In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a Moufang plane, named for
Ruth Moufang
Ruth Moufang (10 January 1905 – 26 November 1977) was a German mathematician.
Biography
Born to German chemist Eduard Moufang and Else Fecht Moufang. Eduard Moufang was the son of Friedrich Carl Moufang (1848-1885) from Mainz, and Elisa ...
, is a type of
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
, more specifically a special type of
translation plane In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desar ...
. A translation plane is a projective plane that has a ''translation line'', that is, a line with the property that the group of automorphisms that fixes every point of the line
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
transitively on the points of the plane not on the line. A translation plane is Moufang if every line of the plane is a translation line.
Characterizations
A Moufang plane can also be described as a projective plane in which the ''
little Desargues theorem'' holds. This theorem states that a restricted form of
Desargues' theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ...
holds for every line in the plane.
For example, every
Desarguesian plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
is a Moufang plane.
In algebraic terms, a projective plane over any
alternative division ring In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all ''x'' and ''y'' in the algebra.
Every associative algebra is ...
is a Moufang plane, and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and Moufang planes.
As a consequence of the algebraic
Artin–Zorn theorem In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin. ...
, that every finite alternative division ring is a field, every finite Moufang plane is Desarguesian, but some infinite Moufang planes are
non-Desarguesian plane
In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective s ...
s. In particular, the
Cayley plane
In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.Baez (2002).
The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describing ...
, an infinite Moufang projective plane over the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s, is one of these because the octonions do not form a division ring.
Properties
The following conditions on a projective plane ''P'' are equivalent:
*''P'' is a Moufang plane.
*The group of automorphisms fixing all points of any given line acts transitively on the points not on the line.
*Some ternary ring of the plane is an alternative division ring.
*''P'' is isomorphic to the projective plane over an alternative division ring.
Also, in a Moufang plane:
*The group of automorphisms acts transitively on quadrangles.
[If transitive is replaced by sharply transitive, the plane is pappian.]
*Any two
ternary ring
In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marsh ...
s of the plane are isomorphic.
See also
*
Moufang loop Moufang is the family name of the following people:
* Christoph Moufang (1817–1890), a Roman Catholic cleric
* Ruth Moufang (1905–1977), a German mathematician, after whom several concepts in mathematics are named:
** Moufang–Lie algebra
** ...
*
Moufang polygon In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of root groups.
In a book on the topic, Tits and Richard Weiss c ...
Notes
References
*
*
*
Further reading
*{{Citation , author1-link=Jacques Tits , last1=Tits , first1=Jacques , last2=Weiss , first2=Richard M. , title=Moufang polygons , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-43714-7 , mr=1938841 , year=2002
Projective geometry