Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

It offers the first systematic representation of a research direction of great depth and beauty - going back to Euler, Gauss, Listing, Riemann, Möbius, Betti and Poincaré - which was then called 'analysis situs' and today probably be called geometric topology.

Written in 1914, not long after the discovery of the fundamental group of a topological space, it tackles a simple and beautiful problem: to confirm a property of the simplest knot which is suggested by five minutes of experimentation: that the right and left trefoil knots are not isotopic. The fundamental group, very new in 1914, was used by Dehn in a sophisticated fashion, via the investigation of its outer automorphism group. Having used it to prove the trefoil is not amphicheiral, Dehn's paper concludes with a brief investigation of the outer automorphism group of the fundamental group of the figure eight knot, which is amphicheiral. He identifies (as was shown later) its generators. ... before 1984 we really didn't have any simple tests for non-amphicheirality, so that Dehn's work (which at first glance looks like the use of a cannon to kill a sparrow) remained central to the subject for nearly 70 years.

A humanistic mathematician who saw mathematics as one chapter - certainly not the least important - in the history of human thought, Dehn could not fail to make an original contribution to the historical study of mathematics, and to involve his colleagues and students in the project. This contribution, or rather this creation, was the historical seminar of the Frankfurt mathematics institute. Nothing could have seemed simpler or less pretentious. A text would be chosen and read in the original, with an effort to follow closely not only the superficial lines but also the thrust of the underlying ideas. ... It was only later that I attended it, on subsequent visits to Frankfurt, a place I made a point of visiting as often as I could. I am not sure whether it was already in the summer semester of 1926 that, during a seminar session devoted to Cavalieri, Dehn showed how this text had to be read from the viewpoint of the author, taking into account both what was commonly accepted in his lifetime and the new ideas that Cavalieri was trying the best of his ability to implement. Everyone participated in the discussion, contributing what he could to the group effort.

In this essay, Dehn attempted, as he wrote "to link the science of mathematics with the whole of human activity". According to Dehn, only a few preconditions are necessary to bring about mathematics. Mathematics, he claimed, arises of its own accord whenever mathematical emotion and logical conclusions come together - and all human beings have both capacities. Dehn thus positioned "mathematical ability" at a basic anthropological level. The belief that mathematics is deeply rooted in human nature, in much the same way that music is based on innate musical abilities, gave him the confidence that non-professional mathematicians could also be made to appreciate this science.

Unforgettable for those who saw him at the time was his calmness, his philosophical composure. For the conversations centred not on the events of the day, but on the relationship of mathematics to art, on problems of archaeology, and finally on the concept of humanity of Confucius.

In 1945, [Dehn] replaced Rudolph Langer, then head of the mathematics department in Madison, Wisconsin, who went on sabbatical to the University of Texas. Prof Dehn came to teach several graduate courses. I attended his course in 'Non-linear Partial Differential Equations'. We were delighted to have a German-speaking instructor since we were to take our German language examinations (required for the doctorate). He, in turn, refused since it was that term that he was to take his examination for US citizenship! Max would hold forth in the Rathskellar (the only campus beer establishment within the Big Ten Universities). We learned more in the beer stube - such as his personal life. He was the black sheep in his family. So long as he remained in his university, he was supported. Thus he received many doctorates. He would declaim in Greek, some passages from the classics, beer stein in hand. At the time for setting out final grades, he merely called the attendance and one by one asked essentially for the titles of the chapters in the textbook. Clearly we could glance ahead to be assured of the correct answer. I believe he gave something like 18 A's and 3-4 B's. Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination.

... most mathematicians fleeing Europe were helped to some sort of position in the United States ... I recall two cases of failures: Max Dehn, noted for work in topology, got only a weak position.

Dehn taught Mathematics, Philosophy, Greek, and Italian. In Mathematics his courses included History of Mathematics and Projective Geometry. He is remembered for his love of nature and the arts. His lectures frequently included tangents on philosophy, the arts, and their connections to mathematics. He was best when lecturing in a socratic style. He was fond of giving these lectures while hiking with his students through the woods. He is remembered as a caring and devoted teacher.

I bear no grudge of any kind. As you may be aware, I am once again in close contact with several mathematicians in Germany, of course primarily with those whom I was particularly close to. But I cannot rejoin the German Mathematical Society. I have lost the confidence that such an association would act differently in the future than it did in 1935. I fear it would, once again, not resist an unjust measure coming from outside. The DMV did not have the custody of very important values. My negative impression is caused by the fact that it did not dissolve itself in 1935 and that not even a considerable number of mathematicians left the association. I am not afraid that the DMV will once again expel Jews, but perhaps next time it will be so-called communists, anarchists or "coloured people". Contact with Germany, especially with German mathematicians, is very dear to my heart.

In 1945, [Dehn] replaced Rudolph Langer, then head of the mathematics department in Madison, Wisconsin, who went on sabbatical to the University of Texas. Prof Dehn came to teach several graduate courses. I attended his course in Non-linear Partial Differential Equations.

We were delighted to have a German-speaking instructor since we were to take our German language examinations (required for the doctorate). He, in turn, refused since it was that term that he was to take his examination for US citizenship !

Max would hold forth in the Rathskellar (the only campus beer establishment within the Big Ten Universities). We learned more in the beer stube - such as his personal life. He was the black sheep in his family. So long as he remained in his university, he was supported. Thus he received many doctorates.

He would declaim in Greek, some passages from the classics, beer stein in hand. At the time for setting out final grades, he merely called the attendance and one by one asked essentially for the titles of the chapters in the textbook. Clearly we could glance ahead to be assured of the correct answer. I believe he gave something like 18 A's and 3-4 B's.

Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination.