An example of this is the following theorem:
In other words, we can decompose the two polygons in lesser polygons, which are congruent to each other. This means that the two polygons can be decomposed equally, and by this they are called "equidecomposable polygons."
In some situations, depending on
the form and the sizing of the polygons, we can discover an equidecomposition.For
example easily, in the pairs of polygons below:
||square and rectangle such that one of the sides of
rectangle is the double of the side of the square
||parallelogram and rectangle with same bases and heights|
To see equidecompositions click here.
In other situations one such equidecompositions is not
obvious. For example, in the situations below:
||square and rectangle with same area|
||square and hexagon with same area|
How to get a equidecomposition for these pairs of polygons?
To develop the theorem we go, step to the step, to construct a puzzle. The material presented here can be used in diverse ways, depending on the audience:
* A playful approach - they are puzzles that transform
triangles into rectangles, rectangles in squares, two squares in an only
one squared and a polygon into a square .
* A intuitive approach - triangles, squares, rectangles, parallelograms, parallelism, perpendicularity are some of the concepts developed in the construction of the puzzles. Here the geometric drawing with ruler and compass is important tool, since the precision of the figures is basic in the assembly of the puzzles.
* A deductive approach - the demonstrations are worked that guarantee that the puzzles are mathematically correct. For this properties of angles and parallelism are used, of congruence and similarity of triangles, length and area.
This theorem was demonstrated by F.Bolyai in 1832 and,
independently, in 1833 for G.Gerwien, an amateur German mathematician.
F.Bolyai was the father of the mathematical celebrity Hungarian Janos Bolyai,
creator of the Hyperbolic Geometry (also created by Lobatchevski and Gauss.
It is natural to ask if the analogous result is true for polyhedrons. Max Dehn, pupil of Hilbert, proved in 1900 that this is not true: In particular, a regular tetahedron and a cube of same volume are not equidecomposable.