EQUIDECOMPOSABILITY

Using the same pieces we can form a hexagon or a square!

Is it also possible to do this with a square and an equilateral triangle?

In fact, it can be shown that any two polygons of the same area are equidecomposable, that is, there is a partition into smaller polygons that is common to both. Finding the way to do this is another problem and can be quite difficult!

Image: Alisson Ricardo.
Image: Alisson Ricardo.

If we increase the dimension and think of the same problem for polyhedra, the answer will change. For example, a regular tetrahedron and a cube of the same volume are not equidecomposable. Who proved this was Max Dehn, in 1900, and it was the quickest answer to one of the problems of Hilbert's famous list, made in the same year.

Video: Rodrigo Tetsuo Argenton.

USEFUL LINKS

This RPM (Mathematics Teacher Magazine) article talks about the equidecomposability theory, known as the Wallace-Bolyai-Gerwien Theorem! The article is quite technical, but despite the more formal demonstration, it has some very intuitive drawings that help you understand.

This UFRGS (Rio Grande do Sul Federal University) article also talks about equidecomposable polygons, in a slightly simpler way, bringing a more intuitive view, some approaches to be used and some examples of equidecompositions.                                                        

And to exemplify this incredible theory, this Minuto Matemática channel video quickly enunciates the theorem and shows some figures "transforming" into others, but in a very special way!                                                                                                                                                                        

Tangram is a Chinese geometric puzzle made up of 7 pieces, which form a square. Your objective is to form figures always with the same pieces. Note that the equidecomposability theory deals with decomposing two distinct geometric figures into the same pieces. Despite this difference, Tangram is very interesting, especially if the goal is to introduce a mathematical theory later! This Wikipedia page tells a little bit of the history of Tangram and shows how you can build your own!

Still on the Tangram and its possibility of being shown as an introduction to something more formal later, this Sandra Gobert - Pensarte e artesanato channel video talks about the "Legend of Tangram", a playful way of explaining and showing how an image can be transformed into others, using the same initial pieces. It is a great option to be presented in basic education classes, and depending on the age of the students, it can even be given as an introduction to the theory of equidecomposability!