Octonion
In mathematics, octonions are the largest of the four normed division algebras over the real numbers, forming an 8-dimensional nonassociative algebra that extends the quaternions through the Cayley-Dickson construction.[1] They consist of elements expressible as a_0 + a_1 e_1 + a_2 e_2 + \dots + a_7 e_7, where the a_i are real numbers and the e_i satisfy specific multiplication rules derived from the Fano plane, a projective plane of order 2 that encodes their nonassociative structure.[2] Discovered independently by John T. Graves in 1843 and Arthur Cayley in 1845, octonions—also known as Cayley numbers—were initially motivated by the quest to generalize complex numbers and quaternions to higher dimensions while preserving a norm that makes them a division algebra, meaning every nonzero element has a multiplicative inverse.[3] Unlike the real numbers (1-dimensional, commutative and associative), complex numbers (2-dimensional, commutative and associative), and quaternions (4-dimensional, noncommutative but associative), octonions lose both commutativity and associativity, yet they retain key properties such as the existence of a Euclidean norm \|x\|^2 = x \bar{x} and the absence of zero divisors, as affirmed by Hurwitz's theorem limiting such algebras to dimensions 1, 2, 4, or 8.[4] Their multiplication can be visualized using the Fano plane, where basis elements correspond to points and lines dictate the rules, such as e_i e_j = -e_j e_i for i \neq j and cyclic permutations for triples.[1] Despite their complexity, octonions underpin significant structures in advanced mathematics and theoretical physics. In algebra, they relate to exceptional Lie groups like G_2 (the automorphism group of the octonions) and appear in Bott periodicity, which describes the topology of stable homotopy groups.[3] In physics, octonions have been explored for unifying fundamental forces, appearing in formulations of string theory, supersymmetry, and models attempting to incorporate exceptional groups into the standard model, though their nonassociativity poses challenges for practical computations.[5] Further iterations via Cayley-Dickson yield sedenions and beyond, but these lose the division property, highlighting the octonions' unique position as the highest-dimensional normed division algebra.[6]Historical Development
Early Formulations
The discovery of octonions traces back to the early 1840s, amid the excitement surrounding William Rowan Hamilton's invention of quaternions in October 1843. John T. Graves, an Irish lawyer and amateur mathematician who was a close friend and correspondent of Hamilton, quickly pursued extensions of this four-dimensional system. Motivated by the desire to generalize algebraic structures for solving higher-degree polynomial equations and developing a theory of imaginaries in dimensions that are powers of two, Graves formulated the octonions— an eight-dimensional algebra—by late December 1843.[7] In a letter to Hamilton dated December 26, 1843, Graves announced this breakthrough, describing it as a natural progression beyond quaternions to what he termed "octaves."[8] Graves' work was deeply influenced by Hamilton's quaternions, which Hamilton had privately communicated to him shortly after their discovery on October 16, 1843. An earlier exchange on October 26, 1843, captured Graves' enthusiasm for further algebraic exploration, where he wrote to Hamilton: "If with your alchemy you can make three pounds of gold, why should you stop there?" This encouragement reflected Graves' interest in constructing normed division algebras capable of representing sums of squares identities, such as the theorem for eight squares, which he proved as part of his octonion formulation.[7] In a follow-up letter on January 18, 1844, Graves elaborated on these ideas, correcting initial errors and emphasizing the eight-dimensional extension's potential for broader algebraic applications, including roots of polynomials.[8] Although Graves delayed publication—sharing details only in private correspondence and later in the Proceedings of the Royal Irish Academy in 1847—his insights laid the groundwork for recognizing octonions as a composition algebra.[8] Independently, Arthur Cayley, a young English mathematician, arrived at a similar eight-dimensional algebra in early 1845 while studying quaternions and hyperelliptic functions. Cayley published his findings in March 1845 in the Philosophical Magazine, naming the system "octaves" and highlighting its non-commutative multiplication, though his initial presentation contained some inaccuracies.[7] He viewed octonions as an extension of quaternionic algebra, motivated by the same quest for higher-dimensional number systems that preserved certain multiplicative norms, akin to those in complex numbers and quaternions. Cayley acknowledged the parallels to Graves' unpublished work after learning of it through Hamilton but proceeded with his own exposition, which popularized the structure and led to octonions often being called Cayley numbers.[7] This independent confirmation underscored the algebraic motivations of the era, focusing on polynomial solvability and geometric interpretations without delving into explicit rules.[8]20th-Century Advances
