# Chapter ᚖ(W): The third landing *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T23, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Its glossary cards](ch23-cards.md) · [Its references](ch23-refs.md) · [Previous chapter](ch22.md) · [Next chapter](ch24.md) *A footnote such as [r-23.3] is reference entry 23.3 and links to it. Section 23 of 28; chapters are named by their Ogham letter, never by number.* ᚖ – [Ór](ch23-cards.md#g-og-or): associated with the spindle tree, Section 23 ## Chapter ᚖ(W): The third landing > “The octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.” > — John C. Baez · The Octonions, *Bulletin of the American Mathematical Society*, 2002 [r-23.1](ch23-refs.md#r-23.1) Formula: ([(e₁e₁)e₂ − e₁(e₁e₂)](ch23-cards.md#g-assoc0) + ([(e₄e₁)(e₂e₄) − e₄((e₁e₂)e₄)](ch23-cards.md#g-moufang)))/([((e₁e₂)e₄ − e₁(e₂e₄))](ch23-cards.md#g-assoc)·([‖1 + e₁ + … + e₇‖²](ch23-cards.md#g-onorm))·([dim G₂](ch14-cards.md#g-g2))) #### Type of number Non-real object · Key concepts: [Octonions](ch23-cards.md#g-octonions): eight real components, the alternative law, and a Moufang identity. It includes symbols you may not have seen before, such as: e₁ … e₇ and [G₂](ch14-cards.md#g-g2). Zero here is an [associator](ch23-cards.md#g-assoc0) that vanishes: moving brackets usually changes an octonion product, except where a surviving law says it cannot. Here we are at 8. Leave the factors in their order and watch what happens when we move a pair of brackets. Octonion multiplication is not generally associative, so that small rearrangement can change the answer. Several more particular agreements survive, though. We can stop at this landing long enough to learn which arrangements still hold. John Graves found [the octonions](ch23-cards.md#g-octonions) shortly after Hamilton’s discovery, and Arthur Cayley discovered them independently and published them in 1845. [The Cayley–Dickson construction](ch24-cards.md#g-cdrule) expresses them as pairs of [quaternions](ch22-cards.md#g-quaternions). An octonion is a₀ + a₁e₁ + ⋯ + a₇e₇, with eight real coefficients. The e’s are basis directions in the algebra. Their multiplication table is [the Fano plane](ch23-cards.md#g-fano): seven points on seven lines, three units to a line, each line a copy of the [quaternions](ch22-cards.md#g-quaternions), and moving a unit off its line is what breaks associativity. [r-23.9](ch23-refs.md#r-23.9) [r-23.2](ch23-refs.md#r-23.2) Field notes Associativity would let us replace (ab)c by a(bc) for any three inputs; [octonions](ch23-cards.md#g-octonions) do not allow that in general. The difference (ab)c − a(bc), called the *associator*, measures what moving the brackets does. The numerator shows two agreements that survive; the denominator shows the failure. #### Numerator [(e₁e₁)e₂ − e₁(e₁e₂)](ch23-cards.md#g-assoc0) = 0; the alternative law, a surviving associativity promise [the octonions](ch23-cards.md#g-octonions) keep. [Octonions](ch23-cards.md#g-octonions) are *alternative*: if two of the three factors agree, the associator vanishes. With e₁ repeated, both groupings give −e₂. More broadly, any subalgebra generated by two [octonions](ch23-cards.md#g-octonions) is associative, even though the full eight-dimensional algebra is not. [r-23.3](ch23-refs.md#r-23.3) [(e₄e₁)(e₂e₄) − e₄((e₁e₂)e₄)](ch23-cards.md#g-moufang) = 0; a Moufang identity, chosen because it holds for three directions that do not associate. [The octonions](ch23-cards.md#g-octonions) obey Ruth Moufang’s identities, among them (zx)(yz) = z((xy)z). Like the alternative law, it repeats a factor, z at both ends, and every algebra that obeys the alternative law obeys Moufang’s identities too. Taking z = e₄, x = e₁, y = e₂ gives zero, even though the same three units, with nothing repeated, fail to associate in the denominator below. [r-23.5](ch23-refs.md#r-23.5) Moufang came to these identities through geometry: she built a projective plane out of the octonions, and the plane, like the algebra, holds together after one familiar law has gone. [r-23.16](ch23-refs.md#r-23.16) #### Denominator A failed regrouping, a length