It was known to Euclid that if two polygons have equal areas, then it is possible to transform one into the other by a cut and paste process (see, e.g., [1]).

(1) Describe a proof of this fact. Also discuss the same problem in spherical and hyperbolic geometries.

An ingenious and impressive illustration of this theorem is H. E. Dudeney's (master puzzler) hinged-square-into-triangle.

In his address to the ICM 1900, D. Hilbert presented the following problem (quoted from J. Gray [2])

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudesIn two letters to Gerling, Gauss expresses his regret that certain theorems in 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 [1]. Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent pairs. 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 succeed in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form to polyhedra which themselves could be split up into congruent tetrahedra.

The problem was solved (before Hilbert's address appeared in print) by Hilbert's student Max Dehn. Dehn found that not only the volume must remain invariant under cut and paste operations, but also certain combination of side lengths and dihedral angles. This combination is now known as the Dehn invariant. (Gray also mentions [2,p. 97] the work of Bricard and of Sforza on this problem, prior to Hilbert's address.)

(1) Define the Dehn invariant of a polyhedron.

(2) Show that the Dehn invariant remains indeed invariant under cut and paste transformations.

(3) Compute some simple examples. In particular, show that the tetrahedron and the cube of equal volumes have different Dehn invariant, thus solving Hilbert's 3rd problem in the negative.

- [1]
- D. Hilbert,
*Foundations of Geometry*(8th Edition), Open Court Publishing Co., La Salle, Il., 1994. - [2]
- J. Gray,
*The Hilbert Challenge*, Oxford University Press, New York, 2000. - [3]
- J. Stillwell,
*Numbers and Geometry*, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1998.