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December 7, 2007

One Geometry (Math Raps)

Posted by John Baez

While Look Around You was a goofy parody of an old-fashioned BBC educational program on math, here’s a rap video with real mathematical content:

  • Stephen Sawin, One Geometry: Perelman’s proof of the Poincaré conjecture, to the tune of Snoop Dogg and Pharrell William’s “Drop it Like it’s Hot”. Lyrics available here.

Steve Sawin is an old friend of mine. I met him when he was a postdoc at MIT — straight outta Berkeley, a student of Vaughan Jones. Since then he’s written some very interesting papers on quantum groups, topological quantum field theory, and path integrals. His latest, with Dana Fine, gives a rigorous version of the path integral for supersymmetric quantum mechanics.

But he also has a sense of fun! This song about Poincaré’s conjecture is not the first of his math raps under the name “essiness”. Here are the lyrics to his proof of the Bolzano–Weierstrass theorem. You can hear it on his website.


My Name Is… Bolzano Weierstrass

by essiness (aka Slim Dorky)


Come on you math majors if you want to be free

From Corporate America you listen to me.

You’ve got a sequence that you built from your approximating tweakins

And you really need to find a convergent subsequence,

So you ask my man Bolzano and his homie Weierstrass,

Who’ve found you a solution with a trick that’s really boss.

Well are you down with that?
We’re down with that!
Well are you down with that?
We’re down with that!

Well you haven’t got much hope unless your sequence is bounded,

So let’s say some interval has got your numbers surrounded.

They’re all greater than aa, they’re all less than some bb,

And it’s right there with those points, that your thoughts have to be.

So to your right is bb, and to your left is aa,

And in between your sequence tries to wind its way,

An infinity of x nx_n’s, this interval has in it,

Still you don’t know where to look, to try to find the limit.

So you stand in the middle, halfway in between,

aa plus bb over two, if you know what I mean,

To your left a line segment, half the big one’s size,

To your right the other half, in the same way lies.

You see every x nx_n lives in the other or the one,

But kid you’ll never believe what this division has done,

Because if every x nx_n lies in one of these, dude,

Then in one or the other an infinitude!

Well are you down with that?
We’re down with that!
Well are you down with that?
We’re down with that!

So you slide to the side where this infinity lies,

To the middle of an interval of half the size,

This new interval (I say it’s half as long),

Contains an infinite subsequence if my logic ain’t wrong.

Now you do it again, divide the line in two,

And if you paid attention, you’ll know just what to do.

You count up all the x nx_n’s, on the left and right,

There’s infinity in one, though the space is getting tight.

So you do this kk times, now we’re really getting small,

One half to the kk, is our interval.

Yet in this little space, within this little bound,

A whole subsequence can still be found.

Well are you down with that?
We’re down with that!
Well are you down with that?
We’re down with that!

Well you can do this forever, until Tishebuv,

Cuz infinite recursion is the thing that we love.

A chain of nested intervals, each inside the last,

Like little Russian dolls, and they’re getting smaller fast.

But what you have to believe, because then we’re nearly done,

Is there’s exactly one point that lives in every one!

See all those left endpoints, they have to have a supremum,

The same way that the right ones have to have an infimum.

Well this sup and this inf, they live in each of these sets,

So the distance that’s between them is as small as it gets.

They are both the same point, so I say what the hell,

I think that its our limit so let’s call it LL!

Well are you down with that?
We’re down with that!
Well are you down with that?
We’re down with that!

Well I promised a subsequence and I never tell a lie,

To distinguish it from xx I’ll call this sequence yy.

Recall the kkth interval, and all the points in its span,

Well I only need one, that’s just how bad I am.

y ky_k’s my name for this point, it lives in interval kk,

Which makes it quite close to LL, you see I planned it that way!

Now you can pick epsilon as small as it wants to be,

Cuz I’ve got nested quantifiers and they’re working for me.

I will come back with an MM, so big I’m sure it will do,

Which of your epsilon is one minus the log base two.

