The octonions are usually represented by the capital letter O, using boldface O or blackboard bold

Octonions have applications in fields such as string theory, special relativity and quantum logic.

The octonions were discovered in December 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions.

"[1] Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 26 December 1843.

The product of each pair of terms can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one (given both by Arthur Cayley in 1845 and John T. Graves in 1843):[7] Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which e0 is an operand.

The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used.

A common choice is to use the definition invariant under the 7 cycle (1234567) with e1e2 = e4 by using the triangular multiplication diagram, or Fano plane below that also shows the sorted list of 1 2 4 based 7-cycle triads and its associated multiplication matrices in both en and

The multiplication is then given by e∞ = 1 and e1e2 = e4, and all expressions obtained from this by adding a constant (modulo 7) to all subscripts: In other words using the seven triples (1 2 4) (2 3 5) (3 4 6) (4 5 0) ( 5 6 1) (6 0 2) (0 1 3) .

These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, GF(2).

and IJKL multiplication matrices also includes the geometric algebra basis with signature (− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element): or alternatively: in which the lower case items {i, j, k, l} are vectors (e.g. {

In keeping ★ = i j k l associative and thus not reducing the 4 dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for ★.

A convenient mnemonic for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves.

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

satisfies This equation means that the octonions form a composition algebra.

The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the sedenions) all fail to satisfy this property.

Wider number systems exist which have a multiplicative modulus (for example, 16 dimensional conic sedenions).

The octonions play a significant role in the classification and construction of other mathematical entities.

[16] The set of self-adjoint 3 × 3 octonionic matrices, equipped with a symmetrized matrix product, defines the Albert algebra.

In discrete mathematics, the octonions provide an elementary derivation of the Leech lattice, and thus they are closely related to the sporadic simple groups.

For example, in the 1970s, attempts were made to understand quarks by way of an octonionic Hilbert space.

[19] It is known that the octonions, and the fact that only four normed division algebras can exist, relates to the spacetime dimensions in which supersymmetric quantum field theories can be constructed.

[20][21] Also, attempts have been made to obtain the Standard Model of elementary particle physics from octonionic constructions, for example using the "Dixon algebra"

[22][23] Octonions have also arisen in the study of black hole entropy, quantum information science,[24][25] string theory,[26] and image processing.

[28] Deep octonion networks provide a means of efficient and compact expression in machine learning applications.

This gives a nonassociative algebra over the integers called the Gravesian octonions.

Label the eight basis vectors by the points of the projective line over the field with seven elements.

Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order.

There are seven ways to do this, giving seven maximal orders, which are all equivalent under cyclic permutations of the seven coordinates 0123456.

(Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were eight maximal orders rather than seven, but as Coxeter (1946) pointed out they are not closed under multiplication; this mistake occurs in several published papers.)

The Kirmse integers and the seven maximal orders are all isometric to the E8 lattice rescaled by a factor of 1⁄√2.