# July 5, 2000 {#week152}

I've been reading about the mathematical physicist William Rowan
Hamilton lately, because I'm writing a review article about the
octonions --- that famous nonassociative $8$-dimensional division algebra.

You see, the day after Hamilton discovered the quaternions and carved
the crucial formula
$$i^2=j^2=k^2=ijk=-1$$
on the Brougham bridge, he mailed a letter explaining his discovery to
his friend John Graves. And about two months later, Graves discovered
the octonions! In December 1843, he sent a letter about them to
Hamilton.

Graves called them "octaves" at first, but later introduced the term
"octonions". He showed they were a normed division algebra and used
this to prove the 8 squares theorem, which says that the product of two
sums of 8 perfect squares is again a sum of 8 perfect squares. The
complex numbers and quaternions allow one to prove similar theorems for
2 and 4 squares. In January 1844, Graves considered the idea of a
general theory of "$2^m$-ions". He tried to construct a $16$-dimensional
normed division algebra and use it to prove a 16 squares theorem, but he
"met with an unexpected hitch" and came to doubt that this was
possible.

(If you read ["Week 59"](#week59) you'll see why.)

Hamilton was the one who noticed that the octonions were nonassociative
--- in fact, he invented the word "associative" right about this time.
He offered to write a paper publicizing Graves' work, and Graves
accepted the offer, but Hamilton kept putting it off. He was probably
busy working on the quaternions!

Meanwhile, Arthur Cayley had heard about the quaternions right when
Hamilton announced his discovery, and he eventually discovered the
octonions on his own. He published a description of them in the March
1845 issue of the Philosophical Magazine. Graves was upset, so he added
a postscript about the octonions to a paper of his that was due to
appear in the following issue of the same journal, asserting that he'd
known about them since Christmas 1843. Also, Hamilton eventually got his
act together and published a short note about Graves' discovery in the
June 1847 issue of the Proceedings of the Royal Irish Academy. But by
then it was too late --- everyone was calling the octonions "Cayley
numbers".

Of course it wasn't *really* too late, since everybody who cares can
now tell that Graves was the first to discover the octonions. And
anyway, it doesn't really make a difference who discovered them first,
except as a matter of historical interest.

But just for the heck of it, I'm trying to find out everything I can
about the early history of the octonions. Hamilton is very famous, and
much has been written about him, but Graves is mainly famous for being
Hamilton's friend --- so to learn stuff about Graves, I have to read
books on Hamilton. In the process, I've learned some interesting things
that aren't really relevant to my review article. And I want to tell
you about some of them before I forget!

Hamilton was a strangely dreamy sort of guy. He spent most of his life
as the head of a small observatory near Dublin, but quickly lost
interest in actually staying up nights to make observations. Instead, he
preferred writing poetry. He was friends with Coleridge, who introduced
him to the philosophy of Kant, which influenced him greatly. He was also
friends with Wordsworth --- who told him to not to write poetry. He fell
deeply in love with a woman named Catherine Disney, who was forced by
her parents to marry a wealthy man 15 years older than her. Hamilton
remained hopelessly in love with her the rest of his life, though he
eventually married someone else. He became an alcoholic, then foreswore
drink, then relapsed. Eventually, many years later, Catherine began a
secret correspondence with him --- she still loved him! Her husband became
suspicious, she attempted suicide by taking laudanum... and then, five
years later, she became ill. Hamilton visited her and gave her a copy of
his "Lectures on Quaternions" --- they kissed at long last --- and then
she died two weeks later. He carried her picture with him ever
afterwards and talked about her to anyone who would listen. A very sad
and very Victorian tale.

He was a bit too far ahead of his time to have maximum impact during his
own life. The Hamiltonian approach to mechanics and the Hamilton-Jacobi
equation relating waves and particles became really important only when
quantum mechanics came along. Luckily Klein liked this stuff, and told
Schroedinger about it. But it's a pity that Hamilton's unification of
particle and wave mechanics came along right when the advocates of the
wave theory of light seemed to have definitively won the battle against
the particle theory --- the need for a compromise became clear only later.

