Vacation is always a good time to catch up on some reading. Besides, there’s very little else you can do at night in a ski-resort… This year, I’ve taken along The Archimedes Codex: Revealing The Secrets Of The World’s Greatest Palimpsest by Reviel Netz and William Noel telling the story of the Archimedes Palimpsest.

The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with his method of double contradiction: assuming that the first is bigger than the second leads to a contradiction, as does the assumption that the first be smaller than the second; so the two must be equal. These proofs, still considered to be rigorous and correct, used what we might now consider secondary-school geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method.
Essentially, the method consists in dividing the two areas or volumes in infinitely many stripes of infinitesimal width, and “weighing” the stripes of the first figure against those of the second, evaluated in terms of a finite Egyptian fraction series. He considered this method as a useful heuristic but always made sure to prove the results found in this manner using the rigorous arithmetic methods mentioned above.
He was able to solve problems that would now be treated by integral calculus, which was formally invented in the 17th century by Isaac Newton and Gottfried Leibniz, working independently. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. Contrary to exaggerations found in some 20th century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. (For explicit details of the method used, see Archimedes’ use of infinitesimals.)
A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler’s Stereometria.
Some pages of the Method remained unused by the author of the Palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid.
In Heiberg’s time, much attention was paid to Archimedes’ brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children’s puzzle. Reviel Netz of Stanford University has argued that Archimedes discussed the number of ways to solve the puzzle. Modern combinatorics leads to the result that this number is 17,152. Due to the fragmentary state of the palimpsest it is unknown whether or not Archimedes came to the same result. This may have been the most sophisticated work in the field of combinatorics in Greek antiquity.

Also I hope to finish the novel Interred with their bones by Jennifer Lee Carrell (though I prefer the Dutch title, “Het Shakespeare Geheim” that is, “The Shakespeare Secret”) on a lost play by Shakespeare, and have a re-read of The music of the primes as I’ll use this book for my course starting next week.

(via the Arcadian Functor) At the time of the doing the Perelman-post someone rightfully commented that “making a voluntary retreat from the math circuit to preserve one’s own well-being (either mental, physical, scientific …)” should rather be called doing the Grothendieck as he was the first to pull this stunt.

On Facebook a couple of people have created the group The Petition for Alexander Grothendieck to Return from Exile. As you need to sign-up to Facebook to use this link and some of you may not be willing to do so, let me copy the description.

Alexander Grothendieck was born in Berlin, Germany on March 28, 1928. He was one of the most important and enigmatic mathematicians of the 20th century. After a lengthy and very productive career, highlighted by the awarding of the Fields Medal and the Crafoord Prize (the latter of which he declined), Grothendieck disappeared into the French countryside and ceased all mathematical activity. Grothendieck has lived in self-imposed exile since 1991.
We recently spotted Grothendieck in the “Gentleman’s Choice” bar in Montreal, Quebec. He was actually a really cool guy, and we spoke with him for quite some time. After a couple of rounds (on us) we were able to convince him to return from exile, under one stipulation - we created a facebook petition with 1729 mathematician members!
If 1729 mathematicians join this group, then Alexander Grothendieck will return from exile!!

1729 being of course the taxicab-curve number. The group posts convincing photographic evidence (see above) for their claim, has already 201 members (the last one being me) and has this breaking news-flash

Last week Grothendieck, or “the ‘Dieck” as we affectionately refer to him, returned to Montreal for a short visit to explain some of the theories he has been working on over the past decade. In particular, he explained how he has generalised the theory of schemes even further, to the extent that the Riemann Hypothesis and a Unified Field Theory are both trivial consequences of his work.

You know what to do!

Mark J. Penn wrote Microtrends: The Small Forces Changing the World. He argues that the most important trends in the world today are the smallest ones. Such as… declining standards in math education!

What should you do on the educational front if you have a child with an aptitude for numbers, as mine does? Both of you had better get cracking, because American college students are studying less math. As an example, “Microtrends” says Harvard has only 77 math majors out of 6,700 undergraduate students.
The math story is different in China and India, which are graduating as many as 950,000 engineers a year. Granted, both nations are far more populous than the United States, but that is a lot of engineers.
Mr. Penn notes that a 2001 bipartisan commission “said that the greatest threat to American national security - behind only terrorist attacks - was the threat of failing to provide sufficient math and science education in America.”

I haven’t read the book yet but it’s high on my wish-list after reading the NYT-article Why There’s Strength in Small Numbers and the Introduction of the book.