Jun 08

Dealing in Triangles:A Little Known Short Proof Of The Existence of A Triangulation Of A Compact Surface And Other Matters Mathematical...........

Holla. Back on the chain gang with my girl bitching I'm broke ass and wondering why she gives me the time of day. OoOoOoO,I can't wait until she's in England next year getting her Master's. She thinks she's going to go to class and work part time since "school is so much easier then working."I can't wait until she calls me at 3 am long distance crying..................LOL

Before we get to the first mathematical post of the blog-I should introduce myself formally. My name is Andrew L. As for the rest:

Name: Oh,wouldn't you like to KNOW..............
Location:The City That Never Sleeps And Hates King George For Letting 9/11 Happen..................
Age:As old as my tongue and a little older then my teeth.

Marital Status::If you can ask,you've never seen a picture of me.......................
Hobbies & Interests:Just about EVERYTHING,really-with particular emphasis on anything mathematical or in the hard sciences.(I don't distinguish between pure and applied mathematics and to me,EVERYTHING other then theoretical mathematics is just applied math-biology,physics,chemistry-EVERYTHING.Sue me............) Studying(naturally),Research;tall,curvaceous girls with brains and hearts (a rare commodity to be sure,but worth the search);writing,debating,compassionate friend to the ungrateful masses and whatever else I can accomplish in this meaningless existence to fill up my time until I join the dinosaurs.
Favorite Gadgets:The internet on whatever PC I can steal...............
Occupation: WAS a double major in mathematics and biochemistry-have since entered Queens College of The City University Of New York as a pure mathematics Master's student and hoping to use it as a new beginning to an Ivy League PHD after wrecking my career caring for my late father.Studying topology with Dennis Sullivan next semester if all goes well,that should get me off on the right foot...............
Personal Quote:"God doesn't exist.GET OVER IT..........."