Well the thing about MM, is that I picked it so good,

That after it the y ky_k’s lie inside of LL’s ‘hood,

LL’s ‘hood is epsilon sized, so all those intervals lie in it,

QED, you’ve got a sequence that converges to a limit!

Well are you down with that?
We’re down with that!
Well are you down with that?
We’re down with that!

Well I am outta here now, because my rap is at its end,

But I’ll leave you with this exercise: to prove it in n\mathbb{R}^n!

Posted at December 7, 2007 4:50 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1529

29 Comments & 1 Trackback

Re: One Geometry

His latest, with Dana Fine, gives a rigorous version of the path integral for supersymmetric quantum mechanics.

As they explain in the introduction, this is an extension of a similar construction by Bär and Pfäffle.

Christian Bär gave a talk on their approach a while ago here in Hamburg. One of the key ingredients is an innocent-looking and very obvious construction: they look at piecewise geodesic paths.

I was thinking about such paths a bit recently, while trying how to prove the statement, which seems to be true, that smooth pseudofunctors from the fundamental groupoid of XX to some nn-group G (n)G_{(n)} Π 1(X)BG (n) \Pi_1(X) \to \mathbf{B}G_{(n)} are equivalent to smooth nn-functors from nn-paths in XX P n(X)BG (n). P_n(X) \to \mathbf{B} G_{(n)} \,.

One direction is easy: you differentiate the pseudofunctor and obtain the data of a Lie nn-algebroid morphism TXg (n) T X \to g_{(n)} from the tangent algebroid of XX to the Lie nn-algebra underlying G (n)G_{(n)}.

But integrating that up a again to a pseudofunctor on Π 1(X)\Pi_1(X) has global subtleties: for any homotopy class of paths, you’d want to pick one representative and then integrate your form data over it. But the result will depend on which representative you choose.

So apparently one has to choose singled out representatives.

Probably choosing a metric and then looking at geodesics might be good way to do this.

But of course that requires a bit care. The best one can hope for is is to deal with piecewise geoedesic paths.

I was wondering if the apparent need to introduce a metric here is a mere nuisance or maybe actually an inidcation that something interesting is going on.

Because the situations where we can make sense of the QM path integral are precisely the situations where we have a particle which is subject to precisely two effects:

a) a gauge background field, b) background gravity.

The first is encoded in a pseudofunctor. The precise abstract right way to encode the latter is less clear, generally.

And it would be nice if both parts could actually be seen to merge nicely. The point is that the path integral is

PXhol(γ)dμ(γ) \int_{P X} \mathrm{hol}(\gamma) d\mu(\gamma)

with the hol-part giving the background gauge field and the measure dμd\mu being the kinetic part, depending on the metric, and coming from a metric. People expect that we should think of this as one measure PXdμ˜(γ). \int_{P X} d\tilde \mu(\gamma) \,.

And when I think of the hol\mathrm{hol} as encoded in a pseudofunctor (pseudo-anafunctor really, but let’s not worry about that right now) it seems I do indeed need think of it always in conjunction with a metric.

Well, this is how far I got with thinking about this right now.

But when I read Bär-Pfäffle and Fine-Sawin, I am feeling there might be something to this.

Posted by: Urs Schreiber on December 7, 2007 10:31 AM | Permalink | Reply to this

Re: One Geometry

Heh that rap is great :-)

Wow I am really excited about this paper making the path integral for susy quantum mechanics rigourous. Actually, I was looking at Christian Bär’s webpage very recently, where I discovered some beautiful expository papers on spin structures, Dirac operators and harmonic spinors. There I also found the paper on calculating the path integral, which seemed great and very readable and I even took it home… but never read it properly. Sadly right now I can’t find his webpage, else I’d give the link. Is he at Hamburg?

This is great though… it seems that the path integral approach to the proof of the index theorem has been made rigorous and elegant! Is that a correct interpretation?

