**Schönflies** worked first on geometry and kinematics but became best known for his work on set theory and crystallography. He classified the 230 space groups in 1891.

- Martin Schönflies also trained as a mathematician and obtained a doctorate.
- Arthur entered the Gymnasium in Landsberg in 1862 and studied there until 1870.
- The Franco-Prussian War began in July 1870 but the French troops had surrendered in September 1870 before Schönflies became a student at the Friedrich-Wilhelms University of Berlin.
- Since the Berlin Industrial Institute could not award doctorates, Schönflies had to have formal advisors at the University of Berlin.
- Following the award of his doctorate, Schönflies continued teaching at a Friedrich-Wilhelms-Gymnasium in Berlin.
- Felix Klein worked to set up a chair of applied mathematics at Göttingen and in 1892 Schönflies was appointed to this chair.
- However, Klein had great difficult with making the appointment due to the fact that Schönflies was Jewish.
- Klein was furious and redoubled his efforts to have Schönflies made an extraordinary professor.
- can only be somewhat remedied by having Schönflies named Extraordinarius.
- Schönflies left Göttingen in 1899 to take up a chair at the University of Königsberg, then in 1911 he became professor at the Academy for Social and Commercial Sciences in Frankfurt.
- Schönflies ended his career at the University of Frankfurt where he served as professor from 1914 until 1922 being rector of the University in the session 1920-21.
- Schönflies worked first on geometry and kinematics but became best known for his work on set theory and crystallography.
- By 1891 Schönflies had found the complete list of 230 such groups.
- The book also includes the Schönflies notation, one of the two conventions still used today to describe crystallographic point groups.
- Schönflies corresponded with Fedorov and they corrected some minor errors in both classifications before publishing their classification.
- Both Fedorov and Schönflies were stimulated by noticing a mistake in Sohncke's work (one symmetry group was listed twice, giving 65) and went on to count all symmetry groups, including those which include some orientation-reversing symmetries.
- an extremely surprising circumstance has come to light, viz a coincidence in the work of two researchers such as has, perhaps, never been observed in the history of science" which has been misinterpreted as a statement that Fedorov and Schönflies worked in complete isolation from each other (and in England further embroidered to include the fiction that Barlow independently arrived at the same result; in fact he knew of the Fedorov/Schönflies papers but tried to obtain the same results direct from Sohncke's work by a different method and obtained a false result 229 even though - as Fedorov pointed out - "...
- Schönflies republished his classification in 1923 in Theorie der Kristallstruktur.
- In around 1895 Schönflies turned his attention towards set theory and topology.
- Brouwer presented counterexamples to some of Schönflies's theorems showing that the notion of closed curve was more complicated than Schönflies had realised.
- We should note the important contributions that Schönflies made to set theory in publishing the two-hundred and fifty page report on set theory Die Entwickelung der Lehre von den Punktmannigfaltigkeiten Ⓣ(The development of the theory of point manifolds) (1899), a substantial part of which studies transfinite numbers.
- Schönflies tells the reader that the proof of the Heine-Borel theorem is one of the most significant applications of transfinite numbers.
- The first of these contains charming personal comments from Schönflies who had witnessed the mathematical revolution at first hand.
- In his 1927 article, Schönflies gives the reader insight into Cantor's thoughts.
- For example, Schönflies puts Cantor's contributions into three categories: general set theory and point-set theory; the theory of transfinite numbers; and his philosophical arguments.
- Schönflies also wrote on kinematics and projective geometry.
- In 1895 Schönflies edited Plücker's complete works.
- The oldest one there was a mathematician named Arthur Schönflies, whom you know, of course, as the man who first formalised the theory of space groups of lattices, but who has many, many other great achievements.
- Among the many honours that Schönflies received we mention his election to membership of the German Academy of Scientists Leopoldina in 1896, to corresponding member of the Royal Society of Sciences in Liege in 1904, and corresponding member of the Prussian Academy of Scientists in 1918 who specifically cited his contributions to promoting Cantor's ideas on set theory.

Born 17 April 1853, Landsberg an der Warthe, Prussia (now Gorzów-Wielkopolski, Poland). Died 27 May 1928, Frankfurt am Main, Germany.

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Group Theory, Origin Poland

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive