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# Glossary cards — Chapter ᚗ(V): The top landing

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T24, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch24.md)

*Sedenions 𝕊: sixteen real components, zero divisors, the loss of a multiplicative norm, and powers that remain unambiguous. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Uilleann — `ᚗ` {#g-og-uilleann}

*Ogham letter*

A supplementary letter whose name means ‘elbow’, and whose kenning is a play on the name.

Uilleand means ‘elbow’, and is the third of the five forfeda. Two lists have it: túthmar fid, “fragrant tree” and cubat oll, “great elbow, or cubit,” the second of which is a play on the letter name rather than a clue to anything beyond it. The Auraicept glosses edleand, honeysuckle. It appears in only one orthodox inscription, at Teeromoyle in County Kerry, though some later scholastic inscriptions use all five forfeda. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.18](ch02-refs.md#r-2.18), [2.19](ch02-refs.md#r-2.19)

### sedenion zero divisors — `(e₃+e₁₀)(e₆−e₁₅)` {#g-zd}

*Integer or rational*

Two nonzero sedenions whose product is 0, impossible in ℝ, ℂ, ℍ, or 𝕆.

The sedenions are the fourth Cayley–Dickson doubling, and they have zero divisors: nonzero elements whose product is zero. The chapter’s example, (e₃ + e₁₀)(e₆ − e₁₅) = 0 under the Cayley–Dickson convention, is verified by exact integer computation; there are 84 ways to split a set of four units into two pairs so that some product (e_{a} ± e_{b})(e_{c} ± e_{d}) vanishes, 42 four-unit sets with two splittings each, and each splitting vanishes for exactly two of its four sign patterns, giving 168 signed unordered pairs and 336 when factor order is counted. Guillermo Moreno showed in 1998 that the space of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact form of the exceptional Lie group G₂, the same group that is the octonions’ symmetry group in Chapter ᚖ(W). The product is exactly 0, a rational number computed in 𝕊, hence the integer color.

Cites: [24.2](ch24-refs.md#r-24.2), [24.9](ch24-refs.md#r-24.9)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [the dimension of G₂](ch14-cards.md#g-g2), [homeomorphism](ch12-cards.md#g-homeo), [the octonions](ch23-cards.md#g-octonions), [the sedenions](ch24-cards.md#g-sedenions)

### norms of the two zero divisors — `‖e₃+e₁₀‖·‖e₆−e₁₅‖` {#g-zdnorm}

*Integer or rational*

√2 · √2 = 2, while the product of the two factors has norm 0: the sedenion norm is not multiplicative.

The norm of a sedenion is the square root of the sum of the squares of its sixteen coordinates, so ‖e₃ + e₁₀‖ = ‖e₆ − e₁₅‖ = √2 and the product of the norms is 2. Hurwitz’s theorem of 1898 says the norm is multiplicative only in dimensions 1, 2, 4, and 8, and this pair is the demonstration: the product of the norms is 2 while the norm of the product is 0. Value 2, an integer, and a factor of the denominator, so it must be nonzero, which it is.

Cites: [24.4](ch24-refs.md#r-24.4), [24.5](ch24-refs.md#r-24.5)

See also: [Hurwitz’s theorem](ch23-cards.md#g-hurwitz), [square root of two](ch04-cards.md#g-sqrt2)

### power associativity survives — `(aa)a − a(aa), a = e₃ + e₁₀` {#g-powassoc}

*Integer or rational*

0: the sedenions are no longer alternative, but powers of a single element are still unambiguous, so (aa)a = a(aa); they are also flexible, (ab)a = a(ba).

Each Cayley–Dickson doubling loses something: ℂ loses order, ℍ commutativity, 𝕆 associativity, 𝕊 the alternative law and the absence of zero divisors. What every rung keeps is power associativity, a·a·a meaning one thing however it is bracketed, and flexibility, (ab)a = a(ba); Schafer proved in 1954 that all Cayley–Dickson algebras are flexible, and all of them are power-associative. For a = e₃ + e₁₀, one of the zero divisors, (aa)a − a(aa) = 0 (verified by exact integer computation). Exactly 0 in 𝕊, read as the scalar 0, an integer, riding beside the zero divisors.

Cites: [24.6](ch24-refs.md#r-24.6), [24.7](ch24-refs.md#r-24.7)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [the sedenions](ch24-cards.md#g-sedenions), [sedenion zero divisors](ch24-cards.md#g-zd)

### the sedenion zero-divisor pairs — `84` {#g-zd84}

*Integer or rational*

The number of ways to choose the two unit pairs {a, b} and {c, d} so that (e_{a} ± e_{b})(e_{c} ± e_{d}) can vanish, ignoring factor order and which ± signs do it: 42 four-unit sets, each split into two pairs in two different ways.

