Gruber P. Convex and Discrete Geometry

16 Volume of Polytopes and Hilbert’s Third Problem 289
In two letters to Gerling, Gauss expresses his regret that certain theorems of solid
geometry depend upon the method of exhaustion, i.e. in modern phraseology, upon
the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in
particular the theorem of Euclid, that triangular pyramids of equal altitudes are to
each other as their bases. Now the analogous problem in the plane has been solved.
Gerling also succeeded in proving the equality of volume of symmetrical polyhedra
by dividing them into congruent parts. Nevertheless, it seems to me probable that a
general proof of this kind for the theorem of Euclid just mentioned is impossible,
and it should be our task to give a rigorous proof of its impossibility. This would
be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases
and equal altitudes which can in no way be split up into congruent tetrahedra, and
which cannot be combined with congruent tetrahedra to form two polyhedra which
themselves could be split up into congruent tetrahedra.
This problem was solved by Hilbert’s student Dehn [250], even before Hilbert’s
list appeared in print.
The problem of necessary and sufficient conditions for G-equidissectability was
studied throughout the whole twentieth century for various groups G of rigid motions.
Hadwiger [465] extended Dehn’s necessary conditions for equidissectability to
all dimensions and Sydler [979] and Jessen [547] showed their sufficiency for d = 3
and d = 4, respectively. For d ≥ 5, the problem is open. For the group of translations
necessary and sufficient conditions were given by Hadwiger and Glur [469] (d = 2)
and Jessen and Thorup [548] (general d). Hadwiger [468], p. 58, showed that two
convex polytopes P, Q ∈ Pp are G-equidissectable if and only if φ(P) = φ(Q)
for all G-invariant valuations φ on Pp. The case d = 3 of this result is due to
Jessen [546].
In this section we first prove the simple result of Bolyai and Gerwien by presenting
several figures. Then Boltyanskiĭ’s [144] concise proof of Dehn’s result is
presented in which he avoids Hamel functions and thus the axiom of choice. See
also the expositions by Boltyanskiĭ [143] and in the nice collection of Aigner and
Ziegler [6].
For more information we refer to the books of Hadwiger [468], Boltyanskiĭ [143]
and Sah [873] and the surveys of McMullen and Schneider [716], Cartier [192],
McMullen [714], Neumann [769], Kellerhals [571] and Dupont [279]. For a popular
presentation of Hilbert’s third problem, see Gray [393].
Equidissectability of Polygons
The following result is due independently to Bolyai [146] and Gerwien [372].
Theorem 16.3. Let P, Q ∈ Pp(E 2 ) such that A(P) = A(Q). Then P and Q are
equidissectable.
Proof. Since equidissectability is a symmetric and transitive relation, it is sufficient
to show that
(1) P is equidissectable to a rectangle of edge-lengths 1 and A(P).