# 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
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