To that,I'd like to add 2 things:
a) I'm learning this career path is MUCH harder then I could have imagined without coffee, which I can't drink anymore. IBS be damned............
b) This past semester,I relearned my love of philosophy under the tutelege of the legendary Saul Kripke in his philosophy of mathematics lectures at the Graduate Center of the City University Of New York.
To expand on these endeared memories-I remember the first day.I showed up with my friend Joey-probably the department's most talented mathematics major-to hear the giant speak. (Check that-I dunno if I'd go so far as to say Joey's the most TALENTED.We have a half a dozen really talented students in our mathematics club. But he's certainly the most advanced of us in his studies and research-and none of us are as dedicated or focused as he is.) Dr.Kripke went on about matters I remembered little about from my philosophy days in his unique,soft spoken and sometimes halting manner-he would stop to think about what he wanted to say and when it did come out,it was amazingly profound. It was clear to me this was a man who cared not only about what he was saying,but took time to stop himself and make sure he got it right.
Joey didn't agree-he looked at me perplexed and disappointed in Kripke's style-and he left and never came back. It was his loss. I'll talk more about it in future posts-but Joey, you missed an experience in this course. Dr.Kripke is giving a second semester by popular demand next semester-sadly,it's at the same time as the deformation theory research seminar myself and several others are already committed to. Unless it can be moved to a half-hour earlier, I'll have to make some hard choices before the fall. It will agonize me to not attend the second semester. But I am a mathematics graduate student and my heart must follow that path for now. I'm torn between my past love and my current path. The fork in the road will have some of my heart's blood on it either way I choose.
Today, I was engrossed with algebraic topology,a subject I swore a blood oath to conquer this summer before I went back. I raced through it last time trying to make a deadline for completion the department forced on me-and ended with a less then stellar grade of B+. Under the circumstances,though-I really should be estatic with it.Considering I crammed most of homology theory in in ONE WEEK,I should be thanking all the Fates and giving my professor John Terilla a kiss for that grade. I'm mad at myself because I know I can do better. Be that as it may-I was looking over Massey's presentation proof of the classification theorum of surfaces. I've always thought it was a beautiful tour-de-force: An incredibly deep result (all compact orientable surfaces(2-manifolds) are homeomorphic to either a) a connected sum of tori ,b) a connected sum of spheres or c) the projective plane) proved with "bare hands" by folding, glueing and pasting carefully selected edges and points on the surface S and showing the resulting quotient spaces have to be equivelent to one of those three. This proof is pretty long-it takes up most of chapter 1 of Massey's Algebraic Topology:An Introduction ( or A Basic Course In Algebraic Topology, the first half of the second book is basically the first with the useless last chapter removed). But it always intrigued me that this proof relies on a fact most topology books take for granted-the fact that all compact surfaces have at least one triangulation. Classically,a trangulation of a surface was exactly that-a homeomorphic decomposition of the surface into a set of oriented disjoint triangles. This step is critical for the classical proof of the classification theorum-there's literally nowhere to begin without it. As I usually do when something mystifies me-I dig into history to see how a concept evolved. Apparently the idea of busting a surface up into a mass of triangles originated with the first rigorous "combinatorial" definition of a surface in the plane by Dehn and Heergard in 1907; they defined a surface S as a simplexical complex where each edge is incident with 2 triangles and each fixed vertex is incident on a set of ordered vertices where each vertex in the set and the fixed vertex define a unique edge of a triangle in S. ( Aren't you glad you're not a topologist living back then?) Classical topology then proceeded on this assumption until mathematicans began to question if the definition was valid i.e. can every surface be covered with edge-pairwise disjoint triangles? The answer turned out to be yes,as was shown by Tibor Rado in 1925-but the proof had 2 major drawbacks: First, it was LONG. 23 PAGES long to be exact. Not exactly the kind of thing you can do in a classroom in a few lectures. The other and much more serious problem was that Rado's proof showed that although a surface can be covered by a set of triangles defined as above, the set need not be finite. Indeed, some mathematicans had already succeeded in producing infinite triangulations of some surfaces before Rado's proof.
It's because of all this BS that modern topologists have come to define triangulations in a much more abstract way: A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h:K to X. Unsurprisingly,even with this definition, most topologists have been reluctant to directly decompose manifolds like they did in the old days;most stick with homology group(or for those sharper tools in the shed. groupoids) based analyses and call it a day. Still-it would be fascinating to have a short,clear proof of the fact to convince oneself that the classification of surfaces isn't mere combinatorial slight of hand.

*blaring of trumpets in the background in anticipation of startling revelation*

Such a simple 2 PAGE proof HAS in fact been found-it was published by P.H. Doyle and D.A. Moran,then both of Michigan State, in an obscure journal in 1968. It's like most good things in life, pretty simple mathematically-and it relies on a very easy generalization of the Jordan curve theorum to convert a covering of 2-cells (open disks) of a surface S into a countably infinite set of simplexical complexes.(Note,though,sadly it hasn't reduced the triangulation to finitely many complexes.Oh well.) I am proud to present the link to the proof,which I have found posted online. It shocks me that this proof is not more commonly known-most mathematicans I've asked referred me with pained expressions to Rado's proof when I asked about taking triangulations on faith. The only mathematican I was able to find in the textbook literature refer to it was James R.Munkres in his book.
So the mystery is no longer so intractable,students! Go forth and add this beautiful result to your toolbox-we no longer need to take this fact on faith! Spread the word, that this momentously practical result can now be shared by all!


Andrew L.
The Mad Mind

2 Responses for "Dealing in Triangles:A Little Known Short Proof Of The Existence of A Triangulation Of A Compact Surface And Other Matters Mathematical..........."

  1. Franklin says:

    Suggestion: Put two spaces between paragraphs. Since you can't indent, it's another way to make the separation obvious.

    EVERYTHING other than theoretical mathematics is just applied math - biology, physics, chemistry - EVERYTHING.Sue me...)

    This reminds me of a comic: http://xkcd.com/435/

    Note,though, sadly it hasn't reduced the triangulation to finitely many complexes. Oh well.

    Well, it's possible that a finate triangulation isn't necessarily computable (i.e. there's no sequence of steps to find it).

  2. Monica says:

    Been a while... going to continue?

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