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November 26, 2007

Poncelet’s Porism

Posted by John Baez

Gavin Wraith is a mathematician with wide-ranging interests and a fondness for mysteries. To get a sense of this, try his article on ‘Ptolemy and non-Archimedes’.

Let me whet your appetite….

The often mocked, vastly underappreciated mathematician and astronomer Claudius Ptolemy proved in his Almagest that:

A quadrilateral can be inscribed in a circle whenever the product of the lengths of the diagonals equals the sum of the products of the lengths of opposite sides.

In modern notation, four points A,B,C,D 2A,B,C,D \in \mathbb{R}^2 lie on a circle if they satisfy this equation:

ACBD=ABCD+ADBC \| A - C \| \; \| B - D\| = \|A - B \| \; \|C - D \| \; + \; \|A - D\| \; \| B - C \|

On the other hand, 4 points A,B,C,DA,B,C,D \in \mathbb{R} always satisfy this strikingly similar equation:

(AC)(BD)=(AB)(CD)+(AD)(BC) (A - C) \; ( B - D) = (A - B)\;(C - D ) \; + \; (A - D)\;(B - C)

A related equation shows up in yet another context: the theory of flows. But here, ‘max’ plays the role of addition and ‘++’ plays the role of multiplication. What’s going on? Gavin explains all.

But now Gavin Wraith has a question that requires help! — help from someone who knows elliptic curves and old-fashioned synthetic geometry. It’s about Poncelet’s Porism.

First of all, what the heck is a ‘porism’?

This is one of those scary Greek math words like ‘syzygy’ and ‘plethysm’ — words that nobody ever seems to explain in a clear, intuitive way. Click on the links and you’ll see what I mean! I’ll explain ‘em someday… but not today.

It’s not promising that the Wikipedia entry for ‘porism’ begins:

The subject of porisms is perplexed by the multitude of different views which have been held by geometers as to what a porism really was and is.

In brief, a porism is something in between a ‘problem’ and a ‘theorem’. Perhaps this explanation is as good as it gets:

The older geometers regarded a theorem as directed to proving what is proposed, a problem as directed to constructing what is proposed, and finally a porism as directed to finding what is proposed.

But never mind. Poncelet’s porism goes like this:

Let CC and DD be two plane conics. If you can find, one nn-sided polygon (n3n \ge 3) that’s inscribed in CC and circumscribed around DD, then you can find infinitely many of them.

A modern treatment quickly gets into elliptic curves, but I don’t understand it yet. Anyway, here’s Gavin’s question:

Dear John -

I hope you do not mind me emailing you to ask for help about a mathematical problem whose answer is probably well known. You may know some algebraic geometer who can give a quick answer. My excuse is that the problem ties in with the beautiful demonstration in week229 of how the torus is a branched double cover of the sphere. You will have seen, I expect, the elegant proof in The Secret Blogging Seminar of Poncelet’s Porism. I raised the problem there, but have not found an answer. Being now retired and having little contact with academic life, I thought I might be impudent enough to see if you could ask your friends - I know you have a very busy life, but then if you do not already know the answer I am guessing you will want to find it.

The background to the problem is: SS and TT are nondegenerate conics having 4 distinct points of intersection (everything over \mathbb{C}, of course). XX is the variety of pairs (s,t)(s,t), ss a point of SS, tt a tangent to TT, such that ss lies on tt. SS is topologically a sphere (projective line) and XX is a double cover of SS with branches over the 4 points of intersection of SS with TT. So XX is a torus. It is therefore a model of the algebraic theory that is the affine part of the theory of abelian groups. This theory is generated by the 3-ary operation which, were we to choose a zero, we might write as

(x 1,x 2,x 3)x 1x 2+x 3.(x_1,x_2,x_3) \mapsto x_1-x_2+x_3.

It is not hard to see from the picture in week229 that this operation takes any 3 of the 4 points of intersection of SS with TT into the fourth. My question is: is there a geometric construction for this operation? I am looking for something old-fashioned; drawn with a stick in the sand, without vulgar reference to anything Poncelet would not have heard of. Given x 1=(s 1,t 1)x_1 = (s_1,t_1) and x 3=(s 3,t 3)x_3 = (s_3,t_3) one is looking for the involution on XX taking x 2x_2 to x 1x 2+x 3x_1-x_2+x_3, described geometrically purely in terms of s 1,s 2,s 3,t 1,t 2,t 3s_1,s_2,s_3,t_1,t_2,t_3 and no messing with algebra if possible. It has to be well known, but I stopped doing this stuff at school.

Best wishes

Gavin Wraith

Posted at November 26, 2007 1:15 AM UTC

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19 Comments & 0 Trackbacks

Re: Poncelet’s Porism

I know about some of Gavin Wraith’s mathematical work, for example, his early pioneering work on elementary topos theory:

  • A. Kock, G.C. Wraith. Elementary Toposes, Aarhus Universitet Lecture Notes Series No. 30, 1971.

