September 18, 2005

This Week's Finds in Mathematical Physics (Week 221)

John Baez

After going to the Streetfest this summer, I wandered around China. I began by going to a big conference in Beijing, the 22nd International Congress on the History of Science. I learned some interesting stuff. For example:

You may have heard of Andalusia, that fascinating melting-pot of cultures that formed when southern Spain was invaded by Muslims. The eleventh century was the golden age of Andalusian astronomy and mathematics, with a lot of innovation in astrolabes. During the Caliphate (929-1031), three quarters of all mathematical manuscripts were produced in Cordoba, most of the rest in Sevilla, and only a few in Granada and Toledo.

I didn't understand the mathematical predominance of Cordoba when I first heard about it, but the underlying reason is simple. The first great Muslim dynasty were the Ummayyads, who ruled from Damascus. They were massacred by the Abbasids in 750, who then moved the capital to Baghdad. When Abd ar-Rahman fled Damascus in 750 as the only Ummayyad survivor of this massacre, he went to Spain, which had already been invaded by Muslim Berbers in 711.

Abd ar-Rahman made Cordoba his capital. And, by enforcing a certain level of religious tolerance, he made this city into the place to be for Muslims, Jews and Christians - the "ornament of the world", and a beacon of learning - until it was sacked by Berber troops in 1009.

Other cities in Andalusia became important later. The great philosopher Ibn Rushd - known to Westerners by the Latin name "Averroes" - was born in Cordoba in 1128. He later became a judge there. He studied mathematics, medicine, and astronomy, and wrote detailed line-by-line commentaries on the works of Aristotle. It was through these commentaries that most of Aristotle's works, including his Physics, found their way into Western Europe! By 1177, the bishop of Paris had banned the teaching of many of these new ideas - but to little effect.

Toledo seems to have only gained real prominence after Alfonso VI made it his capital upon capturing it in 1085 as part of the Christian "reconquista". By the 1200s, it became a lively center for translating Arabic and Hebrew texts into Latin.

Mathematics also passed from the Arabs to Western Europe in other ways. Fibonacci (1170-1250) studied Arabic accounting methods in North Africa where his father was a diplomat. His book Liber Abaci was important in transmitting the Indian system of numerals (including zero) from the Arabs to Europe. However, he wasn't the first to bring these numbers to Europe. They'd been around for over 200 years!

For example: Gerbert d'Aurillac (940-1003) spent years studying mathematics in various Andalusian cities including Cordoba. On his return to France, he wrote a book about a cumbersome sort of "abacus" labelled by a Western form of the Indian numerals - close to what we now call "Arabic numerals". This remained popular in intellectual circles until the mid-12th century.

Amusingly, Arabic numerals were also called "dust numerals" since they were used in calculations on an easily erasable "dust board". Their use was described in the Liber Pulveris, or "book of dust".

I want to learn more about Andalusian science! I found this book a great place to start - it's really fascinating:

1) Maria Rose Menocal, The Ornament of the World: How Muslims, Jews and Christians Created a Culture of Tolerance in Medieval Spain, Little, Brown and Co., 2002.

For something quick and pretty, try this:

2) Steve Edwards, Tilings from the Alhambra,

Apparently 13 of the 17 planar symmetry groups can be found in tile patterns in the Alhambra, a Moorish palace built in Granada in the 1300s.

To dig deeper into the splendors of Arabic mathematics, try these:

3) John J. O'Connor and Edmund F. Robertson, Arabic mathematics: forgotten brilliance?,

John J. O'Connor and Edmund F. Robertson, Biographies of Arab/Islamic mathematicians,

For more on Fibonacci and Arabic mathematics, try this paper by Charles Burnett, who spoke about the history of "Arabic numerals" in Beijing:

4) Charles Burnett, Leonard of Pisa and Arabic Arithmetic,

Another interesting talk in Beijing was about the role of the Syriac language in the transmission of Greek science to Europe. Many important texts didn't get translated directly from Greek to Arabic! Instead, they were first translated into Syriac.

I don't understand the details yet, but luckily there's a great book on the subject, available free online:

5) De Lacy O'Leary, How Greek Science Passed to the Arabs, Routledge & Kegan Paul Ltd, 1949. Also available at

So, medieval Europe learned a lot of Greek science by reading Latin translations of Arab translations of Syriac translations of second-hand copies of the original Greek texts!

George Baloglu recommends this book:

6) Dimitri Gutas, Greek Thought, Arabic Culture: The Graeco-Arabic Translation Movement in Baghdad and Early 'Abbasid Society (2nd-4th/8th-10th Centuries), Routledge, 1998.

