Reidemeister graduated from the Brunswick Gymnasium in 1911 and entered the Albert-Ludwig University in Freiburg im Breisgau, one of the oldest universities in Germany founded in 1457. He studied a broad range of courses including mathematics, philosophy, physics, chemistry and geology. In particular he attended a philosophy course given by Edmund Husserl (1859-1938), the famous philosopher, who, in addition to studying philosophy, had taken courses by Karl Weierstrass and Leo Königsberger. Reidemeister also attended lectures by the philosopher Heinrich Rickert (1863-1936) at Freiburg. As was the custom with German students of this period, Reidemeister attended several different universities, moving to Marburg and the Göttingen after his studies at Freiburg. At Göttingen he attended lectures by Edmund Landau. However, Reidemeister was still a university student when World War I broke out in 1914. He was called up for military service and served for four years, the full duration of the war, reaching the rank of Lieutenant. After being released from military duties he returned to his studies at Göttingen. In 1920, he took the Staatsexamen to qualify as a Gymnasium teacher in mathematics, philosophy, physics, chemistry and geology, and was examined in mathematics by Edmund Landau  (see also ):-
Edmund Landau, not known as an easy examiner, was his mathematics examiner and dismissed him after only 30 minutes with the grade of "distinction".
However, Reidemeister did not become a Gymnasium teacher for, in October 1920, he went to Hamburg to take up the position of assistant to Erich Hecke. This was a particularly difficult period in Germany with the Allied powers imposing harsh reparations on the country in the Treaty of Versailles. Against this background of hunger and poverty, Reidemeister studied algebraic number theory working on his doctorate advised by Hecke. His doctoral thesis was on algebraic number theory, the particular problem having been suggested by Hecke, and the resulting publication Relativklassenzahl gewisser relativquadratischer Zahlköper appeared in 1921. In  it is noted that:-
Of all of the 71 papers listed in Reidemeister's obituary by Artzy (), this is the only one which deals with number theory. It rarely happens that a highly productive mathematician deserts the field of his PhD thesis so consistently later on.Immediately he had written his doctoral thesis, Reidemeister became interested in geometry. It was Wilhelm Blaschke who came up with the particular problems in differential geometry on which Reidemeister began to work :-
In Hamburg, he met Wilhelm Blaschke who turned him toward an interest in geometry, and Blaschke entrusted the brilliant student with cooperation on the second volume of his 'Differential Geometry'. Indeed, just a few months after receiving his doctorate, Reidemeister gave a plenary lecture on this subject, quite different from that of his dissertation, at the annual meeting of the German Mathematical Society [in Jena in September 1921].The perspective and ideas that Blaschke presented, Reidemeister found absolutely fascinating and, quite unexpectedly, he completely changed the topics he was researching. His publications at this time included: Über die singulären Randpunkte eines konvexen Körpers (1921); Über Körper konstanten Durchmessers (1921); Über affine Geometrie. XXXI: Beständig elliptisch oder hyperbolisch gekrümmte Eilinien (1921); and Die Differentialgleichung der Schiebflächen (1922). However, he continued to be interested in philosophy and history, and he was also fascinated to read the book Der Untergang des Abendlandes by the historian and philosopher of history Oswald Spengler (1880-1936). The first volume of the book appeared in 1918 with the second volume coming out in 1922. The book, which had a large section on mathematics, was controversial and prompted much discussion and argument. Reidemeister gave a lecture on Spengler's book but he also wrote short stories and poems, publishing regularly in the leading Hamburg newspaper. He published many reviews both of mathematics texts and texts on philosophy. For example his reviews of Principles of Geometry by Henry Baker and Grundzüge der mehrdimensionalen Differentialgeometrie by Dirk Struik appeared in 1923. He was also planning to habilitate in Hamburg but, while his thesis was still in the planning stage, he received a call to a chair.