Wikipedia’s table lists 84 zero divisors of the shape (e_{a} ± e_{b})(e_{c} ± e_{d}) = 0 among the fifteen imaginary sedenion units. My own exact count agrees, and finds that the 84 come from only 42 distinct four-element sets, none of which contains the real unit: each set, like {e₃, e₁₀, e₆, e₁₅}, splits into a vanishing product in exactly two of its three possible ways, (e₃ + e₁₀)(e₆ − e₁₅) and (e₃ + e₁₅)(e₆ + e₁₀), and each of those splittings vanishes for exactly two of its four sign patterns, which is where the 168 signed pairs and the 336 ordered ones come from. Robert de Marrais’s 2000 catalog reaches the same two numbers by another route: 42 “assessor” planes, each spanned by two units, carrying 168 unit zero divisors, four to a plane. Moreno’s 1998 theorem describes the space of pairs of norm-one sedenions that multiply to zero as homeomorphic to the compact form of G₂. Value 84, an integer, a factor of the denominator.

Cites: [24.12](ch24-refs.md#r-24.12), [24.8](ch24-refs.md#r-24.8), [24.9](ch24-refs.md#r-24.9)

See also: [the dimension of G₂](ch14-cards.md#g-g2), [homeomorphism](ch12-cards.md#g-homeo), [the sedenions](ch24-cards.md#g-sedenions), [sedenion zero divisors](ch24-cards.md#g-zd)

### the Cayley–Dickson doubling rule — `(a, b)(c, d) = (ac − d̄b, da + bc̄)` {#g-cdrule}

*Function or operator*

The one recipe that builds every rung of the ladder: a pair from the rung below multiplies by this rule, with the conjugate (a, b)̄ = (ā, −b).

Leonard Dickson gave the general doubling in 1919, showing how the octonions can be built as a two-dimensional algebra over the quaternions; the construction carries Arthur Cayley’s name as well, for the octonions he printed in 1845. Applied to ℝ it gives ℂ, then ℍ, 𝕆, 𝕊, the 32-dimensional trigintaduonions, and so on forever; the book uses this exact rule, with this sign convention, for every product it computes in Chapters ᚔ(I) to ᚗ(V), which is why its octonion table differs from Baez’s Fano labeling. A function, not a value.

Cites: [24.10](ch24-refs.md#r-24.10), [24.15](ch24-refs.md#r-24.15)

See also: [the octonions](ch23-cards.md#g-octonions), [the quaternions](ch22-cards.md#g-quaternions), [the trigintaduonions](ch24-cards.md#g-trig)

### the trigintaduonions — `𝕋` {#g-trig}

*Non-real object*

The 32-dimensional rung above the sedenions, and the ladder keeps going: 64, 128, and every power of two above, each with zero divisors.

Doubling the sedenions gives a 32-dimensional algebra, the trigintaduonions, written 𝕋, and the construction never stops, giving an algebra of dimension 2ⁿ for every n. From the sedenions on, every rung has zero divisors (the same enumeration that gives the sedenions their 84 gives 1,260 here, which is the number Wikipedia reports, derived from 147 distinct triples), is neither alternative nor associative nor commutative, and keeps power associativity, flexibility, and distributivity. A non-real object, the first rung the book does not climb.

Cites: [24.1](ch24-refs.md#r-24.1), [24.11](ch24-refs.md#r-24.11)

See also: [power associativity survives](ch24-cards.md#g-powassoc), [the sedenions](ch24-cards.md#g-sedenions), [sedenion zero divisors](ch24-cards.md#g-zd)

### the sedenions — `𝕊 = {(a, b) : a,b ∈ 𝕆}` {#g-sedenions}

*Non-real object*

The sixteen-dimensional Cayley–Dickson algebra built from pairs of octonions: noncommutative, nonassociative, and nonalternative, with zero divisors, but still power-associative and flexible.

A sedenion is an ordered pair of octonions and therefore has sixteen real coordinates. It is the fourth Cayley–Dickson doubling of ℝ, after ℂ, ℍ, and 𝕆. Its multiplication is noncommutative, nonassociative, and nonalternative; zero divisors defeat cancellation and unique division, so 𝕊 is not a division algebra. It remains power-associative and flexible, and doubling continues to 32 dimensions and beyond. A non-real object and the ambient algebra of this chapter.

Cites: [24.1](ch24-refs.md#r-24.1), [24.2](ch24-refs.md#r-24.2), [24.6](ch24-refs.md#r-24.6)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [the octonions](ch23-cards.md#g-octonions), [sedenion zero divisors](ch24-cards.md#g-zd)