# Dissections and Lattices

## by Greg Egan

Suppose you are given two polygons with equal areas. Can you cut either of them into a finite number of smaller polygons that can be rearranged to form the other? It turns out that this is always possible: the Wallace–Bolyai–Gerwien theorem shows that any two such polygons can both be cut into pieces that can be arranged into a rectangle whose shape depends only on the area in question. By further subdiving that rectangle into pieces that are the intersections of pieces from both sets, a single set of pieces can be found that can be rearranged to form either of the original polygons.

In more detail:

• Any polygon can be divided into a finite number of triangles.
• Any triangle can be divided into (at most) two right triangles.
• Any right triangle can be cut into two pieces that can be rearranged to form a rectangle.
• Any rectangle can be cut into pieces that can be rearranged into another rectangle with one side of a specified length, say 1.
• All these rectangles with one side of length 1 can be stacked together to form a single rectangle with one side of length 1.

We won’t spell out all the fine points needed to turn this into a completely general algorithm, but we will illustrate the result of cranking the handle and going through these steps for a pair of polygons with no special symmetries.

[More to come soon ...]

Science Notes / Dissections and Lattices / created Thursday, 26 September 2019