Dehn’s theorem

We all know the elementary formulaMathworldPlanetmathPlanetmath to compute the area of a triangleMathworldPlanetmath: basis times height divided by two. This formula can be justified with a scissor type argument: one divides the triangle into smaller polygonsMathworldPlanetmathPlanetmath and rearranges these polygons to obtain a rectangleMathworldPlanetmathPlanetmath which should have the same area.

Can we use the same argument to compute the volume of a pyramidMathworldPlanetmath? This is the third Hilbert’s problem. Quite surprisingly the answer is negative, as states the theorem below. This means that the formulae to compute the volume of polyhedra cannot be proved without a limiting process (for example using integrals).

Definition 1.

We say that two polyhedra P and Q are scissor-equivalent if there exists a finite number P1,,PN of polyhedra and θ1,,θN isometries such that

  1. 1.

    P=k=1NPk and Q=k=1Nθk(Pk);

  2. 2.

    PjPk and θj(Pj)θk(Pk) have empty interior for every kj

The properties given above assure that two scissor-equivalent polyhedra must have the same volume. It is also simple to prove that the scissor-equivalence is indeed an equivalence relationMathworldPlanetmath.

Theorem 1.

The regular tetrahedronMathworldPlanetmathPlanetmathPlanetmath is not scissor-equivalent to any parallelepipedMathworldPlanetmath.

Title Dehn’s theorem
Canonical name DehnsTheorem
Date of creation 2013-03-22 16:18:04
Last modified on 2013-03-22 16:18:04
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 11
Author paolini (1187)
Entry type Theorem
Classification msc 51M04
Classification msc 52B45
Related topic BanachTarskiParadox
Related topic HilbertsProblems
Related topic RegularTetrahedron3
Defines scissor-equivalent