Quaternions, too, might have had more impact if they'd come along
later, when people were trying to understand spin-$1/2$ particles. After
all, the unit quaternions form the group $\mathrm{SU}(2)$, which is perfect for
studying spin-$1/2$ particles. But the way things actually went,
quaternions were not very popular by the time people dreamt of spin-$1/2$
particles --- so Pauli just used $2\times$ complex matrices to describe the
generators of $\mathrm{SU}(2)$.

I like what Hamilton wrote about quaternions, space, and time:

> The quaternion was born, as a curious offspring of a quaternion of
> parents, say of geometry, algebra, metaphysics, and poetry... I have
> never been able to give a clearer statement of their nature and their
> aim than I have done in two lines of a sonnet addressed to Sir John
> Herschel:
>
> >                "And how the One of Time, of Space the Three
> >                 Might in the Chain of Symbols girdled be."

It's also amusing how Hamilton responded when de Morgan told him about
the four-color conjecture: "I am not likely to attempt your
'quaternion of colours' very soon." The pun is ironic, given the
relations people have recently discovered between what is now the
four-color theorem, the vector cross product, and the group $\mathrm{SU}(2)$. (See
["Week 8"](#week8) and ["Week 92"](#week92) for more.)

Of course quaternions were very influential for a while --- they were
taught in many mathematics departments in America in the late 1800s, and
were even a mandatory topic of study at Dublin! But then they were
driven out by the vector notation of Gibbs and Heaviside. If you don't
know this story, you've got to read this book --- it's fascinating:

1) Michael J. Crowe, _A History of Vector Analysis_, University of Notre Dame Press, Notre Dame, 1967.

Check out the graphs showing how many books were written on quaternions:
the big boom in the 1860s, and then the bust!

I hadn't even known about what many people at the time considered
Hamilton's greatest achievement: the prediction of "conical
refraction" by a biaxial crystal like aragonite. Folks compared this to
the discovery of Neptune by Adams and Leverrier --- another triumph of
prediction --- and Hamilton won a knighthood for it.

Does anyone understand how this phenomenon works? I don't.

Personally, I think one of Hamilton's greatest triumphs was his
treatment of complex numbers as pairs of real numbers --- this finally
exorcised the long-standing fears about whether imaginary numbers
"really exist", and helped opened up the way for other algebras.
Interestingly, the person who got him interested in this problem was
John Graves. Graves was the one who introduced Hamilton to John
Warren's book "A Treatise on the Geometrical Representation of the
Square Root of Negative Quantities", which explained the concept of the
complex plane. Hamilton turned this from geometry into algebra.

One of Hamilton's last inventions was the icosian calculus. Faithful
fans of This Week's Finds will remember the icosians from
["Week 20"](#week20). These were invented by Conway and Sloane;
Hamilton's original icosian calculus was a bit different. In August
1856, Hamilton went to the British Association Meeting at Cheltenham and
stayed at the house of his pal John Graves. He enjoyed talking to Graves
and reading his books: "Conceive me shut up and revelling for a
fortnight in John Graves' Paradise of Books! of which he has really an
astonishingly extensive collection, especially in the curious and
mathematical kinds. Such new works from the Continent he has picked up!
and such rare old ones too!" Graves posed some puzzles to Hamilton, and
either Graves or his books got Hamilton to thinking about regular
polyhedra. When Hamilton returned to Dublin he thought about the
symmetry group of the icosahedron, and used it to invent an algebra he
called the "icosians". He then sent a letter to Graves explaining the
icosians.

He basically said: assume we've got three symbols $I$, $K$, and $L$
satisfying these relations:
$$I^2 = 1,\quad K^3 = 1,\quad L^5 = 1,\quad L = IK $$
together with the associative law but not the commutative law. You can
think of $L$ as corresponding to rotating an icosahedron $1/5$ of a turn
around a vertex. $K$ corresponds to rotating it $1/3$ of a turn around a
face, and $I$ corresponds to rotating it $1/2$ of a turn around an edge. The
relations above all follow from this idea.

These days, we would call the icosians the "group algebra of $A_5$". In
modern lingo, the symmetry group of the icosahedron is called $A_5$,
since it's the group of all even permutations of 5 things. If you
don't know why this is true, check this out:

2) John Baez, "Some thoughts on the number six", `http://math.ucr.edu/home/baez/six.html`

We form the "group algebra" of a group by taking all formal linear
combinations of group elements with real coefficients, and defining a
product of such combinations using the product in the group. The
dimension of a group algebra is just the number of elements in the
group. Since $A_5$ has 60 elements, the icosians are a 60-dimensional
algebra. These days this stuff is no big deal. But back then, I bet a
60-dimensional noncommutative algebra was really mindblowing!

