Shape dissection

In mathematics, shape dissection is a theory about subdivision, especially of polygons and polyhedra and in relation to fundamental questions on area and volume. It includes the theory of scissors congruence, which has been shown to have connections with homological algebra.

The dissection definition of area

The area of a rectangle is defined as the product of the lengths of the edges. The definition of the area of a triangle could be given as:

However, a rectangle can be dissected into two right angle triangles by a diagonal:

And adding the axioms that area is not changed by movement, and the principle that the area of a shape assembled from a finite collection of sub-shapes is the sum of the areas of the sub-shapes, the area of a right angle triangle must be half that of the rectangle constructed from two copies, giving:

Any triangle can be dissected into two right angle triangles, as shown in this diagram:

From which the required general expression for the area of a triangle is obtained.

With an arbitrary convex polygon, which, by definition, has no inward pointing vertices, a triangle can be cut off at a vertex, then another, and another, until only a triangle is left. No lines will have crossed each other, so the polygon has been unambiguously broken into (a finite number of) triangles whose areas can be added up to get the area of the polygon.

See also

• Hilbert's third problem
• Bolyai-Gerwien theorem
• Banach-Tarski paradox
• Zylev's theorem
• Sydler's theorem
• shape assembly
• tessellation describes dividing the entire plane, not just a polygon covering a finite part of it.