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.

Notes

References

* * *

Characterizations

Properties

See also

Notes

References

Further reading