Octonion
In mathematics, the octonion number system extends the complex numbers into eight dimensions. It is represented using the symbol . The 16-dimensional sedenions come after the octonions.
History
The octonions were first described by Irish mathematician John T. Graves in 1843, who originally called them "octaves".[1] They were independently described by Arthur Cayley in 1845.
Introduction
The octonions take on the following form, with 8 total elements. Template:Math is a real number, and the others are imaginary units belonging to 7 different dimensions.
Multiplication
The Fano plane is a diagram that shows how another octonion element is obtained when two octonion elements are multiplied with each other.
The two examples below illustrate how a positive product is obtained when moving along with directions of the arrows in the Fano plane.[2]
The two examples below illustrate how a negative product is obtained when moving against the directions of the arrows in the Fano plane.[2]
Both quaternions and octonions are non-commutative, but octonions are also non-associative. However, quaternions are associative. The example below shows how the results of multiplying Template:Math, Template:Math, Template:Math change when they are grouped differently (in order words, when the order of operations differ).[2]