But I was delighted to discover (through the link to his website) some of his fascinating non-mathematical writings, particularly his reminiscences and oneirotopia & fantasies, which I spent a couple of enchanted hours looking at this morning.

Thanks for the post, John!

Posted by: Todd Trimble on November 26, 2007 11:45 PM | Permalink | Reply to this

Re: Poncelet’s Porism

You’re welcome. But, I really want some algebraic geometers to tackle Wraith’s nice puzzle! Has the field really degenerated so much since Poncelet, that nobody can do this kind of stuff?

Posted by: John Baez on November 27, 2007 1:10 AM | Permalink | Reply to this

Re: Poncelet’s Porism

Now what’s that story of famous mathematician, X, declaring how degenerate mathematics has become because of all the leading mathematicians he asked to solve a problem about Y, only Z could do it?

Not much of a story without the particulars, I know. I think X is Arnold, and Y is about some identity involving trigonometric functions. Z is very well known.

Posted by: David Corfield on November 27, 2007 9:15 AM | Permalink | Reply to this

Re: Poncelet’s Porism

Arnold loves to complain about how modern mathematicians are too abstract and can’t solve concrete problems they way they could in the good old days. He often blames the French, and holds up Newton as as his hero. It can get a bit repetitive.

I seem to remember him complaining that most moderns have trouble with limits like

lim x0sin(tanx)tan(sinx)cos(secx)sec(cosx) lim_{x \to 0} {\sin (\tan x) - \tan (\sin x) \over \cos(\sec x) - \sec(\cos x)}

… or something like that! I just made up this particular example, and it may not be as interesting as his, though they share some of irksome features.

Posted by: John Baez on November 27, 2007 7:22 PM | Permalink | Reply to this

Re: Poncelet’s Porism

I don’t have the book to hand, David, but the story’s from Arnold’s book “Huygens and Barrow, Newton and Hooke”. The well-known mathematician who could solve Arnold’s problem (“the exception that proves the rule”) is Gerd Faltings. The problem was pretty close to the one John made up; I just found it on the Web:

lim x0sin(tanx)tan(sinx)arcsin(arctanx)arctan(arcsinx).lim_{x \to 0} \frac{sin(tan x) - tan(sin x)}{arcsin(arctan x) - arctan(arcsin x)}.

Posted by: Todd Trimble on November 27, 2007 7:50 PM | Permalink | Reply to this

Re: Poncelet’s Porism

In a stunning coincidence, I ran across this problem while preparing this morning’s calculus lecture. I opened Stewart’s text to section 11.11 where I’d begin, and on the facing page, after the exercises for section 11.10, was a supplementary assignment on using a computer algebra system to investigate this very limit.

Is it likely Stewart knew of this problem and included it as a specific – though veiled – reference?

Posted by: John Armstrong on November 29, 2007 12:29 AM | Permalink | Reply to this

Re: Poncelet’s Porism

It’s quite possible that either Stewart got it from Arnold or they both got it from a common source.

So what does the bloody thing equal, anyway?

Posted by: John Baez on November 29, 2007 3:58 AM | Permalink | Reply to this

Re: Poncelet’s Porism

Apparently this problem is meant to embarrass us poor modern mathematicians, because back in the day, all Russian schoolchildren were able to solve this with ease, using geometric methods which today are all but forgotten. ;-)

It equals 1. (Exciting, huh?) I can’t find my copy of Arnold’s book, and so (being such a miserable mathematician), I can’t reproduce his proof.

My guess is that Arnold made this problem semi-notorious through his book, and that’s where Stewart got it. It’s ironic that Stewart is using a computer algebra system, i.e., using the very formal symbolic methods that Arnold sneers at.

Posted by: Todd Trimble on November 29, 2007 1:26 PM | Permalink | Reply to this

Re: Poncelet’s Porism

Brings to mind the following:

When I was an undergrad at Mich, there was a postmortem treatment of Putnam problems. Us undergrads couldn’t handle a geometric one: (approximatley) give a Euclidean construction for the intersection of a parabola and a given line, given the focus and directrix of the parabola. Faculty members ducked but sent me to the elder Kazarinoff. After I repeated the problem, he responded roughly as follows:

focus - yes
directrix - yes

and then proceeded to whip out the construction

In re Poncelet’s porism - there’s an excellent article in the math monthly Dec 07 issue.

Posted by: jim stasheff on December 15, 2007 2:35 PM | Permalink | Reply to this

Re: Poncelet’s Porism

Hello John,

This is a bit historical now but was very interested in the Sep 2007 blog re: Deep Beauty symposium. In particular your comment re: Hans Halvorson and his comment that “new mathematics may be needed to make QM more intelligible…”

Are you able to refer me to papers or info generally on this topic?

Any such help much appreciated.