that still multiplies, and symmetries that preserve every product: three views of [octonion multiplication](ch23-cards.md#g-octonions). [((e₁e₂)e₄ − e₁(e₂e₄))](ch23-cards.md#g-assoc) = 2e₇; the failure itself: remove the repetition and associativity breaks. Under the book’s [Cayley–Dickson table](ch24-cards.md#g-cdrule), (e₁e₂)e₄ = e₇ and e₁(e₂e₄) = −e₇, so moving the brackets while keeping the order e₁, e₂, e₄ changes the answer by 2e₇. [r-23.4](ch23-refs.md#r-23.4) [‖1 + e₁ + … + e₇‖²](ch23-cards.md#g-onorm) = 8; the eight-square identity of Degen, Graves, and Cayley, a strong property that survived the loss of associativity. Eight coefficients of 1 give a squared norm of 8. The norm still multiplies: ‖ab‖ = ‖a‖‖b‖. Written in coordinates, that is the eight-square identity—a product of two sums of eight squares is again a sum of eight squares. [r-23.6](ch23-refs.md#r-23.6) [dim G₂](ch14-cards.md#g-g2) = 14; G₂, back from Chapter ᚌ(G), because the connection is literal. [G₂](ch14-cards.md#g-g2) is the symmetry group of the octonions: every way of moving their directions that leaves every product intact, so that if xy = z, the moved x times the moved y is the moved z. It is the smallest of the exceptional groups. 14 is its dimension: the number of independent dials it takes to set one of these symmetries, not a count of them. [r-23.8](ch23-refs.md#r-23.8) Two more rarities come with the octonions. The seven imaginary directions carry a [cross product](ch23-cards.md#g-cross7), a cousin of the one in ordinary three-dimensional space; a product like it exists only in three dimensions and in seven. [r-23.13](ch23-refs.md#r-23.13) And the [integral octonions](ch23-cards.md#g-cayleyint), the octonions’ answer to whole numbers, have exactly 240 units of norm 1, which sit at the 240 roots of [E₈](ch14-cards.md#g-e8), the same E₈ whose 248 dimensions stand in Chapter ᚌ(G)’s denominator. [r-23.11](ch23-refs.md#r-23.11) The only fraction is (0 + 0)/224e₇: 2e₇ × 8 × 14 = 224e₇, a nonzero octonion with inverse −e₇/224. [r-23.15](ch23-refs.md#r-23.15) [r-23.7](ch23-refs.md#r-23.7) **Fine print.** A concrete associator depends on the basis labeling. Other octonion tables rename or re-sign the units and move the nonzero associator to a different triple; the failure itself does not depend on labels. The book keeps one table here and in the sedenion chapter so that a relabeled direction will not masquerade as a new rule. [r-23.4](ch23-refs.md#r-23.4) The real factors cause no regrouping problem, and multiplying the zero numerator by the inverse gives octonion zero on either side. [Hurwitz’s theorem](ch23-cards.md#g-hurwitz) says that among finite-dimensional algebras over the reals with a 1, exactly four have a length that multiplies, ‖ab‖ = ‖a‖‖b‖, so that every nonzero element can be divided by: ℝ, ℂ, ℍ, and 𝕆, of dimensions 1, 2, 4, and 8. [The octonions](ch23-cards.md#g-octonions) are the last of them. The doubling goes on past 8; what ends is the list of algebras whose lengths multiply. [r-23.7](ch23-refs.md#r-23.7) [r-23.14](ch23-refs.md#r-23.14) What this chapter adds to the formula: (0 + 0) / 224e₇ = 0 Eight components kept a multiplicative norm and the alternative law, and gave up associativity. One more landing before we go aboard. Follow the ladder up to 16, beside the airlock. I want to show you two things there that are perfectly substantial on their own, yet multiply to nothing. We have met zero many ways by now; this one still makes me stop and look. Watch and read - [Cohl Furey on the Octonions and Particle Physics](https://www.youtube.com/watch?v=_E2iiuunK-E) Quanta Magazine: what an eight-dimensional number is and why physicists keep looking at it [r-23.17](ch23-refs.md#r-23.17) - [alternative algebra -- featuring the octonions!](https://www.youtube.com/watch?v=ZC7YofZp-cw) Michael Penn: the alternative law, a surviving associativity promise the octonions keep, worked on the board - [Octonion](https://en.wikipedia.org/wiki/Octonion) Wikipedia - [Hurwitz’s theorem (composition algebras)](https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)) Wikipedia