marginalia or many pleasant facts about the square of the hypotenuse

Having taken more than three hundred fifty years to prove since the claim was first coyly presented and in fiction and popular culture, the final, mysterious conjecture by poly-math and number theorist Pierre de Fermat probably did not strike the mathematician himself nor its originally prompter as particularly significant. Fermat’s Last Theorem, as it has come to be known, was inspired by a book of lemmas by an ancient Greek mathematician called Diophantus of Alexandria. For this scholar, considered the father of algebra (not a terror-organisation and ought not to be intimidating to the public like one) for inventing variable notation and despite his monumentally new paradigm of recognising fractions as legitimate numbers, Diophantus (at least in his surviving books) did not break with the traditional penchant for finding whole number solutions for problems.
Finding a nice round solution is much more satisfying and resonates far more—I think, even given computational power that masks the ugly, irrational bits. The book of Diophantus that Fermat was reading, the Arithmetica, was rather a conversational, speculative investigation that proffered that right angled triangles (following the Pythagorean Theorem, a² + b² = c²) exist where the sides of the triangle work out to be whole numbers: 3² + 4² = 5² or 9 + 16 = 25. There seemed to exist as many solutions, however, where the answers were not so tidy. Seeing this, Fermat wondered if the application could be expounded to higher exponents (and thus dimensions—something squared is a flat surface as opposed to a three-dimensional cube) and running with it, asserted that no whole number solutions can exist for a³ + b³ = c³ or higher powers up to infinity. This assertion was scribbled, coyly, in the margin of Diophantus’ ponderings with the aside that there’s a nifty proof for this necessity but not enough room to write it here. Perhaps Fermat felt that the problem was not so pressing and never again returned to that particular problem, leaving generations to wrestle with it after his notes were discovered. There’s a whole cosmos of unsolved equations that might pose more appreciable and immediate significance if explained, and while there’s no obvious application in understanding why what Fermat declared is ultimately true, the insight and techniques developed in trying to find the answer have propelled mathematics forward and have enabled all sorts of progress in understanding and has shaped the modern world. I can’t claim any understanding of the famous proof and my brain starts to hurt from it, but I wonder if it also shows, for this celebrated and veteran conundrum, why it’s the case—that whole numbers are not transcendent.

loose change or standard operating procedure

Surely one of the great tragic coincidences of recent times and a great bounty for conspiracy theorists was that the fledging European Union in late summer of 2001 was poised to assert its supranational judicial rights and challenge the US on certain relics of legislation that gave America relatively unfettered access to European and global channels of communication.
The terror attacks of 9/11 took place and the EU dropped its suit against Project ECHELON, an intelligence scheme, programme stood up by the Five Eyes of the Anglo-Saxon partnership to spy on the Soviets in the late 1960s, once—a week after filing, the whole matter was overshadowed and charges rather reversed. Back in 1998 and the following year, Swiss and then New Zealand (a reluctant junior member of the Five Eyes community herself, though many others I suspect are envious of that cadet role) counter-intelligence suspected that their faxes were being compromised and a series of headlines and nascent exposés (titled among others, “Big Brother without a Cause”) hinted at the existence of this programme and that its mission had expanded far beyond its original reach, snooping on bank activity, internet traffic, satellite telemetry and business communiques. Though progenitor of other initiatives and a mark of enduring awareness of the surveillance state and dragnet and data-warehousing techniques, the existence of ECHELON was not confirmed until August of this year—owing to the disclosures of the Fugitive Snowden.

bib fortuna

H and I are getting very excited about seeing the next instalment of the Star Wars saga. I decided to pose a selection of my collection of action figures from the franchise (old and dear friends, every one), first featuring just a few of those characters with walk-on parts who were only in one scene—albeit no less iconic, and those various costumed variants, suited up for action on different fronts.
I wonder if having a new hat, helmet is still a valid reason for marketers to introduce a new toy, which had little impact on directing decisions—assuredly it is and always was but times seemed simplier back then.  Nonetheless, the merchandising possibilities (though that might well be the unknown power of the Dark Side) seem really enthralling.

kleinstadt

The ever-inspired Nag on the Lake shares a nice travelogue that profiles just a hint of some of the nicest small towns around the world to visit, including her own Niagara-on-the-Lake.
All the destinations look inviting and it is certainly a noble effort in keeping within small-town criteria (rather hard to define, especially considering international variance and considering how small towns grow into big cities) that may be a little of the tourist-trodden path and it invites greater inclusion. I can think of a lot of additions. The only place that H and I know from that list is the magical Rothenburg ob der Tauber here in Middle-Franconia. The immaculately preserved medieval centre of the town is quite a draw for tourists, however, and despite the vociferous authenticity, there’s somewhat of a theme-park, Truman Show atmosphere about it—not that it is not worth seeing and experiencing, quite the opposite. What small towns would you recommend?