Desmond


Posted by: Desmond on November 4, 2008 12:55 AM | Permalink | Reply to this

Re: Poncelet’s Porism

Hello John,

This is a bit historical now but was very interested in the Sep 2007 blog re: Deep Beauty symposium. In particular your comment re: Hans Halvorson and his comment that “new mathematics may be needed to make QM more intelligible…”

Are you able to refer me to papers or info generally on this topic?

Any such help much appreciated.

Desmond


Posted by: Desmond on November 4, 2008 1:08 AM | Permalink | Reply to this

Re: Poncelet’s Porism

…the theory of flows. But here, ‘max’ plays the role of addition and ‘+’ plays the role of multiplication.

Why are flows tropical?

Posted by: David Corfield on November 27, 2007 11:35 AM | Permalink | Reply to this

Re: Poncelet’s Porism

David wrote:

Why are flows tropical?

I’m glad you noticed my subtle hint: hey, guys, here’s that tropical rig again! This time it’s max\mathbb{R}^{max} instead of min\mathbb{R}^{min}, but they’re isomorphic.

Presumably if we stare at the flow on Gavin’s webpage long enough, the answer will become obvious.

Stare.

Stare.

Stare.

It hasn’t happened yet.

It could be a total red herring, but I can’t help but think of the max-flow min-cut theorem — a favorite of my colleague Larry Harper.

Posted by: John Baez on November 29, 2007 4:13 AM | Permalink | Reply to this

Re: Poncelet’s Porism

Two things to note: norms are in the set 0\mathbb{R}^{\ge0}, but calculating the norm has to do with subtraction. So in the tropical rig, are we usually dealing with norms of some kind?

Also, the flow curves behave rather like lines on a topographical map indicating the relative height of the points, their potential difference.

Posted by: Mike Stay on December 1, 2007 3:36 PM | Permalink | Reply to this

Re: Poncelet’s Porism

Mike wrote:

So in the tropical rig, are we usually dealing with norms of some kind?

I don’t know. Doesn’t sound familiar.

I think you can call either max\mathbb{R}^{max} or min\mathbb{R}^{min} ‘the tropical rig’, since they’re isomorphic.

(For max\mathbb{R}^{max} you need to throw in ++\infty to get a unit for max; for min\mathbb{R}^{min} you need to throw in -\infty. But, xxx \mapsto -x is an isomorphism between them.)

I usually think about min\mathbb{R}^{min}. This version of the tropical rig shows up when you’re trying to minimize costs of multi-stage airplane trips, or find the path with least action.

max\mathbb{R}^{max} would therefore show up when you’re trying to maximize something.

Maybe trying to maximize the flow of water through a complicated system of pipes or something? That’s why I keep thinking about the max-flow min-cut theorem for flow networks:

“In layman’s terms, the theorem states that the maximum flow in a network is dictated by its bottleneck. Between any two nodes, the quantity of material flowing from one to the other cannot be greater than the weakest set of links somewhere between the two nodes.”

Maybe a flow network is a string diagram for doing matrix mechanics max\mathbb{R}^{max}, just as a Feynman diagram is a string diagram for matrix mechanics over \mathbb{C}. I don’t know — it’s just a wild speculation.

Posted by: John Baez on December 1, 2007 6:53 PM | Permalink | Reply to this

Re: Poncelet’s Porism

I was looking at flows as topographic lines and figured out how to compute the “flow norm” Gavin uses: you minimize the sum of the absolute differences between stopping points over all piecewise monotonic paths. It’s Maslov dequantization.

For example, given f:f:\mathbb{C}\to \mathbb{R} and four points A,B,C,DA,B,C,D \in \mathbb{C} (labelled clockwise) such that f(A)=5,f(B)=2,f(C)=7,f(D)=4f(A)=5, f(B)=2, f(C)=7, f(D)=4, then to get from BB to DD without leaving the quadrilateral, you have to go up 3 and down 1, so the norm is 3+1 = 4. There’s a saddle point somewhere, so you can’t just go up 2.

Posted by: Mike Stay on December 2, 2007 12:50 AM | Permalink | Reply to this

Re: Poncelet’s Porism

By the way, my analysis about the saddle point above depends on the fact that each of the edges in the graph have flows crossing them in only one direction: it’s easy to construct a surface where you can just go up 2 from BB to DD, but it involves having flow lines go both ways across the edges in question.

Posted by: Mike Stay on December 2, 2007 3:17 AM | Permalink | Reply to this

Re: Poncelet’s Porism

A related equation shows up in yet another context: the theory of flows. But here, ‘max’ plays the role of addition and ‘+’ plays the role of multiplication. What’s going on? Gavin explains…

Did you mean to include the explanation?

Posted by: Mike Stay on November 27, 2007 6:02 PM | Permalink | Reply to this

Re: Poncelet’s Porism

No. Gavin explains… look at his webpage on Ptolemy and non-Archimedes.

I’ve edited the blog entry to make it clearer that I’m really just trying to whet your appetite, and you have to read his stuff to sate it.

Posted by: John Baez on November 27, 2007 7:06 PM | Permalink | Reply to this

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