manifold13
[manifold
home page] [manifold 13 contents] [next
article in manifold 13]
These navigation links
are repeated for convenience at the end of the article.
T'was brillig, and the slithy
toves Lemmawocky
Thus, as all educated persons know, runs the first verse of Lewis Carroll's Jabberwocky. He goes so far, in Alice, as to explain what the words mean. I find his explanations slightly evasive, in that, although they give a meaning to each word seperately, they do not give a coherent structure to the stanza. Yet it clearly possesses such a structure, and consequently must have a deeper significance; but not one which can be readily explained in Alice. Now Carroll was a mathematician  and what could be more difficult to explain to a Victorian child than mathematics? Jabberwocky, then, is mathematical in content. The natural break at the end of line two of the first stanza suggests that it be reinterpreted as a THEOREM It remains only to decipher the content of the theorem. This is the purpose of my article; and we shall see that Carroll has in fact anticipated a considerable section of what, until this article was written, has been thought of as very recent mathematics. Carroll's terminology of course differs from modern usage  or should I say. our usage differs from his? The first clue is the word wabe. Carroll explains this as being derived from 'way beyond' and 'way before'. From his explicit avoidance of 'way.below', it is clear that the wabe is the Euclidean plane R^{^2}. Gyre and gimble,clearly refer to some sort of motion. One is immediately put in mind of a dynamical system, of points flowing in the plane. So gyre and gimble refer to some kind of flow. The next things to consider are slithy toves  what are they? We are told that slithy means 'slippery and slidy', whence the toves slide along the flow. They are manifestly the tangent vectors (see the diagram); a contention which is borne out by the occurrence of the initial t and the ve in both words. Indeed, one might speculate that tove was originally tave, short for TAngent VEctor, but that the printer misspelt the word out of ignorance. T'was brillig  what was brillig? The dynamical system, of course! We can paraphrase Carroll's theorem an follows: Consider a brillig dynamical system whose tangent vectors flow in the plane. Then all the borogroves are mimsy, and the mome raths outgrabe. Mimsy is related, so Carroll says, to flimsy. What is mathematically flimsy? Let's try a few possibilities: measure zero? nowhere dense? finite? We have a brillig dynamical system in the plane, and it has finitely many borogroves.... I stood a while in uffish thought, thumbing through Smale's survey article on differentiable dynamical systems (3). And I found a theorem of Peixoto (1): any structurally stable system in R^2 has finitely many singularities, all generic; the alpha and omegalimit sets of every trajectory are singularities or closed orbits... So brillig means 'structurally stable', while the borogoves are the singularities. Mome is derived from 'far from home', a term clearly applicable to limit sets, particularly if the phrase mome raths is taken to mean 'trajectories far from home'; in which case rath is probably a misspelling of path. Outgrabe is a verb describing the behaviour of the limit sets and must mean 'tend to a closed orbit or singularity'. Now the whole picture emerges, and we render a tentative translation of the entire poem. 



'Twas brillig, and the slithy
toves

A structurally stable dynamical system in the plane has finitely many singularities. and the limit sets of the trajectories are closed orbits of singularities.' 
Beware the Jabberwock, my son!

The hero, a research student. is cautioned by his supervisor against pitfalls. and the work of certain other mathamaticians; and is told to avoid those who might steal ('snatch') his ideas. 
He took his Vorpal sword in
hand:

He acts to work on the problem. but with no success, He pauses to let his subconscious function. 
And as in uffish thought he
stood.

An idea forms in his subconscious and 'burbles' up to the surface of his mind. (it is interesting to compare this with Poincare's discussion of the role of the subconscious in mathematics (2)). 
One two! One, two! And through
and through

He realizes how to solve the problem, perhaps by 'cutting up' the planes and rushes off to tell his supervisor the main idea. 
'And hast thou slain the Jabberwock?

'You've
proved it? Oh, well done, lad! You'll get a Ph.D. out of this!' 
'Twas brillig, and
the slithy toves Did gyre and gimble in the wabe; All mimsy were the borogoves. And the mome raths outgrabe. 
Restatement of the main theorem. 
It is intriguing to speculate on the possible advances that might have been made if Carroll's theorem had been deciphered earlier.
BIBLIOGRAPHY
(1) M.Peixoto: Structural stability on 2din. manifolds Topology 1
(2) H.Poincare in 'The World of Mathematics'. ed. J. Newman
(3) S.Smale Differentiable dynamical systems. Bull.A.M.S. 73.
[manifold home page] [manifold 13 contents] [next article in manifold 13]