# Octonion

In mathematics, the**octonions**are a nonassociative extension of the quaternions. They form an 8-dimensional normed division algebra over the real numbers. The octonion algebra is often denoted

**O**.

Lacking the desirable property of associativity, the octonions receive far less attention then the quaternions. Despite this, the octonions retain importance for being related to a number of exceptional structures in mathematics, among them the exceptional Lie groups.

Table of contents |

2 Definition 3 Properties 4 Related Topics |

## History

The octonions were discovered in 1843 by John T. Graves, a friend of William Hamilton, who called them **octaves**. They were discovered independently by Arthur Cayley, who published the first paper on them in 1845. They are sometimes referred to as **Cayley numbers** or the **Cayley algebra**.

## Definition

The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the **unit octonions** {1, *i*, *j*, *k*, *l*, *li*, *lj*, *lk*}. That is, every octonion *x* can be written in the form

*x*=*x*_{0}+*x*_{1}*i*+*x*_{2}*j*+*x*_{3}*k*+*x*_{4}*l*+*x*_{5}*li*+*x*_{6}*lj*+*x*_{7}*lk*.

*x*

_{a}.

Addition of octonions is accomplished by adding corresponding coefficients, as with the complex numbers and quaternions. By linearity, multiplication of octonions is completely determined by the multiplication table for the unit octonions given below.

1 | i |
j |
k |
l |
li |
lj |
lk |

i |
−1 | k |
−j |
−li |
l |
−lk |
lj |

j |
−k |
−1 | i |
−lj |
lk |
l |
−li |

k |
j |
−i |
−1 | −lk |
−lj |
li |
l |

l |
li |
lj |
lk |
−1 | −i |
−j |
−k |

li |
−l |
−lk |
lj |
i |
−1 | −k |
j |

lj |
lk |
−l |
−li |
j |
k |
−1 | −i |

lk |
−lj |
li |
−l |
k |
−j |
\i |
−1 |

(Note that the basis for the octonions given here is not nearly as universal as the standand basis for the quaternions, however, nearly all other choices differ from this one only in order and sign.)

### Cayley-Dickson construction

A more systematic way of defining the octonions is via the Cayley-Dickson construction. Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. Addition is defined pairwise. The product of two pairs of quaternions (*a*,

*b*) and (

*c*,

*d*) is defined by

- (
*a*,*b*)(*c*,*d*) = (*ac*−*db*^{*})(*a*^{*}*d*+*cb*)

*z*

^{*}denotes the conjugate of the quaternion

*z*. This definition is equivalent to the one given above when the eight unit octonions are idenitifiied with the pairs

- (1,0), (
*i*,0), (*j*,0), (*k*,0), (0,1), (0,*i*), (0,*j*), (0,*k*)

### Fano plane mnemonic

A convenient mnemonic for remembering the products of unit octonions is given by the following diagram:

*i*,

*j*, and

*k*is considered a line) is called the Fano plane. Note that the lines are oriented in this diagram. The seven points correspond to the seven standard basis elements of Im(

**O**). Note that each pair of distinct points lies on a unique line and each line runs through exactly three points.

Let (*a*, *b*, *c*) be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by

*ab*=*c*and*ba*= −*c*

- 1 is the multiplicative identity,
*e*^{2}= −1 for each point in the diagram

**O**isomorphic to the quaternions

**H**.

### Conjugate, Norm, and Inverse

is given by*x*^{*}=*x*_{0}−*x*_{1}*i*−*x*_{2}*j*−*x*_{3}*k*−*x*_{4}*l*−*x*_{5}*li*−*x*_{6}*lj*−*x*_{7}*lk*.

**O**and satisfies (

*xy*)

^{*}=

*y*

^{*}

*x*

^{*}(note the change in order).

The *real part* of *x* is defined as ½(*x* + *x*^{*}) = *x*_{0} and the *imaginary part* as ½(*x* - *x*^{*}). The set of all purely imaginary octonions span a 7 dimension subspace of **O**, denoted Im(**O**).

The *norm* of the octonion *x* is defined as

*x*

^{*}

*x*=

*xx*

^{*}is always a nonnegative real number:

**R**

^{8}.

The existence of a norm on **O** implies the existence of inverses for every nonzero element of **O**. The inverse of *x* ≠ 0 is given by

*xx*

^{−1}=

*x*

^{−1}

*x*= 1.

## Properties

Octonionic multiplication is neither commutative:

*ij*= −*ji*

- (
*ij*)*l*= −*i*(*jl*)

**O**is isomorphic to

**R**,

**C**, or

**H**, all of which are associative.

The octonions do retain one important property shared by **R**, **C**, and **H**: the norm on **O** satisfies

It turns out that the only normed division algebras over the reals are **R**, **C**, **H**, and **O**. These four algebras also form the only alternative, finite-dimensional division algebra over the reals (up to isomorphism).

Not being associative, the nonzero elements of **O** do not form a group. They do, however, form a quasigroup, indeed a Moufang loop.

### Automorphisms

An automorphism,*A*, of the octonions is an invertible linear transformation of

**O**which satisfies

*A*(*xy*) =*A*(*x*)*A*(*y*).

**O**forms a group called

*G*

_{2}. The group

*G*

_{2}is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the five exceptional Lie groups.

*See also*: PSL(2,7) - the automorphism group of the Fano plane.

## Related Topics

### External links and references

- The Octonions - an article by John C. Baez
- Octonion Fractals - fractals generated using octonion mathematics

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