**Friedrich Bachmann**was born in Wernigerode, a little town in the Harz mountains, Northern Germany. He was the son of the Lutheran minister and teacher Hans Bachmann who published works such as

*Das Jesusbild der sogenannten modernen Theologie und der geschichtliche Jesus*Ⓣ (1912). Friedrich's paternal grandfather was the mathematician Paul Bachmann (1837-1920), who had studied with Dirichlet and Dedekind, had Kummer as the advisor for his PhD and, after his Habilitation in 1868 in Breslau, had become a full professor in Münster. Paul Bachmann is well known as the author of several textbooks in Number Theory and has a biography in this archive.

Friedrich Bachmann attended the Gymnasium in Münster and, after graduating, he studied from 1927 in Berlin and Münster. His first area of research was Mathematical Logic and the Foundations of Mathematics, and in 1933 he was awarded his PhD by the Westfälische Wilhelms University of Münster for his thesis *Untersuchungen zur Grundlegung der Arithmetik mit besonderer Beziehung auf Dedekind, Frege, und Russell* Ⓣ. His thesis advisor at Münster had been Heinrich Scholz (1884-1956) who had been appointed there as a full professor of Philosophy in 1928 and later became professor of Mathematical Logic and Foundations of Mathematics.

After further work on logic, including starting to prepare the correspondence between Gottlob Frege and Bertrand Russell for publication, Bachmann turned to questions on geometry. In 1936 he published in *Mathematische Annalen* (volume 113, pages 424-451) the extensive paper *Eine Begründung der absoluten Geometrie der Ebene * Ⓣ in which he investigated plane absolute geometries based upon a set of incidence, orthogonality, and weak congruence axioms. He continued this work in his paper *Stufen der absoluten Geometrie. Die Frage nach der Unabhängigkeit der Anordnung* Ⓣ (1940) published in the same journal. From 1935 he was an assistant to Kurt Reidemeister in Marburg where he habilitated in 1939, becoming a Privatdozent there. In 1941 and 1943-44 he lectured at Königsberg, having lectured at the Humboldt University of Berlin during the academic year 1942-43. He published *Ein lineares Vollständigkeitsaxiom* Ⓣ in 1943 in which he investigated Hilbert's system of axioms for Euclidean geometry. In the same year he began substituting for Fritz Lettenmeyer at the Christian-Albrechts Universität in Kiel where he was promoted to 'Ordentlicher Professor' (Full Professor) on 1 March 1949.

Together with Karl-Heinrich Weise, Bachmann had a main share in rebuilding the 'Mathematisches Seminar' at Kiel after the war. In 1962-63 he was elected Dean of the Mathematics and Science section of the Faculty of Philosophy of Kiel University. Together with others, from 1960 onwards he served as editor of the several volume work *Grundzüge der Mathematik für Lehrer an Gymnasien sowie für Mathematiker in Industrie und Wirtschaft* Ⓣ.

Bachmann's lectures were highly liked and respected for their clarity and for being well structured. He would take extreme care in their preparation. Several of his courses were circulated in the form of very readable lecture notes. Many of the mathematics students at Kiel at that time owe an important part of their mathematical education to him. As demonstrated in the 'Mathematical Genealogy Project', several of his PhD students later became professors of mathematics. Although he did not like publicity much, his profound knowledge of philosophy and logic allowed him to play an important role when a chair in the department of philosophy was dedicated to logic and filled by Paul Lorenzen from 1956 to 1962, and later by Kurt Schütte from 1963 to 1966, broadening the scope of mathematics at Kiel beyond the 'Mathematisches Seminar' proper.

His scientific work at Kiel was mainly devoted to the axiomatic foundation of geometry. In the short note *Zur Begründung der Geometrie aus dem Spiegelungsbegriff* Ⓣ in *Mathematische Annalen *in 1951 he presented his well-known reduced version of A Schmidt's 'Axiomensystem der metrischen (absoluten) Geometrie' which is a system of axioms for absolute geometry based on line reflection only. In 1959 his monograph *Aufbau der Geometrie aus dem Spiegelungsbegriff* Ⓣ appeared as vol. XCVI of the (Springer) 'Grundlehren der Mathematischen Wissenschaften'. In this book Bachmann develops plane metric geometry by systematic use of reflections and the group of motions generated by them. In the preface he summarizes the main idea of his approach:-

Donald Coxeter, in the review [2], writes:-If one coinsiders the reflections themselves as geometric entities, namely as new "points" and "lines", one can define geometric relations such as "incidence" and "orthogonality" between them by group theoretical relations in such a way that the new domain is a faithful image of the originally given points and lines with their incidence, orthogonality etc.

Coxeter ends his review as follows:-This remarkable book is essentially an elaboration of an idea of G Thomsen(The treatment of elementary geometry by a group-calculus, Math. Gaz.17(1933),232). ... One soon begins to realize that such a geometry is not necessarily Euclidean. It is more like the "absolute" geometry of Bolyai, in which a pencil of lines having a common perpendicular is not necessarily the same as a pencil of parallels. In fact, the geometry may be regarded as a special kind of abstract group whose generators, called "points" and "lines," are involutory, with the distinction that , although the product of two lines may be a point, the product of two points is never a line. Even this restriction is later waived so as to cover the case of a generalized elliptic geometry which admits an absolute polarity.

Bachmann published an extended and updated second edition of this book in 1973. Later Bachmann dealt with an extension of his approach to so-called Hjelmslev groups, which is presented in the 1989, posthumously published, bookThe above remarks may serve to suggest something of the flavour of this unusual book, which is well written, well printed, well indexed, and "chock full" of unfamiliar results. All geometers and most algebraists will be glad to keep it on an accessible shelf.

*Ebene Spiegelungsgeometrie. Eine Vorlesung über Hjelmslev-Gruppen*Ⓣ. Dirk Keppens writes:-

In 1970, in collaboration with Eckart Schmidt, Bachmann had publishedThis interesting book deals with a class of plane metric geometries associated to certain groups. ... The book presents a detailed and complete study of the important class of Hjelmslev groups and their associated Hjelmslev group planes. ... It is proved that the groups of plane absolute geometry are Hjelmslev groups. ... . By means of the additional polar triangle axiom the Hjelmslev groups can be divided into two classes: elliptic and nonelliptic ones. Both classes are studied and all elliptic Hjelmslev groups are classified. ... This book is a must for all research workers in metric geometry and for all mathematicians interested in geometry and groups.

*n-Ecke*Ⓣ (an English translation with the title

*n-gons*appeared in 1975). Dirk Keppens, reviewing the 1970 German version, begins his review as follows:-

Friedrich Bachmann retired on 31 March 1977, he died on 1 October 1982 in Kiel, leaving his wife Alexandra née von Bredow, who was a great-granddaughter of Bismarck, and a son Sebastian.This is a delightful little book on polygons or, rather, on mappings between polygons. The authors prefer the title 'n-gons'(in contrast to polygons)for this book, since the mappings under consideration always map n-gons into n-gons for a fixed n. As a consequence the classes that will occur are classes of n-gons, again for the same n.

**Article by:** *J J O'Connor* and *E F Robertson*

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