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August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) 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 ...
and theoretical
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
.


Life and education

Möbius was born in
Schulpforta Schulpforta, otherwise known as Pforta, is a school located in Pforta monastery, a former Cistercian monastery (1137–1540). The school is located near Naumburg on the Saale River in the German state of Saxony-Anhalt. The site has been a sch ...
,
Electorate of Saxony The Electorate of Saxony, also known as Electoral Saxony ( or ), was a territory of the Holy Roman Empire from 1356 to 1806 initially centred on Wittenberg that came to include areas around the cities of Dresden, Leipzig and Chemnitz. It was a ...
, and was descended on his mother's side from religious reformer
Martin Luther Martin Luther ( ; ; 10 November 1483 – 18 February 1546) was a German priest, Theology, theologian, author, hymnwriter, professor, and former Order of Saint Augustine, Augustinian friar. Luther was the seminal figure of the Reformation, Pr ...
. He was home-schooled until he was 13, when he attended the college in Schulpforta in 1803, and studied there, graduating in 1809. He then enrolled at the University of Leipzig, where he studied astronomy under the mathematician and astronomer Karl Mollweide. In 1813, he began to study astronomy under mathematician
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
, while Gauss was the director of the
Göttingen Observatory Göttingen Observatory (''Universitätssternwarte Göttingen'' (Göttingen University Observatory) or ''königliche Sternwarte Göttingen'' (Royal Observatory Göttingen)) is a German astronomical observatory located in Göttingen, Lower Saxony, G ...
. From there, he went to study with Carl Gauss's instructor, Johann Pfaff, at the
University of Halle Martin Luther University Halle-Wittenberg (), also referred to as MLU, is a public research university in the cities of Halle and Wittenberg. It is the largest and oldest university in the German state of Saxony-Anhalt. MLU offers German and i ...
, where he completed his doctoral thesis ''The occultation of fixed stars'' in 1815. In 1816, he was appointed as Extraordinary Professor to the "chair of astronomy and higher mechanics" at the University of Leipzig. Möbius died in
Leipzig Leipzig (, ; ; Upper Saxon: ; ) is the most populous city in the States of Germany, German state of Saxony. The city has a population of 628,718 inhabitants as of 2023. It is the List of cities in Germany by population, eighth-largest city in Ge ...
in 1868 at the age of 77. His son
Theodor Theodor is a masculine given name. It is a German form of Theodore. It is also a variant of Teodor. List of people with the given name Theodor * Theodor Adorno, (1903–1969), German philosopher * Theodor Aman, Romanian painter * Theodor Bluege ...
was a noted
philologist Philology () is the study of language in oral and written historical sources. It is the intersection of textual criticism, literary criticism, history, and linguistics with strong ties to etymology. Philology is also defined as the study of ...
.


Contributions

He is best known for his discovery of the
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...
, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. It was independently discovered by
Johann Benedict Listing Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician. Early life and education J. B. Listing was born in Frankfurt and died in Göttingen. He finished his studies at the University of Göttingen in 1834, and ...
a few months earlier. The Möbius configuration, formed by two mutually inscribed tetrahedra, is also named after him. Möbius was the first to introduce
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
into
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 ...
. He is recognized for the introduction of the
barycentric coordinate system In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The ba ...
.Hille, Einar. "Analytic Function Theory, Volume I", Second edition, fifth printing. Chelsea Publishing Company, New York, 1982, , page 33, footnote 1 Before 1853 and Schläfli's discovery of the
4-polytopes In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), ...
, Möbius (with Cayley and
Grassmann Hermann Günther Grassmann (, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguistics, linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical w ...
) was one of only three other people who had also conceived of the possibility of geometry in more than three dimensions. Many mathematical concepts are named after him, including the
Möbius plane In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any gene ...
, the
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
s, important in projective geometry, and the Möbius transform of number theory. His interest in
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
led to the important
Möbius function The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...
μ(''n'') and the
Möbius inversion formula In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large genera ...
. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.
Howard Eves Howard Whitley Eves (10 January 1911, 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics. Eves received his B.S. from the University of Virginia, an M.A. from Harvard University, and a Ph.D. ...
, A Survey of Geometry (1963), p. 64 (Revised edition 1972, Allyn & Bacon, )


Collected works


Gesammelte Werke erster Band (v. 1)
(Leipzig : S. Hirzel, 1885)
Gesammelte Werke zweiter Band (v. 2)
(Leipzig : S. Hirzel, 1885)
Gesammelte Werke dritter Band (v. 3)
(Leipzig : S. Hirzel, 1885)
Gesammelte Werke vierter Band (v. 4)
(Leipzig : S. Hirzel, 1885)
Die elemente der mechanik des himmels, auf neuem wege ohne hülfe höherer rechnungsarten dargestellt von August Ferdinand Möbius
(Leipzig, Weidmann'sche buchhandlung, 1843) File:Mobius-1.jpg, 1843 copy of ''Die Elemente der Mechanik des Himmels'' File:Mobius-2.jpg, Title page to a 1843 copy of ''Die Elemente der Mechanik des Himmels'' File:Mobius-3.jpg, First page to a 1843 copy of ''Die Elemente der Mechanik des Himmels''


See also

*
Barycentric coordinate system In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The ba ...
*
Collineation In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is ...
*
Homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
* Möbius counter *
Möbius plane In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any gene ...


References


External links

* *
August Ferdinand Möbius - Œuvres complètes
Gallica-Math * A beautiful visualization of Möbius Transformations, created by mathematicians at the University of Minnesota is viewable at https://www.youtube.com/watch?v=JX3VmDgiFnY {{DEFAULTSORT:Mobius, August Ferdinand 1790 births 1868 deaths People from Naumburg (Saale) 19th-century German astronomers 19th-century German mathematicians German number theorists German geometers Leipzig University alumni University of Göttingen alumni University of Halle alumni Academic staff of Leipzig University