**Geometric Dissections**

by

Mircea Pitici

September 2008

This online module presents an overview of mathematiciansâ€™ interest in geometric dissections, from ancient times to the present, as well as the main results and methods employed in geometric dissections.

A *dissection* of a plane geometrical figure is a partition of it by straight lines. The resulting pieces of the decomposition can be rearranged (with no overlap) in various ways to obtain a multitude of different figures, all *equivalent by dissection* to the initial figure and having the same area.

These definitions hold true not only in the Euclidean plane geometry but also on a sphere and on the hyperbolic plane. It can be shown that in all three geometries, any two polygons with the same area are equivalent by dissection. That is, one of the two polygons can be decomposed by straight cuts and the resulting pieces rearranged to obtain the other polygon.

Similar definitions and results can be given for solid Euclidean geometry. We will look into this subject in a separate section reserved for polyhedral dissections.

The history of geometric dissections

Three millennia of geometric dissections

Archimedesâ€™s *Stomachion*

Recent flourishing

Plane dissections

Polygonal dissections

Hinged dissections

3-dimensional (polyhedral) dissections

Non-Euclidean dissections

Resources