Sunday, May 29, 2005

and peace to you

a curious corollary

I emailed Darsh's colleagues & he sent me this

"You emailed a few of my colleagues about one of my
submissions to the "Art of Science" gallery, and they
forwarded me your email. I think you are referring to
the one titled "Strange Crystal" in your email. The
"crystal" depicted is not actually a physical crystal
that I grew, if that's what you thought; it's merely
an image I created on my computer. My use of the word
"crystal" in the description is intended as a metaphor
to help explain the algorithm that generates the
image. Sorry for the confusion, if there was any."

and he did! a 1.8mg file to be exact, very cool that I can see this on the web, in the mountains of Saudi Arabia & in less than one day contact him personally and have him send it to me!

a bright young whipper snapper indeed! keep your eyes peeled to the mathematical journals.....ha!

you never know where it might lead you,.........

............for example the strange story of

Fermat's last theorem


(for a page that can represent all the squigly bits)

(sometimes abbreviated as FLT and also called Fermat's great theorem) is one of the most famous theorems in the history of mathematics. It states that:

There are no positive integers x, y, and z such that in which n is a natural number greater than 2.

The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus: "I have discovered a truly remarkable proof of this theorem that the margin of this page is too small to contain". (Original Latin: "Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.") However, no correct proof was found for 357 years.
This statement is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, or by rigorous proofs found afterwards. Mathematicians were long baffled, for they were unable either to prove or to disprove it. The theorem was therefore not the last that Fermat conjectured, but the last to be proved. The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs.

The Proof!

Using sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the English mathematician Andrew Wiles, from Princeton University, with help from his former student Richard Taylor, devised a proof of Fermat's last theorem that was published in 1995 in the journal Annals of Mathematics.

In 1986, Ken Ribet had proved Gerhard Frey's epsilon conjecture that every counterexample an + bn = cn to Fermat's last theorem would yield an elliptic curve which would provide a counterexample to the Taniyama-Shimura conjecture.

This latter conjecture proposes a deep connection between elliptic curves and modular forms.

Andrew Wiles and Richard Taylor were able to establish a special case of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's last theorem.

The story of the proof is almost as remarkable as the mystery of the theorem itself. Wiles spent seven years working out nearly all the details by himself and with utter secrecy (except for a final review stage for which he enlisted the help of his Princeton colleague, Nick Katz). When he announced his proof over the course of three lectures delivered at Cambridge University on June 21-23 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious error was discovered: it seemed to lead to the breakdown of this original proof.

It began to seem that Wile's proof was destined like so many others to be fatally flawed, and that although he had made many important discoveries, the ultimate goal had eluded him. Wiles was on the point of giving up finally, when he decided to have one last try at solving the last remaining problem in his proof in collaboration with Richard Taylor, one of his former PhD students in 1994. He commented:

"... suddenly, totally unexpectedly, I had this incredible revelation. It was the most important moment of my working life. Nothing I ever do again will mean as much ... it was so indescribably beautiful, it was so simple and so elegant, and I just stared in disbelief for twenty minutes, then during the day I walked round the department. I'd keep coming back to my desk to see it was still there - it was still there."

Praise Be to God (Exalted Be HE!), Al Fatah (The Opener) Who Provided That Experience!

A Muslim, a believer even knows with conviction, the explanation for this remarkable process, and how many different opening's there are......


Coincindentally, I visited Fermat's home in the corner of the delightful village square of Beaumont De Lomagne (the village of Garlic: there's a huge garlic float parade every year!) when I was resident in the very same town for while, I never forgot it! A museum that hardly anyone visits. I had to chase up, with the help of my girlfriend's family, one of the descendents, a kindly, bespectacled madame, to open it up for me, and inside there's a few dusty cabinets holding some of his original papers & some explanations. I tried to imagine what it must be like to be able to encompass all of this in one's head, for some reason it fascinates me.

I don't think anyone had been in there for months when I visited in 1991, I remember the puffs of dust being drawn upwards in small mushroom clouds, as she eased open the door. I remainded interested, of course, in this unsolvable theorem that had stumped every mathematician alive, for hundreds of years.

That was until a year or two later when the above proof (see Wikipedia for details) was furnished by Andrew Wiles in Cambridge & it blew everyone away, even people who didn't understand it...

In fact, there's a gripping BBC horizon doco made about his by a chap called Simon Singh, where Andrew Wiles describes the whole experience & you can see the emotion & lazer like focus he invested for years working it out: absolutely awsome

and so to him and , to Darsh Ranjan, and to all seekers of Truth

May God, All Exalted!, Guide Them to the Source from Hence It Came!

and to you all!

wa salam walykum a rahmatullahi wa barakatuh.....

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