In a way that I don't fully understand, Hamilton connected the icosian
calculus to the problem of travelling along the edges of a dodecahedron,
hitting each vertex just once, and coming back to where you started. In
graph theory, this sort of thing is now called a "Hamiltonian
circuit". Hamilton even invented a puzzle where the first player takes
the first five steps any way they want, and the other player has to
complete the Hamiltonian circuit. He called this the "icosian game".

It was John Graves' idea to actually design a game board with the
dodecahedron graph drawn on it and holes at the vertices that you could
put small cylindrical markers into. In 1859, a friend of Graves
manufactured a version where the game board had legs like a small table,
and sent a copy to Hamilton. Naturally Hamilton was delighted! Graves
put Hamilton in contact with a London toymaker named John Jacques, and
Hamilton sold Jacques the rights to the game for 25 pounds and 6 copies.
Jacques marketed two versions, one for the parlor, which was played on a
flat board, and another for the "traveler", which was played on an
actual dodecahedron --- there was a nail at each vertex, and the players
wound string about these nails as they traced out their Hamiltonian
circuit.

With charming naivete, Hamilton had hopes that the game would sell
wildly. Alas, it did not. Jacques never even recouped his investment.
The problem was that the icosian game was too easy, even for children!
Amusingly, Hamilton had more trouble with it than most people --- perhaps
because he was using the icosian calculus to figure out his moves,
instead of just trying different paths.

By the way, if anyone knows any good source of information about Graves
or the invention of the octonions, I'd love to hear about it. So far
I've gotten most of my stuff from the following sources. First of all,
there's this nice biography of Hamilton:

3) Thomas L. Hankins, _Sir William Rowan Hamilton_, John Hopkins University Press, Baltimore, 1980.

Check out the picture of the icosian game on page 342!

Then there's this much longer biography, which includes lots of
correspondence:

4) Robert Perceval Graves, _Life of Sir William Rowan Hamilton_, 3 volumes, Arno Press, New York 1975.

Robert Perceval Graves was the brother of John Graves! He idolized
Hamilton, so this is not the most balanced account of his work.

Then there is this very helpful summary of the Hamilton-Graves
correspondence on octonions:

5) W. R. Hamilton, "Four and eight square theorems", Appendix 3 of vol. III of _The Mathematical Papers of William Rowan Hamilton_, eds. H. Halberstam and R. E. Ingram, Cambridge University Press, Cambridge, 1967.

Unfortunately this does not include Graves' first letter to Hamilton
about the octonions. Is it lost?

Finally, there's this history of later work on the octonions and the
eight square theorem:

6) L. E. Dickson, "On quaternions and their generalization and the history of the eight square theorem", _Ann. Math._ **20** (1919), 155--171.

It turns out the eight square theorem was proved in 1822, before Graves.
Also, there's some good material in here:

7) Heinz-Dieter Ebbinghaus et al, _Numbers_, Springer, New York, 1990.

This book is a lot of fun for anyone interested in all sorts of
"numbers".

Finally, for an excellent *online* source of information on the history
of quaternions, octonions, and other "hypercomplex number systems",
this is the place to go:

8) Jeff Biggus, "A history of hypercomplex numbers", `http://history.hyperjeff.net/hypercomplex.html`

------------------------------------------------------------------------

**Addendum:** On April 14th, 2005 I received the following email from
Geoff Corbishley in response to my plea for more information about John
Graves:

> John
>
> Your page asks for information on John Graves. I am reading *Goodbye
> to all That*, the autobiography of Robert Graves who also wrote *I
> Claudius* and other books. Chapter 1 (page 14 in my Penguin paperback)
> records that John Thomas Graves helped WR Hamilton with quaternions
> and gives a list of other relatives. Very little extra detail is given
> about JT Graves, sadly.
>
> Hope that has not been reported too often....
>
> Geoff

Fans of Hamilton might like to see my webpage with photos of the plaque
on Brougham Bridge commemorating his discovery of the quaternions:

9) John Baez, Brougham Bridge, `http://math.ucr.edu/home/baez/octonions/node24.html`