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

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

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

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...

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 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...

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

{C} ,} the quaternions H , {\displaystyle \mathbb {H} ,} and lastly the octonions O , {\displaystyle \mathbb {O} ,} where the dimensions of these spaces...

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...

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...

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

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...

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

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...

to construct some of the other lattices of rank 24. If L is the set of octonions with coordinates on the E 8 {\displaystyle E_{8}} lattice, then the Leech...

defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution...

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

Furthermore, in contrast to the octonions, both algebras do not even have the property of being alternative. Whereas octonion unit multiplication patterns...

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...

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

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...

associated lattice. The kissing number in eight dimensions is 240. The octonions are a normed division algebra over the real numbers, the largest such...

Moufang loop. The nonzero octonions form a nonassociative Moufang loop under octonion multiplication. The subset of unit norm octonions (forming a 7-sphere...

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...

spaces have a number of special properties, many of them related to the octonions. An especially distinctive property is that a cross product can be defined...

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

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

Fano plane can be used to describe multiplication of octonions, so G acts on the set of octonion multiplication tables. The Klein quartic is the projective...

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...