mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented...

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...

mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero...

The Geometry of the Octonions is a mathematics book on the octonions, a system of numbers generalizing the complex numbers and quaternions, presenting...

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...

Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in...

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...

sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mathematically expressed as S = C D ( O , 1 ) {\displaystyle...

bimultiplications by unit octonions (a bimultiplication being the composition of a left-multiplication and a right-multiplication by the same octonion and is unambiguously...

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...

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...

also be defined as the following. An algebra of dimension 4 over the octonions O {\displaystyle \mathbb {O} } : ∑ i = 0 3 a i ⋅ e i {\displaystyle \sum...

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...

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...

triangular faces, whose first stellation is the cube-octahedron compound. The octonions are a hypercomplex normed division algebra that are an extension of the...

systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real...

for the hypercomplex numbers of dimension 8 or greater, including the octonions, sedenions, and trigintaduonions, multiplication is generally not associative...

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...

that of the quaternion multiplication table. For further examples, see Octonion § Multiplication, Sedenion § Multiplication, and Trigintaduonion § Multiplication...

not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product...

while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all normed...

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...

The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. The seven-dimensional...

10 {\textstyle 0\leq k\leq n-10} , GCD(k, n) = 1 The quaternions The octonions The sedenions The trigintaduonions The dual numbers (with an infinitesimal)...

Hamilton developed the quaternions, and John T. Graves and Arthur Cayley the octonions. These are normed algebras which extend the complex numbers. Later it...

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...

largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The...

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...

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...

(coquaternions), split-octonions, biquaternions C ⊗ H {\displaystyle \mathbb {C} \otimes \mathbb {H} } , and complex octonions C ⊗ O {\displaystyle \mathbb...