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Hilbert's Problem and Dehn's Theorem

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Hilbert's Problem and Dehn's Theorem Is it p ossible to cut a cub e into pieces and to assemble a tetrahedron Hilb erts problem and Dehns theorem Dmitry FUCHS Hilb erts Problem Three Is it p ossible to cut a cub e by nitely many planes and assemble out of the p olyhedral pieces obtained a regular tetrahedron of the same volume This is a slight mo dication of one of the problems presented by David Hilb ert in his famous talk at the Congress of Mathematicians in Paris on August it go es under the numb er Hilb erts problems had a tremen dous impact on Mathematics Most of them were solved during XX century and each has a very sp ecial history Still Problem Three stands exceptional in many resp ects First this was the rst of Hilb erts problems to b e solved The solution b elonged to a years old German geometer Hilb erts student Max Dehn His article app eared two years after the Paris Congress but the solution existed earlier mayb e even b efore Hilb ert stated the problem or less the same as the one presented b elow was short Dehns pro of more for p opular lectures and clear and it b ecame one of the favorite sub jects articles and b o oks in geometry like the one you are holding in your hands I can recommend the b o ok by Boltianskii for a more extended exp osition But among working mathematicians it was almost forgotten Certainly the name of Dehn was not forgotten He b ecame one of the few top exp erts in top ology of threedimensional manifolds and his work of has b een never regarded as his main achievement it is not even mentioned in Dehns biography available on the web In American Mathematical So ciety published a twovolume collec tion of articles under the title Mathematical Developments Arising from Hilb ert Problems It was a very solid account of the three quarters of century of history of the problems solutions full and partial generaliza tions similar problems and so on This edition contains a thorough analysis of of Hilb erts problems And only Problem Three is not discussed there The opinion of the editors is obvious no developments no inuence on Mathematics nothing to discuss How strange it seemed just a couple of years later Dehns theorem Dehns theory Dehns invariant b ecame one of the hottest sub jects in geom an exciting domain etry This was stimulated by then newb orn Ktheory develop ed along the b orderline b etween algebra and top ology We will not follow this development but will just revise the theorem and its pro of For a similar problem in the plane the an swer is yes Let P P be two polygonal domains in the plane having the Theorem 1 2 same ar ea Then it is possible to cut P into pieces by straight lines and to 1 assemble Of these pieces P 2 Pro of First it is clear that it is sucient to consider the case when P 2 is a rectangle with the sides and area P in doing this we can shorten the 1 notation of P to just P 1 Second since any p olygonal domain can b e cut into triangles we can reduce the general case to that of a triangle see Figure P ! ! ! P area Figure Third we need to remake by cutting and pasting a triangle into a rect angle with one if the sides having length one This is done in four steps on Figure ! ! Step Step rational length pq .
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