Fold-and-cut theorem

The fold-and-cut theorem states that any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut.^[1] Such shapes include polygons, which may be concave, shapes with holes, and collections of such shapes (i.e. the regions need not be connected).

Creating a Koch snowflake curve by the fold-and-cut method

The corresponding problem that the theorem solves is known as the fold-and-cut problem, which asks what shapes can be obtained by the so-called fold-and-cut method. A particular instance of the problem, which asks how a particular shape can be obtained by the fold-and-cut method, is known as a fold-and-cut problem.

History

Creating an anti-Koch snowflake curve by the fold-and-cut method

The earliest known description of a fold-and-cut problem appears in Wakoku Chiyekurabe (Mathematical Contests), a book that was published in 1721 by Kan Chu Sen in Japan.^[2]

An 1873 article in Harper's New Monthly Magazine describes how Betsy Ross may have proposed that stars on the American flag have five points, because such a shape can easily be obtained by the fold-and-cut method.^[3]

In the 20th century, several magicians published books containing examples of fold-and-cut problems, including Will Blyth,^[4] Harry Houdini,^[5] and Gerald Loe (1955).^[6]

Inspired by Loe, Martin Gardner wrote about the fold-and-cut problems in Scientific American in 1960. Examples mentioned by Gardner include separating the red squares from the black squares of a checkerboard with one cut, and "an old paper-cutting stunt, of unknown origin" in which one cut splits a piece of paper into both a Latin cross and a set of smaller pieces that can be rearranged to spell the word "hell". Foreshadowing work on the general fold-and-cut theorem, he writes that "more complicated designs present formidable problems".^[7]

The first proof of the fold-and-cut theorem, solving the problem, was published in 1999 by Erik Demaine, Martin Demaine, and Anna Lubiw and was solved using straight skeleton method.^[8][9]

Solutions

There are two general methods known for solving instances of the fold-and-cut problem, based on straight skeletons and on circle packing respectively.