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# Glossary cards — Chapter ᚕ(J): The second landing

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T22, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch22.md)

*Quaternions ℍ: four real components, a quarter-turn that carries i onto j, and the loss of commutativity. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Ébad — `ᚕ` {#g-og-ebad}

*Ogham letter*

A supplementary letter of unknown meaning; its kennings point at the salmon, and the medieval glossators read it as the aspen.

Ébad is the first of the five forfeda, the supplementary letters invented in the Old Irish period, centuries after the peak of ogham use, and the sounds they stood for are uncertain. The meaning of this one is unknown, and the name looks modeled on edad and idad. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give snámchaín feda, “fair-swimming letter”; cosc lobair, “admonishing of an infirm person,” uncertain; and caínem éco, “fairest fish”—two of the three pointing to éo, the word for salmon. It is the one forfid that is common in the orthodox inscriptions, where it carries a consonant value inside the formula word KOI, ‘here’, the ogham equivalent of hic iacet. The Auraicept glosses crithach, aspen; the elecampane of the letters table is the other meaning alleged for it, though the Irish for elecampane is elenn.

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)

### Hamilton’s identity — `ijk + 1` {#g-ijk}

*Integer or rational*

i² = j² = k² = ijk = −1, carved into Brougham Bridge in 1843.

Hamilton’s fundamental formula, discovered on 16 October 1843 and cut into the stone of Brougham (now Broom) Bridge on the Royal Canal in Dublin: i² = j² = k² = ijk = −1. From it the whole quaternion multiplication table follows, including ij = k and ji = −k. Gauss had found the same multiplication in an unpublished note of 1819, printed only in 1900, and Olinde Rodrigues had parametrized rotations by four numbers in 1840; Hamilton was the one who saw a number system. The numerator adds 1 to ijk, giving exactly 0, a rational number computed in ℍ, hence the integer color.

Cites: [22.1](ch22-refs.md#r-22.1), [22.12](ch22-refs.md#r-22.12), [22.2](ch22-refs.md#r-22.2), [22.8](ch22-refs.md#r-22.8)

### norm of the all-ones quaternion — `‖1+i+j+k‖` {#g-qnorm}

*Integer or rational*

2: the norm √(1² + 1² + 1² + 1²) of the simplest quaternion with every component present.

The quaternion norm ‖a + bi + cj + dk‖ = √(a² + b² + c² + d²) is the four-dimensional Pythagoras, and it is multiplicative: the norm of a product is the product of the norms. That fact is Euler’s four-square identity, sent to Goldbach in a letter of 4 May 1748, a century before Hamilton found it again, as a statement about quaternions in 1843; Lagrange used it in 1770 to prove that every positive integer is a sum of four squares. For 1 + i + j + k the norm is √4 = 2. Value 2, an integer, one of the four nonzero factors of the denominator.

Cites: [22.12](ch22-refs.md#r-22.12), [22.17](ch22-refs.md#r-22.17), [22.3](ch22-refs.md#r-22.3), [22.7](ch22-refs.md#r-22.7)

See also: [the quaternions](ch22-cards.md#g-quaternions)

### the quaternion commutator — `ij − ji` {#g-comm}

*Non-real object*

2k, nonzero: quaternions do not commute.

The commutator ij − ji measures how far two quaternions are from commuting; for the units i and j it is k − (−k) = 2k, nonzero. It is a quaternion, not a real number, so it is tinted as a non-real object. Multiplied by the norm 2, the conjugate product 4 and the order 8 of the quaternion group, it makes the denominator 128k, which has an inverse because every nonzero quaternion does (Hamilton, 1843), so dividing 0 by it is legal and gives 0, the real scalar 0 that ℍ shares with ℝ.

Cites: [22.12](ch22-refs.md#r-22.12), [22.13](ch22-refs.md#r-22.13), [22.17](ch22-refs.md#r-22.17), [22.2](ch22-refs.md#r-22.2), [22.5](ch22-refs.md#r-22.5)

See also: [the order of the quaternion group](ch22-cards.md#g-q8), [a quaternion times its conjugate](ch22-cards.md#g-qconj), [norm of the all-ones quaternion](ch22-cards.md#g-qnorm), [the quaternions](ch22-cards.md#g-quaternions)

### Hamilton’s rotation formula, turning i into j — `q i q^{−1} − j, q = (1 + k)/√2` {#g-qrot}

*Integer or rational*

0: conjugating a vector by the unit quaternion q = cos(π/4) + k sin(π/4) rotates it a quarter turn about the k-axis, and a quarter turn carries i exactly onto j.