In 1923, Adolf Hurwitz's theorem on composition algebras demonstrated that the only real finite-dimensional normed division algebras are those of dimensions 1, 2, 4, and 8, corresponding to the real numbers, complex numbers, quaternions, and octonions, establishing the octonions as the largest such algebra.[9] Building on this, Leonard Dickson in the 1920s formalized the theory of alternative algebras, a class of non-associative algebras satisfying weaker associativity conditions than full associativity; the octonions exemplify this structure, as their multiplication is alternative but not associative.[10] Élie Cartan's early 20th-century work (1908–1920s), particularly his classification of simple Lie algebras and studies on exceptional groups, revealed the intimate connection between octonions and exceptional Lie groups; he noted in 1908 that G₂ is the automorphism group of the octonions, with F₄ incorporating octonionic structures in its representations (detailed in 1925).[1] In the mid-20th century, mathematicians deepened the links between octonions, spinors, and Clifford algebras, as seen in Claude Chevalley's 1954 exposition, which used Clifford algebras to describe spinor representations and highlighted the octonions' role in the triality principle underlying G₂ symmetries.[11] These theoretical advances culminated in John Baez's 2002 survey, which unified the octonions' historical development with their roles in physics and geometry, underscoring their unique position among division algebras.[12]Definition and Construction
Cayley-Dickson Process
The Cayley-Dickson process is a recursive algebraic construction that generates higher-dimensional normed division algebras by doubling the dimension at each step, starting from the real numbers. This method was first introduced by Arthur Cayley in 1845, who used it to define the octonions as pairs of quaternions, and later formalized and generalized by Leonard Dickson in 1919 to produce a sequence of algebras.[13] The construction begins with the real numbers \mathbb{R}, a 1-dimensional algebra over itself with the standard addition and multiplication. Applying the process yields the complex numbers \mathbb{C}, a 2-dimensional algebra. Elements of \mathbb{C} are represented as ordered pairs (a, b) where a, b \in \mathbb{R}, with addition defined componentwise: (a, b) + (c, d) = (a + c, b + d). Multiplication is given by (a, b)(c, d) = (ac - bd, ad + bc), introducing the imaginary unit i via the basis \{1, i\} where i = (0, 1) and i^2 = -1. This algebra is associative and commutative. Doubling again produces the quaternions \mathbb{H}, a 4-dimensional algebra over \mathbb{R}. Here, elements are pairs of complex numbers, (z, w) with z, w \in \mathbb{C}, addition componentwise, and multiplication (z, w)(u, v) = (z u - \bar{v} w, v z + w \bar{u}), where \bar{\cdot} denotes complex conjugation. This introduces basis elements \{1, i, j, k\} with j = (0, 1), k = i j = (i, 1), satisfying i^2 = j^2 = k^2 = i j k = -1. The quaternions are associative but non-commutative, as i j = k while j i = -k. Although the full details of quaternion arithmetic are assumed known, this step preserves the normed division algebra property from \mathbb{C}.[13] The process extends to the octonions \mathbb{O}, an 8-dimensional algebra, by treating elements as pairs of quaternions (p, q) with p, q \in \mathbb{H}. Addition is componentwise, and the general doubling formula, in its parameterized form, defines multiplication for elements from an algebra A with involution (conjugation) as: (a_1, b_1)(a_2, b_2) = (a_1 a_2 - \gamma \overline{b_2} b_1, \, b_2 a_1 + b_1 \overline{a_2}), where \gamma \in \mathbb{R}^\times is a parameter and \overline{\cdot} is the conjugation from A. For the octonions, \gamma = 1, yielding (p, q)(r, s) = (p r - \bar{s} q, s p + q \bar{r}). This introduces seven imaginary units e_1 = i, e_2 = j, e_3 = k, e_4 = (0,1), e_5 = (0,i), e_6 = (0,j), e_7 = (0,k), extending the basis to 8 dimensions. The construction preserves the norm but introduces non-associativity at this stage.[13] Non-associativity arises because the quaternions are non-commutative, causing the associator (x y) z - x (y z) to be nonzero in general for octonions. For example, with basis elements, (e_2 e_4) e_1 = e_7 while e_2 (e_4 e_1) = -e_7, demonstrating (e_2 e_4) e_1 \neq e_2 (e_4 e_1). This property emerges precisely at the octonion level, as prior algebras \mathbb{R}, \mathbb{C}, \mathbb{H} are associative, but the doubling process beyond quaternions sacrifices associativity while maintaining alternative properties and the division algebra structure.[13]Basis Representation
The octonions form an 8-dimensional algebra over the real numbers with a standard basis {1, e_1, e_2, \dots, e_7}, where 1 denotes the multiplicative identity (or scalar unit) and the elements e_1 through e_7 are imaginary units satisfying e_i^2 = -1 for each i = 1, \dots, 7.[14] A general octonion o is then expressed in coordinates aso = a_0 \cdot 1 + a_1 e_1 + a_2 e_2 + \dots + a_7 e_7,
where the coefficients a_0, a_1, \dots, a_7 \in \mathbb{R}.[15] This basis representation provides a direct embedding of the octonions into the space of real 8-tuples, facilitating their study as hypercomplex numbers extending the quaternions. As a vector space over \mathbb{R}, the octonions are isomorphic to \mathbb{R}^8, with the basis {1, e_1, \dots, e_7} serving as a linear coordinate frame that spans this 8-dimensional structure under real scalar multiplication and addition.[14] The choice of seven imaginary units reflects the unique dimensionality of the octonions among normed division algebras, obtained via the Cayley-Dickson doubling process from lower-dimensional algebras.[15] The basis elements are orthogonal with respect to the standard inner product on the octonions, which is the positive definite Euclidean form inherited from \mathbb{R}^8: for octonions o_1 = \sum_{i=0}^7 a_i e_i and o_2 = \sum_{i=0}^7 b_i e_i (with e_0 := 1),
\langle o_1, o_2 \rangle = \sum_{i=0}^7 a_i b_i.
