mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented...
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mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero...
12 KB (1,669 words) - 22:44, 1 September 2024
In mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is...
7 KB (818 words) - 08:14, 5 January 2024
The Geometry of the Octonions is a mathematics book on the octonions, a system of numbers generalizing the complex numbers and quaternions, presenting...
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Cayley–Dickson construction (section Octonions)
Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in...
18 KB (2,224 words) - 05:25, 6 July 2024
Okubo algebra (redirect from Pseudo-octonion algebra)
In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras...
7 KB (830 words) - 15:50, 21 February 2024
Grand Unified Theory (section Octonion representations)
generation of 16 fermions can be put into the form of an octonion with each element of the octonion being an 8-vector. If the 3 generations are then put in...
35 KB (4,567 words) - 11:47, 9 September 2024
Cayley plane (redirect from Octonion projective plane)
the octonions. The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describing the octonions. In...
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triangular faces, whose first stellation is the cube-octahedron compound. The octonions are a hypercomplex normed division algebra that are an extension of the...
62 KB (6,371 words) - 14:04, 16 September 2024
Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not...
23 KB (3,192 words) - 12:50, 13 September 2024
Hamilton's discovery of the quaternions and John T. Graves' discovery of the octonions in 1843 marked the beginning of higher-dimensional geometry. The rest...
34 KB (3,895 words) - 21:48, 16 September 2024
History of quaternions (section Octonions)
nevertheless, octonions are known by the name Cayley gave them – or as Cayley numbers. The major deduction from the existence of octonions was the eight...
19 KB (2,230 words) - 00:14, 14 July 2024
Seven-dimensional cross product (category Octonions)
The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. The seven-dimensional...
34 KB (4,899 words) - 07:33, 5 June 2024
systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real...
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while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all normed...
89 KB (11,605 words) - 23:07, 1 September 2024
as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as an 8‑tuple, and a sedenion can be represented as...
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For example, the tables for Z5 are: For other examples, see group, and octonion. Mokkan discovered at Heijō Palace suggest that the multiplication table...
28 KB (1,335 words) - 04:38, 18 August 2024
SO(8) (section Unit octonions)
bimultiplications by unit octonions (a bimultiplication being the composition of a left-multiplication and a right-multiplication by the same octonion and is unambiguously...
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not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product...
25 KB (2,964 words) - 22:33, 1 September 2024
10 {\textstyle 0\leq k\leq n-10} , GCD(k, n) = 1 The quaternions The octonions The sedenions The dual numbers (with an infinitesimal) Transfinite numbers...
58 KB (3,914 words) - 14:58, 15 September 2024
E8 lattice (section Integral octonions)
real octonions O. It is possible to define the concept of an integral octonion analogous to that of an integral quaternion. The integral octonions naturally...
22 KB (3,560 words) - 01:16, 25 June 2024
u(v+w)=uv+uw,(u+v)w=uw+vw.} In all algebras over a field, including the octonions and other non-associative algebras, multiplication distributes over addition...
19 KB (2,998 words) - 10:38, 19 March 2024
Hamilton developed the quaternions, and John T. Graves and Arthur Cayley the octonions. These are normed algebras which extend the complex numbers. Later it...
58 KB (7,005 words) - 16:53, 15 September 2024
One might think that S7, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic...
28 KB (4,027 words) - 22:10, 31 August 2024
Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M2) and Tr(M3) of the hermitian octonion matrix: M...
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alternative, but so too are some strictly non-associative algebras such as the octonions. Alternative algebras are so named because they are the algebras for which...
7 KB (1,076 words) - 09:57, 24 May 2024
William Rowan Hamilton, in which multiplication is not commutative, the octonions, in which multiplication is not associative in addition to not being commutative...
62 KB (7,747 words) - 11:27, 10 September 2024
Eight-dimensional space (section Octonions)
associated lattice. The kissing number in eight dimensions is 240. The octonions are a normed division algebra over the real numbers, the largest such...
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G2 (mathematics) (category Octonions)
The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen...
15 KB (2,056 words) - 18:40, 24 July 2024
Cross product (section Octonions)
for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector-valued...
75 KB (11,475 words) - 18:39, 5 September 2024