Delian metamorphoses (Q2470899)

Summary: Ernst Specker asked in Aufgabe 1184 of Elem. Math. 57, No. 3, 133 (2002), how to transform the altar of Delos, presumably a cuboid with edge proportions \(2:1:1\), into a cube by cutting and reassembling. In this process only finitely many pieces with finitely many vertices, i.e. polyhedra, should be used. The question of equidecomposable (or scissors equivalent) polyhedra has a long history: Already the ancient Greek determined areas of triangle and parallelograms by this mean. And despite nobody wrote it down in that age: they could have -- with the methods known to Euclid -- proved the theorem of Wallace, Bolyai and Gerwien, that any two polygons (plane polyhedra) of same area are equidecomposable. Here, we consider the corresponding problem in space. Gauß\ wondered 1844 in two letters to Gerling, that the known proofs that two tetrahedra with same bases and same height have the same volume, needed dissections into infinitely many pieces. Hilbert picked this up 1900 at the second International Congress of Mathematicians in Paris in his third problem and asked for two not equidecomposable tetrahedra with same bases and height. Only a few month later Max Dehn (1900) was able to settle the question by defining a dissection invariant, that gave different values for cube and tetrahedron. In this article we describe a solution of Ernst Specker's exercise and discuss the background.