Sunday, February 13, 2011

Topology 101 : Hole-y Polyhedra


The study of Topology begins with such a simple formula that you can explain it to a 3rd grader  ... make that a Kindergartner and maybe an astute pre-schooler. It's this:

V - E + F = 2

Where:

V = The number vertices, that is to say: corners, of a polyhedron (and not just a regular one).
E = The number of edges
F = The number of faces

That was discovered by the incredible Leonhard Euler of Switzerland (and Russia and Prussia and Russia again) in 1750, the proof of which he published one year later.

It is considered the 2nd most beautiful equation in the entire history of Mathematics.

But wait, it gets better, and points out the beauty of a single equation being a special case of a more general equation, an example being in Physics the Special Theory of Relativity (acceleration being zero) being a special case of the General Theory of Relativity (acceleration included).

And remember folks, if there's one thing Mathematicians love, it's backing up, abstracting if you will,  and finding a general case where "a cool thing" is but a special case. We don't have our cell phones or this computer you're using if they didn't.

Here's the general case, and before I state it look at the picture above of the American Pentagon building, which is in the shape of a polyhedron with a hole in it.

V - E + F = 2 - 2R

Where:

V, E, and F are as previously stated
and
R = The number of Holes.

So for one hole as in the case of the U.S. Pentagon building we have:

V - E + F = 2 - 2R = 2 - 2(1) = 0

or

V - E + F = 0

This is always true for a one hole-y polyhedron, and what amazes is that this simple formula was in the grasp of the Ancients, but nobody "got it" until Euler in 1750.

Thus begins our study of Topology.

This and more is clearly stated and explained in the 4-page "Topology" section of Tony Crilly's "50 Mathematical Ideas You Really Need To Know", and for deeper study of this important subject, read "Euler's Gem" by Dave Richeson. After that read the relevant sections of  "Mathematics 1001" then "The Princeton Companion to Mathematics" and the world should be your oyster from there on out.

I'm currently exploring all of those, and will get back when I know more.

Oh, and if you wish to know the MOST beautiful equation, it is also by Euler, the one we've previously talked about that unites the numbers 0, 1, pi, i, and e, which is:

Mathematician's wedding ring, if you're poor and can only afford pewter. What's the difference between a Mathematician and a large pizza? A large pizza can feed a family of four. Euler was an exception (there's always them)

UPDATE: So where is all this leading? Well, it will lead many places. To the early 20th century for starters, when Topology took off, and to the present time with Grigori Perelman's proof of The Poincare Conjecture, and Calabi-Yau manifold theory which is SO important in String Theory, and the world's most excellent rap video, which is this, by Mathematics Professor Steve Sawin, moonlighting as rapper "Slim Dorky", here:



Thanks to Dave Richeson of Division by Zero weblog for turning me on to that. Honestly, how can ANYone be depressed when just THINKing of that video is a drug-free instantaneous cure.

People are funny. :-)

11 comments:

Pat B said...

Steven,
Very nice....
My kids know that I hold Euler in such high regard that I cross myself when I say his name...but in the interest of historical accuracy... in regard to "what amazes is that this simple formula was in the grasp of the Ancients, but nobody "got it" until Euler in 1750."
It can very well be argued that Des Cartes got it somewhat earlier. I have been reading about this only recently and will blog more later (alas, nothing not already known to the world)

Phil Warnell said...

Hi Pat,

Yes good old Rene, who many remain unaware as to his important contributions to both math and science, with the one refer to actually being the Polyhedral Formula, which was lost to obscurity along with many other things Descartes had achieved. As I pointed out in another post to Steve Einstein often paid homage and admitted being inspired by him when it came to GR. These days when Descartes is mentioned most think simply of some wild eyed philosopher, yet I would put him on par with the greats like Newton and in some respects contend he not having an equal, given the fact he added so much in regard to what most consider as three separate disciplines, being science, mathematics and philosophy.

"We may admire Sir Isaac Newton on this occasion, but then we must not censure Descartes.

The opinion that generally prevails in England with regard to these new philosophers is, that the latter was a dreamer, and the former a sage.

Very few people in England read Descartes, whose works indeed are now useless. On the other side, but a small number peruse those of Sir Isaac, because to do this the student must be deeply skilled in the mathematics, otherwise those works will be unintelligible to him. But notwithstanding this, these great men are the subject of everyone's discourse. Sir Isaac Newton is allowed every advantage, whilst Descartes is not indulged a single one. According to some, it is to the former that we owe the discovery of a vacuum, that the air is a heavy body, and the invention of telescopes. In a word, Sir Isaac Newton is here as the Hercules of fabulous story, to whom the ignorant ascribed all the feats of ancient heroes."


