Third problem of Hilbert

The third problem of Hilbert is one of the 23 Problèmes of Hilbert. Regarded as easiest, it treats geometry of the Polyèdre S.

being given two polyhedrons of equal volume, is it possible to cut out the first polyhedron in polyhedrons and to gather them to form the second polyhedron?

Hilbert conjectured that it was not always true. It was confirmed in the year by its pupil, max Dehn, which provides a counterexample.

For the similar problem concerning the Polygon S, the answer is affirmative. The result is known under the name of the Théorème of Wallace-Bolyai-Gerwein.

Answer of Dehn

Dehn uses the algebra to deny the possibility of cutting. When the first polyhedron can be actually cut out in polyhedrons which gather to train the second, the polyhedrons are known as congruent.

With each polyhedron P , one associates a value, called “invariant of Dehn” D ( P ) such as:

D is additive: if P cuts out by section by a single plan in two polyhedrons P ₁ P ₂, D ( P ) =D ( P ₁) +D ( P ₂).

In particular, if two polyhedrons P and Q are congruent, then they have the same invariant of Dehn.

The cube has an invariant of null Dehn, whereas the tetrahedron has an invariant of positive Dehn. These two polyhedrons are not congruent.

The invariant is defined on the lengths and the plane angles. Let us observe:

a cutting by a plan divides the lengths on certain sides into two. It is necessary thus that the invariant is additive in these lengths.
In the same way, if a polyhedron is divided according to a side, the plane angle correspondent is cut into two. It is necessary thus that the invariant is additive in the angles.
Lastly, cutting reveals new sides; thus new lengths, and new plane angles. These contributions must be cancelled.

The invariant of Dehn is defined as an element of the tensorial product of the body of the real numbers R by the quotient R / Q $\ pi$ . The tensorial product is taken on Q .

\ operatorname {D} (P) = \ sum_ {E} L (E) \ otimes (\ theta (E) + \ mathbb {Q} \ pi)

where L ( E ) is the length on the side E , θ ( E ) is the plane angle between the adjacent faces with E , and it sum is taken on all with dimensions polyhedron.

Beyond the problem of Hilbert

Sylder showed in 1965 which two polyhedrons are congruent if and only if they have even volume and even invariant of Dehn.

In 1990, Dupon gives of it a proof simplified while reinterpreting like a theorem on the Homologie of certain traditional groups.

Motivations

The formula of the volume of a pyramid was known of Euclide (proposal 7 of book XII of the Eléments ): 1/3 × surface of the base×hautor. But contrary to the surface of the triangles or the parallelograms which can be compared by simple cutting with that of a rectangle, the demonstration of the volume of the pyramid calls upon the complex Méthode of exhaustion, ancestor of the integral calculus. Gauss deplored it in one of its letters addressed to Gerling.

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