July 5, 2000

This Week's Finds in Mathematical Physics (Week 152)

John Baez

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

                i2 = j2 = k2 = 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 "2m-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 "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 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 x 2 complex matrices to describe the generators of 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 SU(2). (See "week8" and "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 "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:

        I2 = 1,   K3 = 1,   L5 = 1,   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 A5". In modern lingo, the symmetry group of the icosahedron is called A5, 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 A5 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:


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....


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

© 2000 John Baez