Cayley plane
Encyclopedia
In mathematics
, the Cayley plane (or octonionic projective plane) OP2 is a projective plane
over the octonion
s. It was discovered in 1933 by Ruth Moufang
, and is named after Arthur Cayley
(for his 1845 paper describing the octonions).
As a symmetric space
, the Cayley plane is F₄ / Spin(9), where F₄ is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space
(realized in F₄). As a homogeneous space
it is also the quotient of a noncompact form of the group E₆ by a parabolic subgroup P1.
. It is one of the rare projective spaces, where the Desargues' theorem
does not hold.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Cayley plane (or octonionic projective plane) OP2 is a projective plane
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 that do not intersect...
over the octonion
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s. It was discovered in 1933 by Ruth Moufang
Ruth Moufang
Ruth Moufang was a German mathematician.Born to a German chemist Dr. Eduard Moufang and Else Fecht Moufang, she studied mathematics at the University of Frankfurt. In 1931 she received her Ph.D. on projective geometry under the direction of Max Dehn, and in 1932 spent a fellowship year in Rome...
, and is named after Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
(for his 1845 paper describing the octonions).
As a symmetric space
Symmetric space
A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...
, the Cayley plane is F₄ / Spin(9), where F₄ is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-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...
(realized in F₄). As a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
it is also the quotient of a noncompact form of the group E₆ by a parabolic subgroup P1.
Properties
In the Cayley plane, lines and points may be defined in a natural way so that it becomes a projective spaceProjective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
. It is one of the rare projective spaces, where the Desargues' theorem
Desargues' theorem
In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
does not hold.