ABSTRACT
There is a lovely and little-known theorem due to the great mathematician David Hilbert () which states that if two plane polygonal figures A and B have the same area, then there is a dissection of A into a finite number of pieces which can be reassembled to form B. It is an amazing fact that an analogous result is not true in three dimensions.
However, while Hilbert's theorem says that a dissection exists, its proof gives us no clue how to achieve it. Some dissection problems can be very difficult, but most are fun, and the amateur has often just as much chance of being successful as the skilled professional. Here an experimental approach is absolutely vital and the final solution should be displayed as if plucked out of the air, without any mention of the many failed attempts in the recycling bin.
Squares and rectangles are the most popular shapes to dissect, often masquerading as pieces of carpet. But there are also dissection “proofs” of geometric results, including the famous theorem of Pythagoras. And a huge and some would say a virtually impossible task is to show that the number of pieces involved in a given dissection is as small as possible.
