Quaternions and octonions
Between October 1843 and July 1844, letters between Hamilton and Graves produced the 4-dimensional and the 8-dimensional number systems, and the word “associative”. An 8-dimensional algebra cannot be drawn as a space, but its multiplication can: as seven points and seven lines.
1843: the quaternionsWhy did Hamilton need four dimensions to multiply in three?
Hamilton had treated complex numbers as pairs of reals (1835) and spent years looking for “triplets”: a way to multiply vectors in \(\R^3\) that keeps the length multiplicative, \(|xy|=|x||y|\). There is none in dimension 3. On 16 October 1843 he found one in dimension 4,
\[\Ha=\R\oplus\R i\oplus\R j\oplus\R k,\qquad i^2=j^2=k^2=ijk=-1,\]and the next day described it in a letter to John T. Graves [Hamilton 1844]; the date and the carving of the formula on Brougham (Broome) Bridge come from his later letters, quoted in [Baez, §1, pp. 1–2]. The product is not commutative: \(ij=k=-ji\).
How to picture \(\Ha\). The unit quaternions form the group \(\mathrm{SU}(2)\), the double cover of the rotation group \(\SO_3\) [Baez, §1, p. 2]: a unit \(q\) rotates the imaginary quaternions \(\R i\oplus\R j\oplus\R k=\R^3\) by \(v\mapsto qv\bar q\), and \(q\) and \(-q\) give the same rotation. These are the rotations \(k\) that turned the ellipsoids on the \(\SL_3/\SO_3\) page. For the multiplication itself, the picture is a triangle \(i\to j\to k\to i\): going along an arrow gives \(+\), against it \(-\).
1843–1845: the octonionsIf four works, what about eight?
| date | event | source |
|---|---|---|
| 17 Oct 1843 | Hamilton to Graves: the quaternions, found “yesterday” | [Hamilton 1844] |
| 26 Dec 1843 | Graves to Hamilton: an 8-dimensional normed algebra, the “octaves”, and the eight squares identity | [Baez, §1, p. 2], [Dickson, §8, p. 164] |
| Jan 1844 | Graves tries 16 dimensions and meets “an unexpected hitch” | [Baez, §1, p. 2] |
| Jul 1844 | Hamilton to Graves: “\(A\cdot BC=AB\cdot C\) … if \(A,B,C\) be quaternions, but not so, generally, with your octaves”; around this time Hamilton coins “associative” | [Baez, §1, p. 2] |
| Mar 1845 | Cayley publishes the octonions as a postscript, “with seven imaginary roots of \((-1)\)” | [Cayley 1845, pp. 210–211] |
| Apr 1845 | Graves's postscript: octads “applied by me about Christmas 1843” | [Graves 1845, p. 320] |
Cayley's priority in print is why the octonions were long called Cayley numbers [Baez, §1, p. 2].
Picture 1: seven points, seven linesHow do you see a multiplication table of 64 entries at once?
With the basis \(1,e_1,\dots,e_7\) and the rule \(e_ie_{i+1}=e_{i+3}\) (indices mod 7) of the Albert algebra page, the seven triples \(\{i,i+1,i+3\}\) are the lines of the Fano plane, the projective plane with seven points. Each line is oriented: along an arrow \(e_ae_b=+e_c\), against it \(-e_c\). This is Baez's mnemonic [Baez, §2.1, pp. 6–7].
Three things to find with the widget:
- Every line together with \(1\) spans a copy of \(\Ha\); the octonions contain seven of them, one per line.
- Three units on one line associate; three units not on one line anti-associate, \((e_ae_b)e_c=-e_a(e_be_c)\). Try \(e_1,e_2,e_3\).
- Any two units lie on a common line. This is the shadow, on basis units, of alternativity: any two octonions generate an associative subalgebra [Baez, §1.1], while three need not.
Picture 2: doubling, and what each step costsWhy does the list stop at 8?
Each algebra in the list is pairs from the one before. With conjugation \(\overline{(a,b)}=(\bar a,-b)\), one common convention is
\[(a,b)(c,d)=(ac-\bar d\,b,\ da+b\,\bar c).\]Dickson wrote this formula for the step \(\Ha\to\Oc\) in 1919 [Dickson, §4, p. 158, eq. (6)]; iterated from \(\R\) it gives \(\C,\Ha,\Oc\), and then the 16-dimensional sedenions. Baez proves what each step keeps [Baez, §2.2, Props. 1–5, p. 9] (his formula differs from this one by a convention).
| algebra | dim | lost at this step |
|---|---|---|
| \(\C\) | 2 | order |
| \(\Ha\) | 4 | commutativity |
| \(\Oc\) | 8 | associativity (alternativity stays) |
| sedenions | 16 | alternativity, \(\Nm(xy)=\Nm(x)\Nm(y)\), no zero divisors |
1898: the list is completeCould some other construction give a normed algebra of dimension 16?
(Hurwitz) [Hurwitz 1898, p. 309], [Baez, §1.1, Thm 1, p. 4] Proved (cited)
An identity \((x_1^2+\dots+x_n^2)(y_1^2+\dots+y_n^2)=z_1^2+\dots+z_n^2\) with each \(z_k\) bilinear in \(x,y\) exists only for \(n=1,2,4,8\). Equivalently, the normed division algebras over \(\R\) are \(\R,\C,\Ha,\Oc\).
Hurwitz stated it as a theorem on composing quadratic forms; the division-algebra form is the later reformulation in Baez. These four algebras are the four members of the \(A_2\) row, and \(1,2,4,8\) are its multiplicities \(\mult\): the list stops at \(\Oc\) for the same reason the row does.