Gruber P. Convex and Discrete Geometry

290 Convex Polytopes
P clearly can be dissected into triangles. For the proof of (1) it is thus sufficient to
show that
(2) Each triangle T is equidissectable to a rectangle of edge-lengths 1 and
A(T ).
The following figure shows how one can obtain (2) and thus (1) (Fig. 16.1). ⊓⊔
1
Fig. 16.1. Equidissectability of a triangle and a rectangle with one edge-length equal to 1
Hilbert’s Third Problem
Let α1,...,αm ∈ R be rationally independent and let β1,...,βm ∈ R. The corresponding
Hamel quasi-function f is then defined by:
f (r1α1 +···+rmαm) = r1β1 +···+rmβm for rational ri.
f is compatible with a proper convex polytope P in E3 if the dihedral angles
ϑ1,...,ϑm of P at its edges are all rational linear combinations of α1,...,αm. The
f -Dehn invariant of P is
m�
D f (P) = li f (ϑi),
i=1
where li is the length of the edge of P corresponding to ϑi. Note, if α1,...,αm
and β1,...,βm are extended to rationally independent α1,...,αm,...,αk ∈ R and
β1,...,βm,...,βk ∈ R and, correspondingly, f to a function g, then
Dehn’s theorem is the following.
D f (P) = Dg(P).
Theorem 16.4. Let P, Q be equidissectable, proper convex polytopes in E 3 . Then
D f (P) = D f (Q) for each Hamel quasi-function f which is compatible with P and
Q and such that f (π) is defined and equal to 0.
Proof. Let f be such a Hamel quasi-function. Since D f is rigid motion invariant, it
is sufficient to prove the following proposition.
1