A unit quaternion q = cos(θ/2) + u sin(θ/2), with u a unit pure quaternion, rotates any vector v by the angle θ about the axis u through the map v ↦ q v q^{−1}. Olinde Rodrigues had the composition rule for rotations, in four parameters, in 1840, three years before quaternions existed; Cayley published the conjugation form in 1845. Here θ = 90° about k, so i becomes j, and the difference q i q^{−1} − j is exactly 0 (verified by computation). The unit quaternions form the 3-sphere S³ and the group SU(2); because q and −q give the same rotation, S³ double-covers the rotation group SO(3), which is why a spinor needs a 720° turn to come home. Games and spacecraft use this formula instead of Euler angles because it has no gimbal lock. Exactly 0, an integer, riding beside Hamilton’s identity.

Cites: [22.10](ch22-refs.md#r-22.10), [22.4](ch22-refs.md#r-22.4), [22.6](ch22-refs.md#r-22.6), [22.8](ch22-refs.md#r-22.8)

See also: [Hamilton’s identity](ch22-cards.md#g-ijk), [the quaternions](ch22-cards.md#g-quaternions), [the unit quaternions as a group](ch22-cards.md#g-s3su2), [a versor, a unit quaternion](ch22-cards.md#g-versor)

### a quaternion times its conjugate — `(1 + i + j + k)(1 − i − j − k)` {#g-qconj}

*Integer or rational*

4: the conjugate q̄ reverses the signs of i, j, and k, and q·q̄ = ‖q‖² always, so this product is 2² = 4.

The conjugate of q = a + bi + cj + dk is q̄ = a − bi − cj − dk, and q·q̄ = q̄·q = a² + b² + c² + d² = ‖q‖² is real; this is how quaternions are divided, since q^{−1} = q̄/‖q‖². For the all-ones quaternion the product is 4 (verified by computation). Hurwitz’s quaternion integers, the a + bi + cj + dk whose coordinates are all integers or all half-integers, use this norm to run a Euclidean algorithm that proves Lagrange’s four-square theorem: every positive integer is a sum of four squares. Hurwitz published them in the Göttingen Nachrichten of 1896 and set them out at length in his Vorlesungen of 1919. Value 4, an integer, a factor of the denominator.

Cites: [22.12](ch22-refs.md#r-22.12), [22.7](ch22-refs.md#r-22.7)

See also: [the Euclidean algorithm](ch06-cards.md#g-euclidalg), [the quaternions](ch22-cards.md#g-quaternions)

### the order of the quaternion group — `|Q₈|` {#g-q8}

*Integer or rational*

8: the eight elements ±1, ±i, ±j, ±k close under multiplication and form the smallest group in which Hamilton’s minus sign lives.

The quaternion group Q₈ = {±1, ±i, ±j, ±k} is a non-abelian group of order eight, generated by Hamilton’s relations i² = j² = k² = ijk = −1. It and the symmetry group of the square are the only two non-abelian groups of order eight; every subgroup of Q₈ is normal, which makes it the model Hamiltonian group, a name Dedekind gave in 1897. Chapter ᚌ(G) holds the larger groups. Value 8, an integer, a factor of the denominator.

Cites: [22.16](ch22-refs.md#r-22.16), [22.5](ch22-refs.md#r-22.5)

See also: [Hamilton’s identity](ch22-cards.md#g-ijk)

### a versor, a unit quaternion — `‖(1 + i + j + k)/2‖` {#g-versor}

*Integer or rational*

1: Hamilton’s name for a quaternion of norm one; the versors form the 3-sphere S³, and every rotation of space is represented by exactly two of them, q and −q.

