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 Elisab ...
, 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 ...
, more specifically a special type of
translation plane. 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
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 t ...
'' holds. This theorem states that a restricted form of
Desargues' theorem holds for every line in the plane.
For example, every
Desarguesian plane 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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galoi ...
, 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 describi ...
, 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 hav ...
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 rings 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
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 ...
, location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-43714-7 , mr=1938841 , year=2002
Projective geometry