I want to read this book, too:

7) Scott L. Montgomery, Science in Translation: Movements of Knowledge through Cultures and Time, U. of Chicago Press, 2000. Review by William R. Everdell available at MAA Online,

The historian of science John Stachel, famous for his studies of Einstein, says this book "strikes a blow at one of the founding myths of 'Western Civilization'" - namely, that Renaissance Europeans single-handedly picked up doing science where the Greeks left off. As Everdell writes in his review:

Perhaps the best of the book's many delightful challenges to conventional wisdom comes in the first section on the translations of Greek science. Here we learn why it is ridiculous to use a phrase like "the Renaissance recovery of the Greek classics"; that in fact the Renaissance recovered very little from the original Greek and that it was long before the Renaissance that Aristotle and Ptolemy, to name the two most important examples, were finally translated into Latin. What the Renaissance did was to create a myth by eliminating all the intermediate steps in the transmission. To assume that Greek was translated into Arabic "still essentially erases centuries of history" (p. 93). What was translated into Arabic was usually Syriac, and the translators were neither Arabs (as the great Muslim historian Ibn Khaldun admitted) nor Muslims. The real story involves Sanskrit compilers of ancient Babylonian astronomy, Nestorian Christian Syriac-speaking scholars of Greek in the Persian city of Jundishapur, and Arabic- and Pahlavi-speaking Muslim scholars of Syriac, including the Nestorian Hunayn Ibn Ishak (809-873) of Baghdad, "the greatest of all translators during this era" (p. 98).

And now for something completely different: the Langlands program! I want to keep going on my gradual quest to understand and explain this profoundly difficult hunk of mathematics, which connects number theory to representations of algebraic groups. I've found this introduction to be really helpful:

8) Stephen Gelbart: An elementary introduction to the Langlands program, Bulletin of the AMS 10 (1984), 177-219.

There are a lot of more detailed sources of information on the Langlands program, but the problem for the beginner (me) is that the overall goal gets swamped in a mass of technicalities. Gelbart's introduction does the best at avoiding this problem.

I've also found parts of this article to be helpful:

9) Edward Frenkel, Recent advances in the Langlands program, available at math.AG/0303074.

It focuses on the "geometric Langlands program", which I'd rather not talk about now. But, it starts with a pretty clear introduction to the basic Langlands stuff... at least, clear to me after I've battered my head on this for about a year!

If you know some number theory or you followed recent issues This Week's Finds (especially "week217" and "week218") it should make sense, so I'll quote it:

The Langlands Program has emerged in the late 60's in the form of a series of far-reaching conjectures tying together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic forms. To motivate it, recall the classical Kronecker-Weber theorem which describes the maximal abelian extension Qab of the field Q of rational numbers (i.e., the maximal extension of Q whose Galois group is abelian). This theorem states that Qab is obtained by adjoining to Q all roots of unity; in other words, Qab is the union of all cyclotomic fields Q(11/N) obtained by adjoining to Q a primitive Nth root of unity


The Galois group Gal(Q(11/N)/Q) of automorphisms of Q(11/N) preserving Q is isomorphic to the group (Z/N)* of units of the ring Z/N. Indeed, each element m in (Z/N)*, viewed as an integer relatively prime to N, gives rise to an automorphism of Q(11/N) which sends




Therefore we obtain that the Galois group Gal(Qab/Q), or, equivalently, the maximal abelian quotient of Gal(Q-/Q), where Q- is an algebraic closure of Q, is isomorphic to the projective limit of the groups (Z/N)* with respect to the system of surjections

(Z/N)* → (Z/M)*

for M dividing N. This projective limit is nothing but the direct product of the multiplicative groups of the rings of p-adic integers, Zp*, where p runs over the set of all primes. Thus, we obtain that

Gal(Qab/Q) = ∏ Zp*.

The abelian class field theory gives a similar description for the maximal abelian quotient Gal(Fab/F) of the Galois group Gal(F-/F), where F is an arbitrary global field, i.e., a finite extension of Q (number field), or the field of rational functions on a smooth projective curve defined over a finite field (function field). Namely, Gal(Fab/F) is almost isomorphic to the quotient A(F)*/F*, where A(F) is the ring of adeles of F, a subring in the direct product of all completions of F. Here we use the word "almost" because we need to take the group of components of this quotient if F is a number field, or its profinite completion if F is a function field.

When F = Q the ring A(Q) is a subring of the direct product of the fields Qp of p-adic numbers and the field R of real numbers, and the quotient A(F)*/F* is isomorphic to

R+ × ∏p Zp*.

where R+ is the multiplicative group of positive real numbers. Hence the group of its components is

p Zp*

in agreement with the Kronecker-Weber theorem.

