Given a certain hexagon, a fairly simple way to obtain a new hexagon is to join the midpoints of its sides, as exemplified in the following picture:

This construction will be referred to as bisection. Is there any relation between the given hexagon and the new hexagon? Can the new hexagon be arbitrary or does it have some particular characteristics? And what happens with other polygons (triangles, quadrilaterals, pentagons, etc.)?

Note that by applying the bisection construction to the new hexagon, we get another hexagon. Continuing the procedure indefinitely we get a sequence of polygons (hexagons, in this case). In the hexagon case, it seems that not only do the hexagons in the sequence appear to be smaller and smaller, but they also have alternately the same shape. Is this always true, even for other polygons?

In this page we will answer these questions. We shall cover not only the bisection case but also some more general processes. For that, we will need some basic knowledge of Linear Algebra.

Translated for Atractor by a CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

(*) This work was carried out under the guidance of Professor Maria Carvalho from the Universidade of Porto, under a grant by the Calouste Gulbenkian Foundation to develop a project for the promotion of Mathematics in Atractor.
Since many browsers are blocking Java nowadays, it was decided (in 2020) to convert to Javascript the original applets of this section. This conversion was carried out by a high-school teacher, who is working full-time in Atractor with the support of the Ministry of Education.

Difficulty level: University