August Ferdinand Möbius
Encyclopedia
August Ferdinand Möbius was a German
mathematician
and theoretical astronomer
.
He is best known for his discovery of the Möbius strip
, a non-orientable two-dimensional surface with only one side when embedded
in three-dimensional Euclidean space
. It was independently discovered by Johann Benedict Listing
around the same time. The Möbius configuration
, formed by two mutually inscribed tetrahedra, is also named after him. Möbius was the first to introduce homogeneous coordinates
into projective geometry
.
Many mathematical concepts are named after him, including the Möbius transformations, important in projective geometry, and the Möbius transform of number theory. His interest in number theory
led to the important Möbius function
μ(n) and the Möbius inversion formula
. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.
Möbius was born in Schulpforta, Saxony-Anhalt
, and was descended on his mother's side from religious reformer Martin Luther
. He studied mathematics under Carl Friedrich Gauss
and Johann Pfaff. Möbius died in Leipzig
in 1868 at the age of 77.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and theoretical astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
.
He is best known for his discovery of the Möbius strip
Möbius strip
The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
, a non-orientable two-dimensional surface with only one side when embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in three-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. It was independently discovered by Johann Benedict Listing
Johann Benedict Listing
Johann Benedict Listing was a German mathematician.J. B. Listing was born in Frankfurt and died in Göttingen. He first introduced the term "topology", in a famous article published in 1847, although he had used the term in correspondence some years earlier...
around the same time. The Möbius configuration
Möbius configuration
In geometry, the Möbius configuration is a certain configuration in Euclidean space consisting of two mutually inscribed tetrahedra: each vertex of one tetrahedron lies on a face plane of the other tetrahedron and vice versa...
, formed by two mutually inscribed tetrahedra, is also named after him. Möbius was the first to introduce homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
into projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
.
Many mathematical concepts are named after him, including the Möbius transformations, important in projective geometry, and the Möbius transform of number theory. His interest in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
led to the important Möbius function
Möbius function
The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
μ(n) and the Möbius inversion formula
Möbius inversion formula
In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. Other Möbius inversion formulas are obtained when different local finite partially ordered sets replace the classic case of the natural numbers ordered by...
. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.
Möbius was born in Schulpforta, Saxony-Anhalt
Saxony-Anhalt
Saxony-Anhalt is a landlocked state of Germany. Its capital is Magdeburg and it is surrounded by the German states of Lower Saxony, Brandenburg, Saxony, and Thuringia.Saxony-Anhalt covers an area of...
, and was descended on his mother's side from religious reformer Martin Luther
Martin Luther
Martin Luther was a German priest, professor of theology and iconic figure of the Protestant Reformation. He strongly disputed the claim that freedom from God's punishment for sin could be purchased with money. He confronted indulgence salesman Johann Tetzel with his Ninety-Five Theses in 1517...
. He studied mathematics under Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
and Johann Pfaff. Möbius died in Leipzig
Leipzig
Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...
in 1868 at the age of 77.
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)
Tributes
- The character of Johann Wilhelm Möbius—a particular scientist who tries to evade his own inventions by pretending to be insane—in Dürrenmatt's satiric drama The Physicists is named after him.
- The crater MöbiusMöbius (crater)Möbius is a lunar crater that is located on the Moon's far side, beyond the eastern limb and northeast of the Mare Marginis. It lies less than one crater diameter to the northwest of the larger, 90-km-diameter Hertz, and just to the southeast of Popov...
on the Moon's far side is named for him. - The asteroidAsteroidAsteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
28516 Möbius28516 Möbius28516 Möbius is a main belt asteroid with an orbital period of 1571.4431759 days .The asteroid was discovered on February 27, 2000....
is also named after him. - There is a tribute in Serious Sam: The Second Encounter, in the form on the deathmatch map dm_mobius (which has since spawned various spin-offs such as in the Unreal Tournament series).
- Möbius plays a pivotal part in Brian LumleyBrian LumleyBrian Lumley is an English horror fiction writer.Born in County Durham, he joined the British Army's Royal Military Police and wrote stories in his spare time before retiring with the rank of Warrant Officer Class 1 in 1980 and becoming a professional writer.He added to H. P...
's Necroscope book series.
External links
- Möbius math genealogy
- A beautiful visualization of Möbius Transformations, created by mathematicians at the University of Minnesota is viewable at http://youtube.com/watch?v=JX3VmDgiFnY
- Middle School Mathematician Project Short biography of Mobius by middle school students.