Posted by: Bruce Bartlett on December 7, 2007 5:33 PM | Permalink | Reply to this

Re: One Geometry

Bruce wrote:

This is great though… it seems that the path integral approach to the proof of the index theorem has been made rigorous and elegant! Is that a correct interpretation?

I believe so. It’s sort of weird that they don’t even bother to point it out. I’ll send him an email about this.

Posted by: John Baez on December 7, 2007 6:53 PM | Permalink | Reply to this

Re: One Geometry

Okay, I’ve had occasion to look at these two papers again:

Bar and Pfaffle, Path integrals on manifolds by finite dimensional approximation.

Fine and Sawin, A rigorous path integral for supersymmetric quantum mechanics and the heat kernel.

I must say that for some reason I found the first one much easier to read than the second one, which has some pretty gnarly statements and conventions… but anyhow. Hooray! Path integrals are now rigorous pic!

I was wondering: to what extent might this approach generalize to the kind of functional integrals occuring in quantum field theory (i.e. an integral over fields, not just paths)? Couldn’t you play a similar game: just as there is a unique geodesic between any two points (as long as they’re close enough), there’s probably a unique field (section of some vector bundle) satisfying the relevant differential equation when some thingy and some other thingy are close enough.

What I mean is, in the nice cases the time development of a field is freely specified by its initial and final configurations. And these boundary configurations usually form a finite dimensional vector space - or at worst a tame infinite sum of such spaces. So you could play the same game, couldn’t you?

Another question: to what extent does Bar and Pfaffle’s rigorous nonperturbative definition of the path integral agree with perturbation theory expansions?

Posted by: Bruce Bartlett on July 10, 2008 5:55 PM | Permalink | Reply to this

Re: One Geometry

to what extent might this approach generalize to the kind of functional integrals occuring in quantum field theory

I wish I knew. But the canonical guess is that various of the many norm estimates involved in the Bär-Pfäffle work are either ill-defined or fail. (?)

I recently asked somebody who new a bit about it a similar question:

another way to make the (Euclidean) quantum mechanical path integral rigorous is to realize that the kinetic energy contribution in the integrand dimes the heuristic path integral measure is really to be read as Wiener measure dW(γ)d W(\gamma) on stochastic paths γ\gamma. This also exists, dW gd W_g for Riemannian spaces with metric gg, as far as I know. In that language the (Euclidean) path integral for the particle coupled to gravity and Yang-Mills forces (the charged 1-particle) is the well-defined (as far as I am aware) stochastic Wiener path integral

LXhol (γ)dW g(γ). \int_{L X} hol_\nabla(\gamma) \; d W_g(\gamma) \,.

(here written for closed paths, for notational simplicity, holhol is the holonomy of the background connection along the loop taken in your favorite representation).

So from this point of view a natural question is: what is known about Wiener-like measures for higher-dimensional paths?

The answer was something like: a bit is actually known about Wiener measures for string-paths, i.e. for paths in loop space. But lots of technical problems kick in that for instance make everything break down immediately unless the Riemannian metric on XX is flat.

But I am already forgetting any further details.

(i.e. an integral over fields, not just paths)

By the way (as you know of course, but just for the record): a path is a field(-configuration), too. It just happens to be a field with very low dimensional domain.

Posted by: Urs Schreiber on July 11, 2008 12:46 AM | Permalink | Reply to this

Re: One Geometry

As a general rule of thumb: making the quantum theory of point particles rigorous is relatively easy.

The real work in making path integrals for point particles rigorous was done a long time ago by people like Wiener, Mark Kac, and Bismut. The new papers we’re takling about here just build on that old work. For the Hamiltonian approach, the old theorem of Kato and Rellich is key.

Superrenormalizable quantum field theories, like 2d quantum field theories with polynomial interactions, are harder. People like Segal and his student Ed Nelson did the Hamiltonian approach; Glimm, Jaffe and others worked out the path integral approach.

Renormalizable but not superrenormalizable field theories are where it gets really hard. People have been working on them for many decades, so far without definitive success!

The problem, of course, is dealing with renormalization in a rigorous way.