Hamilton coined versor in the 1840s, and fixed it in his Lectures on Quaternions of 1853, for a quaternion of norm one, a pure turning with no stretching, just as e^{iθ} is in the plane. (1 + i + j + k)/2 is a versor; as a rotation it turns space by 120° about the axis (1, 1, 1), cycling i → j → k → i (verified by computation). The versors form the unit 3-sphere S³ ⊂ ℝ⁴, which is the Lie group SU(2), the double cover of the rotation group SO(3). A key entry, not in the factor: it is the unit that Hamilton’s rotation formula rides on. Value 1, an integer.

Cites: [22.10](ch22-refs.md#r-22.10), [22.11](ch22-refs.md#r-22.11)

See also: [the unit quaternions as a group](ch22-cards.md#g-s3su2)

### the Pauli matrices — `σ₁, σ₂, σ₃` {#g-pauli}

*Non-real object*

Three 2 × 2 complex matrices from Wolfgang Pauli’s 1927 theory of electron spin; sending i, j, k to −iσ₁, −iσ₂, −iσ₃, where the i in front is the complex √−1, turns Hamilton’s table into matrix multiplication.

σ₁ = [[0, 1], [1, 0]], σ₂ = [[0, −i], [i, 0]], σ₃ = [[1, 0], [0, −1]]: traceless, Hermitian, unitary, and σ₁σ₂ = iσ₃. Pauli introduced them in 1927 to describe the spin of the electron. The algebra generated by −iσ₁, −iσ₂, −iσ₃ is isomorphic to the quaternions, so ℍ sits inside the 2 × 2 complex matrices, and the spin of the electron can be described with quaternions; the same matrices return in Chapter ᚍ(K)’s Hilbert space. Non-real objects and a tool of the chapter, not in the factor.

Cites: [22.9](ch22-refs.md#r-22.9)

See also: [a Hermitian operator](ch15-cards.md#g-hermitian), [Hilbert space](ch15-cards.md#g-hilbert), [the quaternions](ch22-cards.md#g-quaternions)

### the unit quaternions as a group — `S³ ≅ SU(2)` {#g-s3su2}

*Non-real object*

The quaternions of norm 1 form the 3-sphere, which is the Lie group SU(2); it double-covers the rotation group SO(3), q and −q giving the same rotation.

The unit quaternions {q : ‖q‖ = 1} are the 3-sphere S³ in ℝ⁴, closed under multiplication, and as a group they are SU(2), the 2 × 2 unitary matrices of determinant 1. The map q ↦ (v ↦ q v q^{−1}) sends S³ onto the rotation group SO(3) two-to-one, since q and −q rotate identically; that double cover is why a spinor returns to itself only after 720°, and S³ = SU(2) is the Lie group whose Lie algebra su(2), of dimension 3, opens Chapter ᚌ(G)’s brace. A non-real object and a tool of the chapter, not in the factor.

Cites: [22.10](ch22-refs.md#r-22.10)

See also: [a Lie algebra](ch14-cards.md#g-liealg), [the quaternions](ch22-cards.md#g-quaternions), [the dimension of su(2)](ch14-cards.md#g-su2)

### the quaternions — `ℍ = {a + bi + cj + dk : a,b,c,d ∈ ℝ}` {#g-quaternions}

*Non-real object*

The four-dimensional real algebra generated by i² = j² = k² = ijk = −1: associative, with every nonzero element invertible, but multiplication is not commutative.

A quaternion has the form a + bi + cj + dk with real coefficients. Addition is coordinatewise, while Hamilton’s relations determine multiplication; in particular ij = k but ji = −k. Multiplication remains associative, and conjugation gives q·q̄ = ‖q‖² and q⁻¹ = q̄/‖q‖² for q ≠ 0. Thus ℍ is a four-dimensional real division algebra, the second rung of the ladder, after ℂ: it keeps division but gives up commutativity. Quaternions were influential in mathematical physics until vector analysis displaced them in many applications. A non-real object and the ambient number system of this chapter.

Cites: [22.15](ch22-refs.md#r-22.15), [22.2](ch22-refs.md#r-22.2), [22.3](ch22-refs.md#r-22.3)

See also: [Hamilton’s identity](ch22-cards.md#g-ijk)