On Hans Hahn's recommendation, despite having never habilitated, Reidemeister was appointed as associate professor of geometry at the University of Vienna in October 1923. Here he became a colleague of Hans Hahn, Wilhelm Wirtinger and Philipp Furtwängler. Two brilliant students were also about to complete their doctorates, Otto Schreier and Karl Menger. It was Wirtinger who interested Reidemeister in knot theory, the topic for which he is best remembered today. In particular Wirtinger showed Reidemeister how to compute the fundamental group of a knot from its projection. This method, originally due to Wirtinger, appears in work of Artin which was published in 1925. In Vienna, he met Elisabeth Wagner who was a professional photographer and the daughter of a Protestant minister from Riga; they married in 1924. In fact this move to Austria was very beneficial for Reidemeister since it allowed him to escape from the misery of hyper-inflation in Germany with the strikes and unrest which resulted. While in Vienna, Reidemeister came across the Tractatus by Wittgenstein and joined the Vienna Circle of Logical Positivists. Led by Reidemeister, the group of mathematicians at Vienna spent a year studying the deep ideas on logic and mathematics in the Tractatus.
In 1925 Reidemeister was offered the chair in Königsberg which had been left vacant when Wilhelm Meyer retired, which he accepted. There he worked with several young mathematicians including Ruth Moufang, Richard Brauer, Werner Burau, and Rafael Artzy (the author of ). He continued to publish reviews, for example his reviews of Formalismus und Intuitionismus in der Mathematik by Richard Baldus, Begriffsbildung by Karl Boehm, and Systematische Axiomatik der Euklidischen Geometrie by Moritz Geiger appeared in 1926. In 1930 the German Mathematical Congress met in Königsberg and Reidemeister organised the first international conference on the philosophy of mathematics to be a part of the larger Congress. Reidemeister worked on the foundations of geometry and he wrote an important book on knot theory Knoten und Gruppen (1926). He established a geometry and topology based on group theory without the concept of a limit. He published other important books while at Königsberg: Vorlesungen über Grundlagen der Geometrie (1930), Einführung in die kombinatorische Topologie (1932) and Knotentheorie (1932). The authors of  comment on this 1932 text on combinatorial topology:-
Although Reidemeister ... was, above all, a geometer, his book on 'combinatorial topology' contains hardly any drawings. Abstraction and rigor were very much in fashion.His knot theory book of 1932 was short, being only 74 pages long, but was highly significant. It was reprinted in 1974 and translated into English in 1983. After an Introduction, the book contained the following chapters: Knots and their projections; Knots and matrices; Knots and groups; and Tables of knots. Reidemeister's Grundlagen der Geometrie was reviewed by Robin Robinson who writes :-
Reidemeister's work is restricted almost entirely to the plane, and his chief aim seems to be the setting up of an axiomatic basis for the study of affine and projective geometries. Two distinct approaches to this task are considered - the analytic and the axiomatic, and they motivate the division of the book into two parts. In the first part Reidemeister discusses the foundations of algebra, and bases affine and projective geometries on them. In the second part, a purely axiomatic development of geometry is based on the concept of a 3-web, and its completeness and consistency are demonstrated by forming from it a number system satisfying the postulates embodied in the first part. ... Both as an original composition and for its collection of interesting developments, this work is a distinct addition to geometrical literature, and should appeal strongly to anyone deeply interested in the logical foundations of geometry.Although academically he was achieving great things in Königsberg, life there for him and his wife was not easy, and they found it impossible to maintain the wide-ranging cultural life they had previously enjoyed. He was forced to leave his chair in Königsberg in 1933 by the Nazis, whom he strongly opposed, who classed him as 'politically unsound' :-
In January 1933, shortly before Hitler's accession to power, National Socialist students at Königsberg fomented a disturbance directed against the university Rektor. Reidemeister devoted a whole mathematics lecture to explaining why the behaviour of these students was totally unsupportable and not compatible with rational thinking.He only learnt that he had been dismissed when he read it in the local newspaper. He was shattered by this treatment :-
The experience of Königsberg left serious wounds in him that never quite healed. The aging Reidemeister always had an attitude of being easily prone to moral outrage, an attitude which had its root here.Blaschke immediately tried to help his colleague and collected signatures on a petition seeking to reinstate Reidemeister. After being suspended from his chair, Reidemeister went to Rome where he continued to undertake research. Perhaps due to Blaschke's efforts, he was appointed to Kurt Hensel's chair in Marburg at what was considered a smaller and less prestigious university. He took up the position in the autumn of 1934. The next years were difficult ones for him living under Nazi rule and then living through the horrors of World War II. Of course he had to keep his views from becoming public since he had learned the consequences when in Königsberg. However, he had a small circle of friends in Marburg whom he could trust and with whom he could express his views. In 1938 he published the book Topologie der Polyeder und kombinatorische Topologie der Komplexe . The review  begins by putting this work in context:-