Posted by: John Baez on July 11, 2008 8:12 AM | Permalink | Reply to this

Re: One Geometry

The problem, of course, is dealing with renormalization in a rigorous way.

The week after next week Walter van Suijlekom, whose work I had mentioned here will give us an informal lecture series or the like on the Hopf-algebraic approach to renomrlaization. I’ll try to keep you informed.

Posted by: Urs Schreiber on July 11, 2008 10:04 AM | Permalink | Reply to this

Re: One Geometry

This Hopf algebra business concerns the combinatorial bookkeeping of perturbative renormalization. I wish I understood this better, because it seems really cool. But there’s a big step from this to constructing a quantum field theory. That requires summing the formal power series provided by perturbation theory — or since they usually don’t converge, taking some other approach.

In some cases, like the ϕ 4\phi^4 theory in 3d spacetime, the formal power series provided by perturbative renormalization are Borel summable. In this situation we can hope that the Hopf-algebraic approach to perturbative renormalization will someday provide a simplified proof that the field theory exists… and that would be a very good thing, since the current proofs are only comprehensible after huge amounts of effort. (More than I’ve ever made, anyway.)

But for the really interesting field theories, like Yang–Mills theory in 4d spacetime, I don’t know if people expect the power series to be Borel summable. I guess for a field theory with truly nonperturbative effects, we can’t expect to construct the theory just by pondering perturbative renormalization.

Someday soon I want to write about Stephen Summer’s nice description of the state of the art in constructive quantum field theory. There are some new ideas out there…

Posted by: John Baez on July 12, 2008 9:05 AM | Permalink | Reply to this

Re: One Geometry

One thing I’m interested in of course is whether the new approach to the point particle path integral by Bar and Pfaffle, with the supersymmetric plug-in by Fine and Sawin, is able to make rigorous the sort of manipulations of the path integral needed in the classical physicist’s proof of the index theorem. I’m talking about the kind of manipulations performed for instance by Witten in the IAS lectures,

Lectures on The Dirac Index on Manifolds and Loop Spaces, Part I, notes by John Morgan.

Let me for instance quote the last paragraphs of the proof of the index theorem given in those notes:

We have now established that

(1)index( +)=C 2n YA^(R(Y)) index(\partial^+) = C^{2n} \int_Y \hat{A}(R(Y))

where CC is a universal constant. Of course, the usual formula of Atiyah-Singer is

(2)index( +)= YA^(R(Y)). \index(\partial^+) = \int_Y \hat{A}(R(Y)).

In this derivation we have been careless with universal constants depending only on dimension (infinite but regularized in some manner). To evaluate the constant CC one can either follow through the regularization procedure or simply evaluate it on one example where the index is non-trivial.

Posted by: Bruce Bartlett on July 11, 2008 5:04 PM | Permalink | Reply to this

Re: One Geometry

Bruce wrote:

One thing I’m interested in of course is whether the new approach to the point particle path integral by Bar and Pfaffle, with the supersymmetric plug-in by Fine and Sawin, is able to make rigorous the sort of manipulations of the path integral needed in the classical physicist’s proof of the index theorem.

The classical physicist’s proof? I think you mean the quantum physicist’s proof!

I’m friends with Steve Sawin and Dana Fine, and I think the whole point of their paper was to make Witten’s proof of the index theorem rigorous. Maybe this was so obvious to them that they forgot to mention it. You know how mathematicians are sometimes. Did they really not say that their work achieves this goal?

Posted by: John Baez on July 12, 2008 9:11 AM | Permalink | Reply to this

Re: One Geometry

You might well be right John, but my somewhat confused impression is that they only show that the kernel of the time evolution operator in a certain supersymmetric quantum mechanical system is given by Bar and Pfaffle’s rigorous expression for the path integral, and hence deduce as a corollary that the time evolution kernel equals the heat kernel.

In other words: Bar and Pfaffle showed that

The heat kernel has a rigorous path integral formula

and (I think?) Fine and Sawin showed as a corollary that

The time evolution operator in a SUSY quantum mechanical system equals the heat kernel.

