
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, known for his contributions to the application of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
to
crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
, and for work in
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
.
Schoenflies was born in
Landsberg an der Warthe (modern
Gorzów, Poland). Arthur Schoenflies married Emma Levin (1868–1939) in 1896. He studied under
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...
and
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
,
and was influenced by
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
.
The
Schoenflies problem is to prove that an
-
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
in Euclidean ''n''-space bounds a topological ball, however
embedded. This question is much more subtle than it initially appears.
He studied at the
University of Berlin
The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany.
The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...
from 1870 to 1875. He obtained a doctorate in 1877,
and in 1878 he was a teacher at a school in Berlin. In 1880, he went to
Colmar
Colmar (; ; or ) is a city and commune in the Haut-Rhin department and Alsace region of north-eastern France. The third-largest commune in Alsace (after Strasbourg and Mulhouse), it is the seat of the prefecture of the Haut-Rhin department ...
to teach.
Schoenflies was a frequent contributor to
Klein's ''Encyclopedia of Mathematical Sciences'': In 1898 he wrote on
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, in 1902 on
kinematics
In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics.
Kinematics is concerned with s ...
, and on
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
in 1910.
He was a great-uncle of
Walter Benjamin
Walter Bendix Schönflies Benjamin ( ; ; 15 July 1892 – 26 September 1940) was a German-Jewish philosopher, cultural critic, media theorist, and essayist. An eclectic thinker who combined elements of German idealism, Jewish mysticism, Western M ...
.
Selected works
* ''Geometrie der Bewegung in synthetischer Darstellung.'' Teubner, 1886; translated by Charles Speckel as ''La Géométrie du Mouvement. Exposé synthétique.'' Gauthier-Villars 1893
* ''Einführung in die mathematische Behandlung der Naturwissenschaft.'' 1st edition, Dr. E. Wolff, 1895; 2nd editions 1931 (with
Walther Nernst
Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German physical chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid-state physics. His formulation of the Nernst heat theorem helped ...
)
* ''Entwicklung der Mengenlehre und ihrer Anwendungen.'' Teubner, 1913 (with Hans Hahn).
*''Kristallsysteme und Kristallstruktur'', Teubner 1891
* ''Theorie der Kristallstruktur. Ein Lehrbuch.'' Gebr. Borntraeger, 1923.
*''Einführung in die Hauptgesetze der zeichnerischen Darstellungsmethoden'', Teubner 1908
Project Gutenberg ebook*Articles: ''Mengenlehre'' (1898), ''Projektive Geometrie'' (1909), ''Kinematik'' (1902), ''Kristallographie'' (with
Theodor Liebisch,
Otto Mügge), in Klein's encyclopedia.
See also
*
Fyodorov–Schoenflies–Bieberbach theorem
*
Jordan–Schoenflies theorem
*
Schoenflies notation
*
Schoenflies displacement
*
Heine–Borel theorem
In real analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:
For a subset S of Euclidean space \mathbb^n, the following two statements are equivalent:
*S is compact, that is, every open cover of S has a finite s ...
References
External links
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{{DEFAULTSORT:Schoenflies, Arthur Moritz
1853 births
1928 deaths
People from Gorzów Wielkopolski
People from the Province of Brandenburg
19th-century German mathematicians
20th-century German mathematicians
Jewish German scientists
Humboldt University of Berlin alumni
Academic staff of the University of Königsberg
Burials at Frankfurt Main Cemetery
German topologists
German geometers
Presidents of the German Mathematical Society