The first systematic exposition of combinatory topology was made by Dehn and Heegaard in a section of the "Enzyklopadie der mathematischen Wissenschaften", in which the exact concepts involved are only developed as far as the third dimension. The extension of the theory to complexes of higher dimensions was fraught with considerable difficulties; and it was due to the work of M A H Newman that the gap was finally bridged. In the section of the encyclopaedia referred to above, he showed that combinatory topology was one of the most primitive branches of geometry, in which the concept of limiting values had as yet no place. In fact, as Professor Reidemeister describes in detail in the present treatise, combinatory topology turns out to be a well-defined partial domain of the theory of polyhedra - a fact well calculated to demonstrate the elementary character of the subject and to justify Professor Reidemeister in providing a new and systematic presentation of it.Reidemeister had an important influence on group theory, partly through his work on knots and groups, partly through his influence on Otto Schreier. Talking of this influence on group theory, Chandler and Magnus write in :-
Reidemeister was ... essentially a geometer. His influence on combinatorial group theory is largely that of a pioneer. His ideas were stimulating and had, at least in some cases, a long-lasting effect.After he went to Marburg, Reidemeister's interests became almost exclusively philosophy, the foundations of mathematics and the history of mathematics. For example he published Die Arithmetik der Griechen (1940), Mathematik und Logik bei Plato (1942), and Das System des Aristoteles (1943). He published two volumes of his essays and poems: Figuren (1946) and Von dem Schönen (1947). He translated poems by the French poet and critic Stéphane Mallarmé which he published in 1948 as the book Dichtungen . Reidemeister spent the two years 1948-50 at the Institute for Advanced Study at Princeton in the United States. There he enjoyed talking to Oswald Veblen, Carl Siegel and Hermann Weyl. His interest in algebraic topology was renewed and he gave the specially invited plenary address Complexes and homotopy chains to the Philadelphia Meeting of the American Mathematical Society on 30 April 1939. It was later published in the Bulletin of the Society.
Returning to Marburg, he published books Das exakte Denken der Griechen (1949), Geist und Wirklichkeit, Kritische Essays (1953), and Die Unsachlichkeit der Existenzphilosophie (1954). In 1955 Reidemeister left Marburg when he was appointed to the University of Göttingen. His most important publication while at Göttingen was Raum und Zahl (1957). Donald Coxeter reviewed this work in :-
This book introduces several branches of mathematics, not only for their intrinsic interest but as background for philosophical discussions. The first chapter, on the origin of geometrical thought, contains a set of axioms for the affine plane, a description of some theorems of closure, and a well-chosen quotation from Plato's 'Menon', where Socrates is teaching his slave that the square on the diagonal of a given square has twice the area of the given square. This is followed by a delightful chapter on linkages, instruments for trisecting a given angle and for drawing ellipses and other special curves. The chapter on analytic geometry shows how translations, rotations and dilatations are represented by linear transformations of a complex variable, and how the scope is extended by considering linear fractional transformations. The fourth chapter gives a system of axioms for Euclidean geometry, using distance as a primitive concept. ... The fifth chapter describes some classical paradoxes and introduces the theory of transfinite ordinals and combinatorial topology. The sixth (on geometry and logic) begins with Hjelmslev's idea of representing the points and lines of the Euclidean (or non-Euclidean) plane by involutory transformations that leave them invariant ... This is followed by remarks about Russell's paradox and about affine geometry over an arbitrary field. The seventh chapter clarifies some prevalent obscurities in the foundations of differential and integral calculus. The eighth is a tribute to Gauss. The ninth (on geometry and number theory) develops the theory of algebraic numbers, leading to rigorous proofs of the impossibility, by Euclidean constructions, of duplicating the cube and trisecting an angle of 600. The tenth and last chapter (Prolegomena to a critical philosophy) is the text of a lecture to the "Kongress des Internationalen Forums" (Zürich, 1954).In Göttingen, both Reidemeister's own health and that of his wife began to deteriorate. He was unhappy at his fading abilities and isolated himself more and more. His friends tried to help him but they reached a stage when they could do no more and he spent his last years in loneliness. He retired and reached the age of seventy without any celebration. His final scholarly contribution was his work as editor of Hilbert-Gedenkband (1971).
Article by: J J O'Connor and E F Robertson