But that’s only the first step in proving the index theorem… now we actually have to calculate that path integral! That’s the part I’m asking about: the kind of manipulations of the path integral that Witten does in order to explicitly compute it - ‘differentiate under the path integral’ and those kinds of things - can they be shown to be rigorous with respect to the rigorous definition of the path integral? Don’t get me wrong - I’m sure they can be… but those are the crucial steps one needs to justify in order to have a path integral proof of the index theorem.

Posted by: Bruce Bartlett on July 12, 2008 11:26 AM | Permalink | Reply to this

Re: One Geometry

Cheese whizz, and this link from the guy who would not include my “Quantum Gravity Topological Quantum Gravity Blues,” in his Oxford University Press book “Knots and Quantum Qravity.”

Lyrics below:

I’ve been calculating
I said I’ve been calculating
calculating all night long
Got a quasi-triangular Hopf algebra
and I wrote down the coproduct wrong.

I’ve been integrating
integrating the whole day through
I said I’ve been integrating
integrating the whole day through
Got a Chern-Simons functional integral
and its convergent, too.

I’ve been writing down knot diagrams
converting them to braids
Using the Alexander isotopy
you know I’m not afraid
I’ve been
assigning modules
to each of these six strings
been doin’ it for weeks now
and I still don’t understand a thing.

I’ve got them old Quantum Gravity
Topological Quantum Field Theory Blues
I’ve got them old Quantum Gravity
Topological Quantum Field Theory Blues
And without NSF funding I think that you would, too.


********
Now I think I have to vid the calculus rap:
********

I’m a going to tell you all you need to know
To be the master rapper of the calculus show:
Now the first thing you do is to take the limit;
Yes that’s right, that’s how to begin it.
You don’t need to use epsilon delta stuff,
Four simple rules are going to be enough:

The limit of a constant is still a constant
And the limit of a sum is the sum of the limits
The limit of a product is the product of the limits
And the limit of the quotient is the quotient of the limits
provided that the limit of the denominator is not zero.
This last condition is certainly necessary
for in this case you must be clever!

Now the thing upon which we will take the limit
is an object known as the Newton quotient
It’s a simple object so will you repeat it:
Say ef of ex plus delta ex minus ef of ex over delta ex
Say ef of ex plus delta ex minus ef of ex over delta ex
Now this here limit is evavaluated as delta ex approaches zero
Given a function when you take this limit
The resulting quantity is called the derivative
But in general it is quite inefficient
To take the derivative by using the limit
SO a few simple rules will help you in fact
Master the world of the calculus rap.

The power rule is very simple
using it will cure your pimples
See the derivative of ex to the en
is en times ex to the en minus one
en times ex to the en minus one
don’t be frightened don’t be scared
differentiate the function ex squared
Yes the derivative of ex to the two
is two times ex

And any old fat headed thick skulled rube
Can differentiate the function ex cubed
Yes the derivative of ex to the three
is three times ex to the two

The next of the rules that we will employ
Is the product rule so stop calling me “Boy!”
The derivative of how times doo
is do dee how plus how dee doo

Where the conotation of the notation dee
Is to differentiate, naturally.

Now the next of the rules ain’t a rule for fools
this here rule is the quotient rule
Yes the derivative of hi over ho
is ho dee hi minus hi dee ho
over ho ho

And the next few functions will not make you go mental
these are the functions that are called transcendental
Say derivative of the sine of ex is
the cosine of ex.
And the derivative of the cosine of ex is
minus the sine of ex.
And the derivative of ee to the ex is
ee to the ex.
And the derivative of the natural log of ex is
one over ex.

