proof of Dehn’s theorem

We define the Dehn’s invariantMathworldPlanetmath, which is a number given to any polyhedron which does not change under scissor-equivalence.

Choose an additive function f: such that f(π)=f(0)=0 and define for any polyhedron P the number (Dehn’s invariant)

D(P)=e{edges of P}f(θe)(e)

where θe is the angle between the two faces of P joining in e, and (e) is the length of the edge e.

We want to prove that if we decompose P into smaller polyhedra P1,,PN as in the definition of scissor-equivalence, we have

D(P)=k=1ND(Pk) (1)

which means that if P is scissor equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to Q then D(P)=D(Q).

Let P1,,PN be such a decomposition of P. Given any edge e of a piece Pk the following cases arise:

  1. 1.

    e is contained in the interior of P. Since an entire neighbourhood of e is contained in P the angles of the pieces which have e as an edge (or part of an edge) must have sum 2π. So in the right hand side of (1) the edge e gives a contribution of f(2π)(e) (recall that f is additive).

  2. 2.

    e is contained in a facet of P. The same argument as before is valid, only we find that the total contribution is f(π)(e).

  3. 3.

    e is contained in an edge e of P. In this case the total contribution given by e to the right hand side of (1) is given by f(θe)(e).

Since we have choosen f so that f(π)=0 and hence also f(2π)=0 (since f is additive) we conclude that the equivalence (1) is valid.

Now we are able to prove Dehn’s Theorem. Choose T to be a regular tetrahedronMathworldPlanetmathPlanetmathPlanetmath with edges of length 1. Then D(T)=6f(θ) where θ is the angle between two faces of T. We know that θ/π is irrational, hence there exists an additive function f such that f(θ)=1 while f(π/2)=0 (as there exist additive functions which are not linear).

So if P is any parallelepipedMathworldPlanetmath we find that D(P)=0 (since each angle between facets of P is π/2 and f(π/2)=0) while D(T)=6. This means that P and T cannot be scissor-equivalent.

Title proof of Dehn’s theorem
Canonical name ProofOfDehnsTheorem
Date of creation 2013-03-22 16:18:07
Last modified on 2013-03-22 16:18:07
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Proof
Classification msc 51M04
Defines Dehn invariantMathworldPlanetmath
Defines Dehn’s invariant