- Voltaire, (from a letter written after the passing of Newton)

Best,

Phil

Steven Colyer said...

My kids know that I hold Euler in such high regard that I cross myself when I say his name

Since Euler was a strict Calvinist, and ticked off Frederick the Great the hedonist who preferred hangin' with Voltaire thereby, you stand in good stead by doing so. I bet Euler puts in a good word for you with the Big Guy. Me? I pray to St. Anthony the patron saint of lost objects, usually concerning my lost keys, and I'll be danged if that doesn't work in 5 minutes, sometimes 5 seconds.

From the first paragraph of the Introduction of Dave Richeson's "Euler's Gem", emphasis mine:

"They all missed it. The ancient Greeks -- mathematical luminaries such as Pythagoras, Theaetetus, Plato, Euclid, and Archimedes, who were infatuated with polyhedra -- missed it. Johannes Kepler, the great astronomer, so in awe of the beuty of polyhedra that he based an early model of the solar system on them, missed it. In his investigation of polyhedra the mathematician and philosopher Rene Descartes was but a few logical steps away from discovering it, yet he too missed it. These mathematicians and so many others , missed a relationship that is so simple that it can be explained to any schoolchild, yet is so fundamental that it is part of the fabric of modern mathematics."

Steven Colyer said...

Phil wrote:

As I pointed out in another post to Steve Einstein often ...

(emphasis mine)

"Steve Einstein", eh? I like that! lol , yeah, I wish! Does have a nice ring to it though. :-)

Yet another case where quoting a person out of context gives a completely different meaning. :-0

Phil Warnell said...

Hi Steve,

“Yet another case where quoting a person out of context gives a completely different meaning. :-0”

I would say whichever way you take it is perfectly fine; that is unless with taking Einstein as the premise Albert is had then to be deduced. Then again Rene would be looking down to have certain this not be the case:-)

Oh by the way, as Pat eluded there has been some evidence surface over the years that Descartes had realized Euler’s Formula before the great one and then again with the revelations as to how close Archimedes had come to developing calculus with the discovery of Archimedes Palimpsest; aided by using recent imaging technology, I wouldn’t be surprised to learn one day a clever ancient Greek beat them all to it. That is he may have not found it as significant or interesting ;-)

... certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

-Archimedes, in relation to what he called “the method” as discovered in Archimedes Palimpsest

Best,

Phil

Steven Colyer said...

Thanks, Phil. You always get my gears turning, brah.

UPDATE: So where is all this leading? Well, it will lead many places. To the early 20th century for starters, when Topology took off, and to the present time with Grigori Perelman's proof of The Poincare Conjecture, and Calabi-Yau manifold theory which is SO important in String Theory, and the world's most excellent rap video, which is this, by Mathematics Professor Steve Sawin, moonlighting as rapper "Slim Dorky", here:

Jérôme CHAUVET said...

Did anyone find some property of matter which has e^(iPi)+1 = 0 in the theory ?

If not, then what a shame!

Steven Colyer said...

Euler's identity (sometimes called Euler's formula) is a special case of Euler's formula (sometimes called Euler's trigonometric formula).

Confused yet? Well I am, I wish people would find a convention for the confusion.

In any event, Euler's trigonometric formula, which beautiful unites the exponential function and trigonometric functions, is:

e^(iz)= cos z + i sin z

So in the special case where z = Pi we get Euler's identity. It is rather amazing what can be done in Physics with the ETF, but the identity is beautiful in and of it self.

Jérôme CHAUVET said...

My question was indeed rather naive, since complex numbers are the necessary mathematical background in many area of physics... I wondered if there was a special case in which the system would stick to Euler's idendity, or some sort of thing like that ;)

Best,

Steven Colyer said...

Nerdy T-shirts and coffee cups? And from Wiki:

After proving Euler's Identity during a lecture, Benjamin Peirce, a noted American 19th century philosopher/mathematician and a professor at Harvard University, stated that "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

The Stanford University mathematics professor, Dr. Keith Devlin, said, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."

I think in France you would replace Shakespeare with Victor Hugo, yes? :-)

Jérôme CHAUVET said...

Nice answer.