The next of the rules is the golden rule
This here rule is called the chain rule
We apply it to the composite of functions
it is very much like peeling an onion
layer by layer oh yes it’s a fun one:
Now suppose that a function ach of ex
can be written in the form ef of gee of ex
Then the derivative of ach of ex
is the derivative of ef at gee of ex
times the derivative of gee at ex
We often say dee why dee ex
is dee why dee you times dee you dee ex
where why is ef and you is gee
why is ef and you is gee
but it’s alphabet soup if you want to ask me

Now put on your coat and put on your hat
Since you have completed the calculus rap
Yes put on your shoes and put on your pants
Because in the next song we do the calculus dance!

Copywrite 1988 By J. Scott Carter (P. E. Zap)
Published Lobe Current Music and BMI.
All right reserved.

Posted by: Scott Carter on December 7, 2007 4:35 PM | Permalink | Reply to this

Re: One Geometry

Sorry, Scott… but this blog ain’t Oxford U. Press. Thanks for posting those lyrics. If you do a video of a math song, I’ll blog it and propel you to stardom.

Posted by: John Baez on December 7, 2007 6:32 PM | Permalink | Reply to this

Re: One Geometry

Yes, you should get the Calculus Rap videoed. We were looking for it when we were doing our research into maths on YouTube.

Posted by: The Catsters on December 7, 2007 8:46 PM | Permalink | Reply to this

Re: One Geometry

I think I’ll stick with Maxwell:

My soul’s an amphicheiral knot
Upon a liquid vortex wrought
By Intellect in the Unseen residing,
While thou dost like a convict sit
With marlinspike untwisting it
Only to find my knottiness abiding,
Since all the tools for my untying
In four-dimensioned space are lying,
Where playful fancy intersperses
Whole avenues of universes,
Where Klein and Clifford fill the void
With one unbounded, finite homoloid,
Whereby the infinite is hopelessly destroyed.

Posted by: David Corfield on December 7, 2007 4:50 PM | Permalink | Reply to this

Re: One Geometry

The original Maxwell to Tait manuscript can be found at Maxwell’s house in a collection of letters that Tait’s grandson donated. When visiting, ask the curator. There is a treasure there!

Posted by: Scott Carter on December 7, 2007 5:16 PM | Permalink | Reply to this

Re: One Geometry

Where’s Maxwell’s house?

(And what kind of coffee do they serve there?)

Posted by: John Baez on December 7, 2007 6:37 PM | Permalink | Reply to this

Re: One Geometry

Do you mean 14 India Street in Edinburgh – the house in which he was born – which is now the home of the International Centre for Mathematical Sciences (ICMS)? The building is owned by the Maxwell Foundation and there are Maxwell bits and bobs in the front room.

Posted by: Simon Willerton on December 7, 2007 8:42 PM | Permalink | Reply to this

Re: One Geometry

What’s a “homoloid”, and what is Maxwell talking about here?

Whole avenues of universes,
Where Klein and Clifford fill the void
With one unbounded, finite homoloid,
Whereby the infinite is hopelessly destroyed.

Posted by: John Baez on December 7, 2007 6:35 PM | Permalink | Reply to this

Re: One Geometry

My Colleague Dan Silver (seen here chilling with one of his buddies) cowrote an eassay, The Last Poem of James Clerk Maxwell in which he writes, “The original version of `A Paxadoxical Ode’ is contained in a large scrapbook recently donated to the James Clerk Maxwell Foundation by a relative of Tait, and stored in a locked cabinet.” The cabinet is in the James Clerk Maxwell House in Edingburgh. I don’t think they serve coffee there. “Homaloid,” is footnoted there to mean “Three dimensional space in which the axioms and postulates of Euclid hold.”

Posted by: Scott Carter on December 7, 2007 7:49 PM | Permalink | Reply to this

Re: One Geometry

Alfred M. Bork in ‘The Fourth Dimension in Nineteenth-Century Physics’, Isis, Vol. 55, No. 3. (Sep., 1964), pp. 326-338, has a variant:

My soul is an entangled knot,
Upon a liquid vortex wrought
By Intellect, in the Unseen residing,
And thine doth like a convict sit,
With marlinspike untwisting it,
Only to find its knottiness abiding;
Since all the tools for its untying
In four-dimensioned space are lying,
Wherein thy fancy intersperses
Long avenues of universes,
While Klein and Clifford fill the void
With one finite, unbounded homaloid,
And think the Infinite is now at last destroyed.

