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Arthur Schönflies was a student at the University of Berlin from 1870 until 1875 working under Kummer and Weierstrass. He obtained a doctorate from Berlin in 1877 and the following year he obtained a post as a teacher at a school in Berlin. In 1880 he went to Colmar in Alsace to teach.
He then wrote his Habilitation thesis which he presented to Göttingen, the qualification being awarded in 1884. Klein worked to set up a chair of applied mathematics at Göttingen and in 1892 Schönflies was appointed to this chair.
He left Göttingen in 1899 to take up a chair at Königsberg, then in 1911 he became professor at the Academy for Social and Commercial Sciences in Frankfurt. This Academy became a University in 1914. 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. Klein suggested the problem of finding the crystallographic space groups in the late 1880s. By 1891 he had found the complete list of 230 such groups. His presentation of crystallographic space groups published in 1892 used the latest aspects of group theory and became a classic on the subject.
In fact the classification of the crystallographic space groups was made independently by E S Fedorov. Schönflies corresponded with Fedorov and corrected some minor errors in his classification. He republished his classification in 1923 and in the same year he published a book on crystallography.
In around 1895 Schönflies turned his attention towards set theory and topology. He wrote many works which were important at the time they were published but they were rather superseded by Hausdorff's Grundzüge der Mengenlehre in 1914. Three important papers on plane topology proved the topological invariance of the dimension of the square. His work contains gaps and errors which were investigated by Brouwer who made some deep discoveries from studying these errors.
Schönflies also wrote on kinematics and projective geometry. He wrote textbooks on descriptive geometry and analytic geometry and a calculus textbook jointly with Nernst. In 1895 Schönflies edited Plücker's complete works.
Article by: J J O'Connor and E F Robertson
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