Octonion

In mathematics, the octonion number system extends the complex numbers into eight dimensions. It is represented using the symbol ${\displaystyle \mathbb {O} }$. 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. e₀ is a real number, and the others are imaginary units belonging to 7 different dimensions.

x = e₀ + e₁ + e₂ + e₃ + e₄ + e₅ + e₆ + e₇

Multiplication

The Fano plane

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]

e₅e₃ = e₆

e₇e₆ = e₁

The two examples below illustrate how a negative product is obtained when moving against the directions of the arrows in the Fano plane.^[2]

e₂e₄ = -e₆

e₇e₅ = -e₂

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 e₅, e₂, e₄ change when they are grouped differently (in order words, when the order of operations differ).^[2]

(e₅e₃)e₁ = (e₆)e₁ = e₇

e₅(e₃e₁) = e₅(e₂) = -e₇