Another gem from Maxwell’s pen – ‘To the Committee of the Cayley Portrait Fund’ (1874) – has the lines:

March on, symbolic host! with step sublime,
Up to the flaming bounds of Space and Time!
There pause, until by Dickenson depicted,
In two dimensions, we the form may trace
Of him whose soul, too large for vulgar space,
In n dimensions flourished unrestricted.

Posted by: David Corfield on December 7, 2007 8:33 PM | Permalink | Reply to this

Re: One Geometry

My favorite piece of Maxwell doggerel is the following which is hidden below the knot on my webpage. To understand it, it helps to have read (Peter) Tait’s knot theory papers. Tait has various procedures (which I’ve forgotten) of placing copper coins and silver coins at certain places in knot diagrams. Tait was also great with coming up names for things like “beknottedness” and “flypes” and Maxwell’s taking the mickey somewhat.

(Cats) Cradle Song, By a Babe in Knots.

Peter the Repeater
Platted round a platter
Slips of silvered paper,
Basting them with batter.

Clear your coil of kinkings
Into perfect plaiting,
Locking loops and linkings
Interpenetrating

Why should a man benighted,
Beduped, befooled, besotted,
Call knotful knittings plighted,
Not knotty but beknotted?

It’s monstrous, horrid, shocking,
Beyond the power of thinking,
Not to know, interlocking
Is no mere form of linking.

But little Jack Horner
Will teach you what is proper,
So pitch him, in his corner,
Your silver and your copper.

James Clerk Maxwell

Posted by: Simon Willerton on December 7, 2007 9:05 PM | Permalink | Reply to this

Re: One Geometry

Mmm… Maxwell is my favourite scientist and mathematician of all time, but I have to say I find his poetry slightly, um… :-)

And think the Infinite is now at last destroyed.

Is it just me, or is this line a bit pokey, sort of ignoring the rhythm of the previous lines too… perhaps tongue in cheek?

Posted by: Bruce Bartlett on December 8, 2007 1:25 AM | Permalink | Reply to this

Re: One Geometry

Ok, I’ve done an about turn on Maxwell’s poetry. They’re genius! Besides, I was critique-ing the wrong poem (woops) :-(

I see now that Maxwell was the Weird Al of his day - just add a Victorian sensibility, a gentleman-ly disposition, and a touch of Scottish hoe-down.

My soul’s an amphiceiral knot

What a great opening line… you’ve got to love Maxwell’s spelling of amphiceiral too, very exotic! It appears to be the original spelling too, from Maxwell’s scrapbook (thanks to Scott Carter for these links):

pic of maxwell's letter

Definitely better than the more sanitized version “My soul’s an entangled knot” from Alfred Bork’s book : it doesn’t have the same geekiness.


Posted by: Bruce Bartlett on December 8, 2007 10:52 AM | Permalink | Reply to this

Re: One Geometry

Bruce said

you’ve got to love Maxwell’s spelling of amphiceiral too, very exotic! It appears to be the original spelling too,

Are you sure that isn’t a transcription error by Dan Silver? I can’t see the manuscript terribly clearly. Given that the root is Greek it seems unlikely that Maxwell wouldn’t have put that extra ‘h’ in – that is to say, spell it ‘amphicheiral’. The only two google hits for ‘amphiceiral’ are Dan Silver’s pdf files.

Talking of Maxwell and Greek words, hands up all of you who knew that Maxwell was responsible for naming \nablanabla’.

Posted by: Simon Willerton on December 8, 2007 2:53 PM | Permalink | Reply to this

Re: One Geometry

Maxwell was also fond of parody. In the first of his Lectures to Women on Physical Science, he has some high-spirited fun parodying Tennyson’s The Splendour Falls on Castle Walls:

PLACE. – A small alcove with dark curtains. The class consists of one member.

