proof of Dehn’s theorem
Choose an additive function such that and define for any polyhedron the number (Dehn’s invariant)
where is the angle between the two faces of joining in , and is the length of the edge .
We want to prove that if we decompose into smaller polyhedra as in the definition of scissor-equivalence, we have
which means that if is scissor equivalent to then .
Let be such a decomposition of . Given any edge of a piece the following cases arise:
Since we have choosen so that and hence also (since is additive) we conclude that the equivalence (1) is valid.
Now we are able to prove Dehn’s Theorem. Choose to be a regular tetrahedron with edges of length . Then where is the angle between two faces of . We know that is irrational, hence there exists an additive function such that while (as there exist additive functions which are not linear).
|Title||proof of Dehn’s theorem|
|Date of creation||2013-03-22 16:18:07|
|Last modified on||2013-03-22 16:18:07|
|Last modified by||paolini (1187)|