With a single cut

Let's take a sheet of paper and draw an arbitrary polygon. Is it possible to fold the sheet so that the polygon could be cut out with a single straight slit?

Consider the simplest case of an arbitrary triangle.

We draw a straight cut along it.

If you unfold the hole has the form of the initial triangle.

Let's draw a star.

It is also possible to cut out by one straight slit the polygon drawn in the beginning of the film. In 1998 the general theorem was proven.


Given any polygon there exist a folding of a sheet of paper and a line on this folding such that scission along the line removes the (folded) polygon.

The proof of the theorem is algorithmical, that is the authors present a way to find a folding of a sheet of paper so that one straight cut creates any desired polygon.

See also in section ”Other interesting subjects”15


Other projects by Mathematical Etudes