Fold and One Cut

What’s going on

Fold a square of paper flat as many times as you like, make one straight cut right across it, and unfold. What can fall out?

Press Fold, cut, unfold above. The paper is folded four times, the scissors go along a single straight line, and when the paper opens, a perfect star drops out.

A fold is a mirror

The whole trick rests on one fact: a fold line works like a mirror. A cut through both layers of a folded sheet makes two cuts on the open sheet: the one you made, and its reflection in the fold.

Try the Square hole first, step by step with Next step:

1. Fold along one diagonal. Two layers.
2. Fold along the other diagonal. Now there are four layers, in the shape of a triangle.
3. Cut straight across the triangle. The cut goes through four layers at once.
4. Unfold. Each fold reflects the cut, so the one cut turns into four: the four sides of a square.

The further the cut is from the centre, the bigger the square. In the scene the cut is halfway, so the hole is half as wide as the sheet.

Why the star needs those folds

A five-pointed star has 10 equal edges and five mirror lines, which cut it into 10 identical slices of 36° (360 ÷ 10 = 36). Each slice contains exactly one edge, from a point of the star to a notch.

So the folds have one job: stack all 10 slices on top of each other. Every fold goes along one of the star’s mirror lines. After four folds the paper is a thin 36° wedge, 10 layers thick, with every layer’s edge in the same place. One cut from tip to notch cuts all 10 edges at once.

The six-pointed star works the same way with 12 slices of 30°, and the triangle with 6 slices of 60°.

The plus sign: a fold that is not a mirror line

In the Plus sign, three folds through the centre stack its 8 slices, but each slice still has two edges at a right angle: the end of an arm and its side. The fourth fold is not a mirror line of the plus. It lays one edge exactly along the other, so both lie on one straight line, and one cut does it.

Any shape at all

This is not a trick for a few lucky shapes. Any shape with straight sides can be cut out with one straight cut, if you fold the sheet the right way: a letter, a swan, even several shapes at once with holes in them. This is the fold-and-cut theorem. The folds for a complicated shape can be very clever, but they always exist.

Choose Your own folds to experiment: drag a line across the paper to fold along it, then switch to Draw the cut. The cut on the open sheet is always your cut line, reflected in every fold.

Try it yourself

1. One fold, one cut

Fold a sheet just once and make one straight cut. Which shapes can fall out?

Answer

The cut goes through two layers, so on the open sheet you get the cut and its mirror image in the fold. If the cut crosses the fold line, the two cuts meet there and the piece that falls out is an isosceles triangle, a triangle with two equal sides. If the cut is parallel to the fold, you cut off a strip. A square needs at least the second fold.

2. Two folds in half

Fold a square sheet in half, and in half again, so it becomes a smaller square with the centre of the sheet at one corner. Now cut straight across that corner. What hole appears?

Answer

A rhombus, a diamond with four equal sides. The cut piece is a right-angled triangle at the centre, and the two folds reflect it into four copies around the centre. If the cut is at 45°, cutting off equal lengths of both folded edges, the rhombus is a square standing on one corner.

3. Count the edges

A sheet is folded so that it lies in 12 layers, and a straight cut goes right across all of them. At most how many straight cuts can you see on the open sheet? Which shape in the scene uses exactly this?

Answer

At most 12: each layer gets one cut, and unfolding sends each one to a different place. Sometimes two of them line up and look like one longer cut. The six-pointed star uses all 12: its 12 edges come from one cut through 12 layers.

Go deeper

• A Japanese puzzle book from 1721 already asks how to cut out a family crest with one straight cut. The magician Harry Houdini described cutting a five-pointed star this way in his book Paper Magic (1922).
• The fold-and-cut theorem was proved in the late 1990s by Erik Demaine, Martin Demaine and Anna Lubiw, and also by a second team, Marshall Bern, Erik Demaine, David Eppstein and Barry Hayes, using a different method.