Hilbert's third problem

Hilbert's third problem is the third of the problems posed by David Hilbert in his famous report at the II International Congress of Mathematicians in Paris in 1900. This problem is devoted to the issues of equal polyhedra : the possibility of cutting two polyhedra of equal volume into a finite number of equal parts-polytopes.

The formulation of such a question was connected with the fact that, on the one hand, on the plane, any two polygons of equal area are equally spaced — as the Boyaya – Gervin theorem states. On the other hand, the available methods of proving the formula for the volume of the tetrahedron (1/3 of the product of height by the area of the base) were somehow connected with the limiting transitions, and thus with the axiom of Archimedes ^[1] . Although literally in the formulation proposed by Hilbert it was a question of the equal composition of tetrahedra (and, more precisely, the proof of the impossibility of such a partition in the general case), it immediately and naturally expands to the question of the equalness of arbitrary polytopes of a given volume (or rather, the necessary and sufficient for this conditions).

The third problem turned out to be the simplest of Hilbert's problems: an example of unequal tetrahedra of equal volume was presented already a year later, in 1901, in the work ^{[2] of a} student of Hilbert M. Den . Namely, he constructed (taking values in a certain abstract group ) a quantity - the Dehn invariant - whose values are equal on equal polyhedra, and presented an example of tetrahedra of equal volume, for which the values of the Dehn invariant are different.

Subsequently, Saydler in his work ^{[3] of} 1965, he showed that the coincidence of the volume and the Dehn invariant are not only necessary, but also sufficient conditions for the equality of polyhedra.

Problem

Hilbert's third problem is formulated as follows:

Gauss, in his two letters to Gerling, deplores the fact that some well-known provisions of stereometry depend on the method of exhaustion, that is, in modern terms, on the axiom of continuity (or on the axiom of Archimedes).

Gauss specifically notes the Euclidean theorem, according to which the volumes of triangular pyramids having equal heights are referred to as the areas of their bases. A similar problem of planimetry is now completely solved. Gerling also managed to prove the equality of volumes of symmetric polyhedra by breaking them into congruent parts.

Nevertheless, it seems to me that in the general case it is impossible to carry out the proof of the mentioned Euclidean theorem in this way, and this, apparently, can be confirmed by rigorous proof of the impossibility.

Such proof could be obtained if it were possible to indicate two tetrahedra with equal bases and equal heights that cannot be decomposed into congruent tetrahedra in any way and which cannot also be supplemented by congruent tetrahedra to polyhedra for which decomposition into congruent tetrahedra possibly.

- David Hilbert (cited from the book of V. G. Boltyansky ^[4] )

Den invariant

The invariant constructed by Den takes values in an abstract group (and, moreover, a vector space over $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ )

V=\mathbb {R} \otimes _{\mathbb {Q} }\mathbb {R} /\langle \{l\otimes \pi \mid l\in \mathbb {R} \}\rangle .

{\ displaystyle V = \ mathbb {R} \ otimes _ {\ mathbb {Q}} \ mathbb {R} / \ langle \ {l \ otimes \ pi \ mid l \ in \ mathbb {R} \} \ rangle. }

Namely, for a polyhedron P with edge lengths $l_{1},\dots ,l_{n}$ ${\ displaystyle l_ {1}, \ dots, l_ {n}}$ and corresponding dihedral angles $\alpha _{1},\dots ,\alpha _{n}$ ${\ displaystyle \ alpha _ {1}, \ dots, \ alpha _ {n}}$ Dehn's invariant D (P) is set equal to

D(P):=\sum _{i}l_{i}\otimes \alpha _{i}\in V

{\ displaystyle D (P): = \ sum _ {i} l_ {i} \ otimes \ alpha _ {i} \ in V}

When cutting a polyhedron into parts, the value of the sum “edge length $\otimes$ ${\ displaystyle \ otimes}$ adjoining angle ”may change only when new edges appear / disappear arising inside or at the border. But for such edges, the sum of the dihedral angles adjacent to them is equal to $2\pi$ ${\ displaystyle 2 \ pi}$ or $\pi$ ${\ displaystyle \ pi}$ accordingly, therefore, as an element of factor V, the Dehn invariant does not change.

Example

An example of the application of the Dehn invariant is the inequality of a cube and a regular tetrahedron of equal volume: for a cube with an edge l, the Dehn invariant is $12l\otimes {\frac {\pi }{2}}=6l\otimes \pi =0$ ${\ displaystyle 12l \ otimes {\ frac {\ pi} {2}} = 6l \ otimes \ pi = 0}$ , and for a regular tetrahedron with edge a -

6a\otimes 2\arctan {\frac {1}{\sqrt {2}}}\neq 0,

{\ displaystyle 6a \ otimes 2 \ arctan {\ frac {1} {\ sqrt {2}}} \ neq 0,}

insofar as $\arctan {\frac {1}{\sqrt {2}}}\notin \mathbb {Q} \pi .$ ${\ displaystyle \ arctan {\ frac {1} {\ sqrt {2}}} \ notin \ mathbb {Q} \ pi.}$

Notes

↑ Hilbert Problems / Ed. P.S. Aleksandrova . - M .: Nauka, 1969 .-- S. 28, 92-94. - 240 p. - 10,700 copies. Archived on October 17, 2011. Archived October 17, 2011 on Wayback Machine
↑ Max Dehn: Über den Rauminhalt, Mathematische Annalen 55 (1901), no. 3, pages 465-478.
↑ Sydler, J.-P. “Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidean à trois dimensions.” Comment. Math. Helv. 40, 43-80, 1965.
↑ Boltyanskii V.G. The Third Hilbert Problem . - M .: Nauka, 1977 .-- S. 46. - 208 p.

Links

Weisstein, Eric W. Dehn Invariant ( Wolfram ) at Wolfram MathWorld .

Literature

Hilbert Problems / ed. P.S. Aleksandrova . - M .: Nauka, 1969 .-- 240 p. - 10,700 copies. Archived on October 17, 2011. Archived October 17, 2011 on Wayback Machine
Dehn, M. "Über raumgleiche Polyeder." Nachr. Königl. Ges. der Wiss. zu Göttingen fd Jahr 1900, 345-354, 1900.
Dehn, M. "Über den Rauminhalt." Math. Ann. 55, 465-478, 1902.
Sydler, J.-P. “Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidean à trois dimensions.” Comment. Math. Helv. 40, 43-80, 1965.
P. Cartier, Décomposition des polyèdres: le point sur le troisième problème de Hilbert , Séminaire Bourbaki, 1984-85, n ° 646, p. 261-288.

[1] Hilbert Problems / Ed. P.S. Aleksandrova . - M .: Nauka, 1969 .-- S. 28, 92-94. - 240 p. - 10,700 copies. Archived on October 17, 2011. Archived October 17, 2011 on Wayback Machine

[2] Max Dehn: Über den Rauminhalt, Mathematische Annalen 55 (1901), no. 3, pages 465-478.

[3] Sydler, J.-P. “Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidean à trois dimensions.” Comment. Math. Helv. 40, 43-80, 1965.

[4] Boltyanskii V.G. The Third Hilbert Problem . - M .: Nauka, 1977 .-- S. 46. - 208 p.