram here, and fold along all the lines. Dotted lines should be folded the opposite way to the solid ones.

Fold the piece of paper in half lengthways, so it’s long and thin.

Fold the top layer of paper so that the long edge goes along the fold you just made. Then turn the paper over and do the same on the other side.

Rotate the page so that the long edge of the paper is closest to you. The other side should be two long folds.

Measure how wide the strip of paper is. Draw a faint line one quarter of the way across the strip, closer to you. If you don’t have a pencil or ruler, fold the paper in half, and then in half again to find the right line. Make these folds softly, and only fold the top layer.

Fold the strip so that the bottom left corner meets the line. You should try to make the fold about four centimetres from the left end of the strip. Make sure you fold all the layers of paper.

Turn the paper over.

Fold the strip about four centimetres from the last fold. Make sure this fold is parallel with the last fold, and not the same direction as the end of the paper. If you have a wide ruler, use it to make sure the folds are in line. The top of your previous fold should be parallel to the top edge of the paper.

Keep folding the paper left and right at four centimetre intervals until you run out of paper, keeping the folds in line and parallel. If you have folded it at the same angle, the zig-zag at the top should be evenly spaced – see the picture.

Unfold the paper completely.

All the creases you have are in the right places, but some of them are bent the wrong way. Softly bend and refold the paper so that the first zig-zag fold going across the page is all folded one way (like a mountain), and the next all the other way (like a valley). Keep alternating the zig zag folds down the page. Here’s a crease diagram to help:

The solid lines need to be folded one way, and the dotted lines the other.

Once you have finished refolding the paper, it should collapse down into a small, flat pad. Gently unfold and refold it a couple of times. Congratulations! You’ve just made a Miura fold.

To use your Miura fold, grab two corners of the paper that are diagonally opposite. If you push them slowly together the paper will fold up. If you pull them apart, it will unfold.

What’s happening:

The creases that you just made are called the Miura-ori or Miura map fold. It was invented by an astrophysicist named Koryo Miura, who wanted to find a good way of folding solar panels for space ships. The Miura-ori has many benefits over other methods of folding, which make it good for such missions.

Firstly, it can be unfolded just by moving the two corners apart. This means fewer moving parts need to be used to unfold the panels. Secondly, the stresses on the edges and corners are smaller than with other folding methods. Finally, and most importantly, the folds can be made without stretching the solar panel in any way. A lot of other methods of folding large paper sheets into smaller shapes involve stretching the paper very slightly so that the folds can be made. This is okay with paper, which is thin and very flexible, but solar panels are a lot thicker and more fragile.

The Miura-ori has been used on solar panels for several space missions, and there are plans to use it for other spaceship parts, such as solar sails. It is also used for some maps to make them easier to fold and unfold, but the process is complicated, so it makes the maps more expensive.

Applications:

There are a lot of interesting maths problems about folding. One of the most famous is the napkin folding problem, which is also called Arnold’s rouble problem. Arnold’s problem asks if it is possible to fold a piece of paper so that the final flat shape has a longer perimeter than the unfolded sheet had at the beginning.

Robert Lang found a series of solutions in 1997 that could have as long a perimeter as you wanted, but it involved stretching the paper in order to make the folds. In 2004, Alexey Tarsov found a solution that didn’t involve any stretching.