Flächeninhaltgleichheit

Flächeninhaltgleichheit, also known as equidecomposability or scissors congruence, is a concept in geometry that deals with whether two shapes can be cut into a finite number of congruent pieces and reassembled to form each other. If such a dissection and reassembly is possible, the two shapes are said to be flächeninhaltgleich, meaning they have the same area.

The fundamental theorem related to flächeninhaltgleichheit is the Wallace-Bolyai-Gerwien theorem. This theorem states that two polygons are equidecomposable if and only if they have the same area. This means that for polygons, having the same area is a necessary and sufficient condition for them to be flächeninhaltgleich. This theorem is significant because it provides a way to compare the areas of polygons without directly measuring them, by relying on the ability to dissect and rearrange them.

The concept of flächeninhaltgleichheit is not directly applicable to all geometric figures. For example, not all shapes with the same area are necessarily equidecomposable. The most famous example is the Hilbert's third problem, which asks whether a cube and a regular tetrahedron of the same volume are equidecomposable in three dimensions. Dehn proved in 1900 that they are not, demonstrating that flächeninhaltgleichheit in higher dimensions is a more complex notion than in two dimensions. The existence of equidecomposability depends on more than just having the same measure.