One can obtain complete information about the maximal abelian quotient of a group by considering its one-dimensional representations. The above statement of the abelian class field theory may then be reformulated as saying that one-dimensional representations of Gal(F-/F) are essentially in bijection with one-dimensional representations of the abelian group

A(F)* = GL(1,A(F))

which occur in the space of functions on

A(F)*/F* = GL(1,A(F))/GL(1,F)

A marvelous insight of Robert Langlands was to conjecture that there exists a similar description of n-dimensional representations of Gal(F-/F). Namely, he proposed that those may be related to irreducible representations of the group GL(n,A(F)) which are automorphic, that is those occurring in the space of functions on the quotient


This relation is now called the Langlands correspondence.

At this point one might ask a legitimate question: why is it important to know what the n-dimensional representations of the Galois group look like, and why is it useful to relate them to things like automorphic representations? There are indeed many reasons for that. First of all, it should be remarked that according to the Tannakian philosophy, one can reconstruct a group from the category of its finite-dimensional representations, equipped with the structure of the tensor product. Therefore looking at n-dimensional representations of the Galois group is a natural step towards understanding its structure. But even more importantly, one finds many interesting representations of Galois groups in "nature".

For example, the group Gal(Q-/Q) will act on the geometric invariants (such as the etale cohomologies) of an algebraic variety defined over Q. Thus, if we take an elliptic curve E over Q, then we will obtain a two-dimensional Galois representation on its first etale cohomology. This representation contains a lot of important information about the curve E, such as the number of points of E over Z/p for various primes p.

The point is that the Langlands correspondence is supposed to relate n-dimensional Galois representations to automorphic representations of GL(n,A(F)) in such a way that the data on the Galois side, such as the number of points of E over Z/p, are translated into something more tractable on the automorphic side, such as the coefficients in the q-expansion of the modular forms that encapsulate automorphic representations of GL(2,A(Q)).

More precisely, one asks that under the Langlands correspondence certain natural invariants attached to the Galois representations and to the automorphic representations be matched. These invariants are the Frobenius conjugacy classes on the Galois side and the Hecke eigenvalues on the automorphic side.

Since I haven't talked about Hecke operators yet, I'll stop here!

But, someday I should really explain the ideas behind the baby "abelian" case of the Langlands philosophy in simpler terms than Frenkel does here. The abelian case goes back way before Langlands: it's called "class field theory". And, it's all about exploiting this analogy, which I last mentioned in "week218":

 NUMBER THEORY                 COMPLEX GEOMETRY                                 

 Integers                      Polynomial functions on the complex plane
 Rational numbers              Rational functions on the complex plane
 Prime numbers                 Points in the complex plane            
 Integers mod p^n              (n-1)st-order Taylor series
 p-adic integers               Taylor series
 p-adic numbers                Laurent series
 Adeles for the rationals      Adeles for the rational functions
 Fields                        One-point spaces
 Homomorphisms to fields       Maps from one-point spaces
 Algebraic number fields       Branched covering spaces of the complex plane

Addendum: I thank Fabien Besnard for some suggestions on how to improve this Week's Finds. Bruce Smith, Noam Elkies, and Miguel Carrión-Álvarez had some things to say about the history of science. In response to this comment of mine:

So, medieval Europe learned a lot of Greek science by reading Latin translations of Arab translations of Syriac translations of second-hand copies of the original Greek texts!
my friend Bruce wrote:
This all seems so precarious a process that it makes me wonder whether there was ten times as much valuable ancient math and philosophy as we know about, most of which got completely lost.
Something like this almost certainly true.

Like Plato, Aristotle is believed to have written dialogs which presented his ideas in a polished form. They were all lost. His extant writings are just "lecture notesquot; for courses he taught!

Euripides wrote at least 75 plays, of which only 19 survive in their full form. We have fragments or excerpts of some more. This isn't philosophy or math, but it's still incredibly tragic (pardon the pun).

The mathematician Apollonius wrote a book on Tangencies which is lost. Only four of his eight books on Conics survive in Greek. Luckily, the first seven survive in Arabic.

The burning of the library of Alexandria is partially to blame for these losses.

There's some good news, though:

Archimedes did more work on calculus than previously believed! We know this now because a manuscript of his on mechanics that had been erased and written over has recently been read with the help of a synchrotron X-ray beam! This is a great example of modern science helping the history of science.

This manuscript, called the Archimedes Palimpsest, also reveals for the first time that he did work on combinatorics:

10) Nova, The Archimedes Palimpsest,

11) Heather Rock Woods, Placed under X-ray gaze, Archimedes manuscript yields secrets lost to time, Stanford Report, May 19, 2005,

12) Erica Klarreich, Glimpses of genius: mathematicians and historians piece together a puzzle that Archimedes pondered, Science News 165 (2004), 314. Also available at

Also: a team using "multispectral imaging" has recently been able to read parts of a Roman library in the town of Herculaneum. The books in this library were "roasted in place" - heavily carbonized - during the eruption of Vesuvius that destroyed Pompeii. By distinguishing between different shades of black, researchers were able to reconstruct the entire book On Piety by one Philodemus:

13) Julie Walker, A library of mud and ashes, BYU Magazine, Spring 2001,

I can't resist quoting a bit:

A sister city to Pompeii that was also buried in the volcanic eruption of A.D. 79, Herculaneum was a seaside town that sat between Vesuvius' fertile foot and the gleaming Bay of Naples. The collection of 2,000 carbonized Greek and Latin scrolls, primarily Epicurean philosophical writings, was found in a luxurious Herculaneum house known as the Villa of the Papyri, which was discovered in 1752.