This bilinear form renders the basis orthonormal, as \langle e_i, e_j \rangle = \delta_{ij} for i, j = 0, \dots, 7, where \delta_{ij} is the Kronecker delta.[15] The restriction to 8 dimensions for the octonions arises because, by Hurwitz's theorem on the composition of quadratic forms, finite-dimensional normed division algebras over \mathbb{R} exist only in dimensions 1, 2, 4, and 8, with the octonions realizing the maximum.[9] This theorem underscores the exceptional nature of the octonion basis, whose 7-dimensional imaginary subspace connects to deeper geometric structures like the Fano plane in defining the algebra.[14]
Arithmetic Operations
Addition and Subtraction
Octonions form an 8-dimensional vector space over the real numbers, with addition and subtraction defined component-wise in the standard basis \{1, e_1, e_2, e_3, e_4, e_5, e_6, e_7\}.[15][14] Any octonion can be expressed as o = a_0 + a_1 e_1 + a_2 e_2 + a_3 e_3 + a_4 e_4 + a_5 e_5 + a_6 e_6 + a_7 e_7, where each a_i \in \mathbb{R}. For two octonions o = (a_0, a_1, \dots, a_7) and p = (b_0, b_1, \dots, b_7) in this basis representation, their sum is o + p = (a_0 + b_0, a_1 + b_1, \dots, a_7 + b_7), or equivalently, o + p = (a_0 + b_0) + (a_1 + b_1) e_1 + (a_2 + b_2) e_2 + \cdots + (a_7 + b_7) e_7. This operation is bilinear and commutative.[15] Subtraction follows similarly as component-wise difference: o - p = (a_0 - b_0, a_1 - b_1, \dots, a_7 - b_7). The additive inverse of an octonion o is -o = (-a_0, -a_1, \dots, -a_7) = -a_0 - a_1 e_1 - \cdots - a_7 e_7, allowing subtraction to be expressed as addition with the inverse. Under addition, the set of octonions forms an abelian group, with the zero octonion $0 = (0,0,\dots,0) as the identity element; addition is associative, commutative, and every element has an inverse.[14][15] Scalar multiplication by a real number r \in \mathbb{R} distributes over the components: r o = (r a_0, r a_1, \dots, r a_7) = r a_0 + r a_1 e_1 + \cdots + r a_7 e_7. This linearity confirms the octonions' structure as a real vector space, enabling standard linear combinations.[15][14] For example, consider the octonions o = 1 + 2 e_1 and p = 3 + 4 e_2. Their sum is o + p = (1 + 3) + 2 e_1 + 4 e_2 = 4 + 2 e_1 + 4 e_2, while o - p = (1 - 3) + 2 e_1 - 4 e_2 = -2 + 2 e_1 - 4 e_2. Scaling o by r = 5 yields $5 o = 5 + 10 e_1. These operations illustrate the straightforward, vector-like arithmetic.Multiplication Rules
Octonion multiplication is defined to be bilinear over the reals and extends the quaternion multiplication while introducing a new imaginary unit. An arbitrary octonion can be written as o = x_0 + \sum_{i=1}^7 x_i e_i, where x_0, x_i \in \mathbb{R} and \{1, e_1, \dots, e_7\} is the standard basis, with the scalar part x_0 multiplying as real numbers and commuting with all elements. The product o_1 o_2 for two octonions is then (x_0 y_0 - \sum x_i y_i) + \sum_k z_k e_k, where the coefficients z_k are determined by the basis multiplications.[16] The multiplication of basis elements satisfies e_i^2 = -1 for i = 1, \dots, 7, and for i \neq j, e_i e_j = -\delta_{ij} + \sum_{k=1}^7 f_{ijk} e_k, where f_{ijk} are the totally antisymmetric structure constants of the octonion algebra (with f_{ijk} = 0 if i = j). These constants encode the non-commutativity, as e_j e_i = -e_i e_j for i \neq j. The explicit products are given by the following table for the imaginary units:| e_i \backslash e_j | e_1 | e_2 | e_3 | e_4 | e_5 | e_6 | e_7 |
|---|---|---|---|---|---|---|---|
| e_1 | -1 | e_4 | e_7 | -e_2 | e_6 | -e_5 | -e_3 |
| e_2 | -e_4 | -1 | e_5 | e_1 | -e_3 | e_7 | -e_6 |
| e_3 | -e_7 | -e_5 | -1 | e_6 | e_2 | -e_4 | e_1 |
| e_4 | e_2 | -e_1 | -e_6 | -1 | e_7 | e_3 | -e_5 |
| e_5 | -e_6 | e_3 | -e_2 | -e_7 | -1 | e_1 | e_4 |
| e_6 | e_5 | -e_7 | e_4 | -e_3 | -e_1 | -1 | e_2 |
| e_7 | e_3 | e_6 | -e_1 | e_5 | -e_4 | -e_2 | -1 |