SUBJECT.– Thomson’s Mirror Galvanometer.

The lamp-light falls on blackened walls,
And streams through narrow perforations,
The long beam trails o’er pasteboard scales,
With slow-decaying oscillations.
Flow, current, flow, set the quick light-spot flying,
Flow current, answer light-spot, flashing, quivering, dying,

O look! how queer! how thin and clear,
And thinner, clearer, sharper growing
The gliding fire! with central wire,
The fine degrees distinctly showing
Swing, magnet, swing, advancing and receding,
Swing magnet! Answer dearest, What’s your final reading?

O love! you fail to read the scale
Correct to tenths of a division.
To mirror heaven those eyes were given,
And not for methods of precision.
Break contact, break, set the free light-spot flying;
Break contact, rest thee, magnet, swinging, creeping, dying.

Elsewhere (perhaps in report to the British Association?), Maxwell opens a discussion of mechanics in a manner recalling Burns’s Comin’ thro’ the Rye (“In Memory of Edward Wilson, Who Repented of what was in his Mind to Write after Section”):

Rigid Body (sings).

Gin a body meet a body
Flyin’ through the air
Gin a body hit a body,
Will it fly? and where?
Ilka impact has its measure,
Ne’er a ane hae I,
Yet a’ the lads they measure me,
Or, at least, they try.

Gin a body meet a body
Altogether free,
How they travel afterwards
We do not always see.
Ilka problem has its method
By analytics high;
For me, I ken na ane o’them,
But what the waur am I?

You can find more of Maxwell’s poetry than you can shake a stick at, starting on page 577 (page 637 of 727) here (The Life of James Clerk Maxwell, by Lewis Campbell, 1882).

Posted by: Todd Trimble on December 8, 2007 6:45 PM | Permalink | Reply to this

Maxwell’s Silver Hammer; Re: One Geometry

His publications have stood the test of time. And he got to market effectively. Nobody can beat Maxwell’s Distribution.

Posted by: Jonathan Vos Post on December 8, 2007 10:18 PM | Permalink | Reply to this

Re: One Geometry

Regarding ‘homaloid’, Helmholz in an article – The Origin and Meaning of Geometrical Axioms (1876) – in the first volume of Mind, now a very different journal, writes:

Now whenever the value of this measure of curvature in any space is everywhere zero, that space everywhere conforms to the axioms of Euclid; and it may be called a flat (homaloid) space in contradistinction to other spaces, analytically constructible, that may be called curved because their measure of curvature has a value other than zero. Analytical geometry may be as completely and consistently worked out for such spaces as ordinary geometry for our actually existing homaloid space.

You see in that article an expression of a revised Kantianism:

To sum up, the final outcome of the whole inquiry may be thus expressed :-

(1.) The axioms of geometry, taken by themselves out of all connection with mechanical propositions, represent no relations of real things. When thus isolated, if we regard them with Kant as forms of intuition transcendentally given, they constitute a form into which any empirical content whatever will fit and which therefore does not in any way limit or determine beforehand the nature of the content. This is true, however, not only of Euclid’s axioms, but also of the axioms of spherical and pseudospherical geometry.

(2.) As soon as certain principles of mechanics are conjoined with the axioms of geometry we obtain a system of propositions which has real import, and which can be verified or overturned by empirical observations, as from experience it can be inferred. If such a system were to be taken as a transcendental form of intuition and thought, there must be assumed a pre-established harmony between form and reality.

Those were the days when philosophy journals contained reflection on geometry.

Posted by: David Corfield on December 7, 2007 8:52 PM | Permalink | Reply to this

Re: One Geometry

Having just read through the math raps in the post, I heard this poem in my head in Snoop Dogg’s voice, with a rap beat.

Posted by: yagwara on December 7, 2007 10:22 PM | Permalink | Reply to this
Read the post Elvis Zap does the Calculus Rap
Weblog: The n-Category Café
Excerpt: Professor Elvis Zap explains the rules of differential calculus.
Tracked: August 20, 2008 11:07 PM

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