The scrolls have endured a destructive path through history: first, rain soaked the papyri, then a 570-degree swell of molasses-thick mud engulfed the villa and charred the scrolls. They would remain buried under 65 feet of mud for hundreds of years.

As a result, many of the fragile scroll cylinders are pressed into trapezoidal columns; some are bowed and snaked into half-moons, others folded into v-shapes.

After their discovery the mortality rate for the scrolls continued to climb as would-be conservators struggled to find a way to unroll the fragile manuscripts. Some scrolls were turned to mush when they were painted with mercury; many were sliced down the middle and cut into fragments. Early transcribers would copy the visible outer layer of a scroll, then scrape it off and discard it to read the next layer.

Even today, scholars use metaphors of near impossibility to describe the scroll unrolling process. It is like "flattening out a potato chip" without destroying it, or like "separating (burned) layers of two-ply tissue," says Jeffrey Fish of Baylor University.

The current unrolling methoddeveloped by a team of Norwegian conservators involves applying a gelatin-based adhesive to the scroll's outer surface. As the adhesive dries, the outer shell - which bears the text on its interior - can be slowly peeled off. It can take days to remove a single fragment, months or years to process a complete scroll. Some 300 of the library's scrolls have yet to be unrolled, and many more scrolls are in various stages of conservation and repair.

On the Herculaneum project, CPART researchers Steve and Susan Booras conducted multispectral imaging (MSI) on 3,100 trays of papyrus fragments and photographed them with a high-quality digital camera. The images will be used to create a digital library that can be accessed by scholars worldwide. Developed for NASA scientists, the imaging technique has only recently been applied to the study of ancient texts. Rather than focusing on light that is seen at wave lengths visible to the eye, MSI uses filters to focus on nonvisible portions of the light spectrum. In the nonvisible infrared spectrum, the black ink on a blackened scroll can be clearly differentiated. In some cases clear, legible writings have been found on fragments that researchers believed were completely blank.

The same team is now studying over 400,000 fragments of papyrus found in an ancient garbage dump in the old Egyptian town of Oxyrhynchus. They've pieced together new fragments of plays by Euripides, Sophocles and Menander, lost lines from the poets Sappho, Hesiod, and Archilocus, and most of a book by Hesiod:

14) Oxyrhynchus Online, multispectral imaging,

If you just want to look at a nice "before and after" movie of what multispectral imaging can do, try this link.

Finally, in response to this remark of mine:

Amusingly, Arabic numerals were also called "dust numerals" since they were used in calculations on an easily erasable "dust board". Their use was described in the Liber Pulveris, or "book of dust".
Noam Elkies wrote:
This is even more amusing than you may realize: the word "abacus" comes from a Greek word "abax, abak-" for "counting board", which conjecturally might come from the Hebrew word (or a cognate word in another semitic language) for "dust"! See for instance:

So these "dust numerals" replaced a reckoning device whose name may also originate with calculation a dust board...

Interesting! While "calculus" refers back to pebbles.

My erstwhile student Miguel Carrión-Álvarez clarified the issue somewhat:

The first abaci were drawn in the sand with sticks. The next step was to carve grooves in a board (wooden, or clay: think cuneiform tablets) and place beads in them. Pierced beads moving on beams (wood, later metal) must have been a pretty recent development, relatively speaking.

Remember that Archimedes was studying geometry by drawing figures in the sand when he was slain. If a sand abacus is the precursor of the modern calculator, Archimedes' sandbox is the precursor of GUI geometry software.

One of Archimedes' most fanciful works is "The Sand Reckoner". Here the reckoner can be understood to be himself, as he is counting the grains of sand which fit inside the sphere of fixed stars, but it can also refer to a sand abacus (reckoner = calculator). In fact, romance translations of this title that I've seen (French: L'arenaire, Spanish: El arenario, etc.) unambiguously refer to an object, not a person. It is easy to imagine Archimedes inventing his positional number system on a sand abacus, and using the counting of grains of sand as an excuse to write about it.

We avail ourselves of what our predecessors may have said. That they were or were not our coreligionists is of no account.... Whatever accords with the truth, we shall happily and gratefully accept, and whatever conflicts, we shall scrupulously but generously point out. - Averroes

© 2005 John Baez