Also available as http://math.ucr.edu/home/baez/week221.html
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 meltingpot
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 (9291031), 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 Ummayads, who ruled from
Damascus. They were massacred by the Abbasids in 750, who then
moved the capital to Baghdad. When Abd arRahman 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 arRahman 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 linebyline 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 (11701250) 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 (9401003) 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 mid12th 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,
http://www2.spsu.edu/math/tile/grammar/moor.htm
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?,
http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Arabic_mathematics.html
John J. O'Connor and Edmund F. Robertson,
Biographies of Arab/Islamic mathematicians,
http://wwwgroups.dcs.stand.ac.uk/~history/Indexes/Arabs.html
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,
http://muslimheritage.com/topics/default.cfm?ArticleID=472
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
http://www.aina.org/books/hgsptta.htm
So, medieval Europe learned a lot of Greek science by reading Latin
translations of Arab translations of Syriac translations of
secondhand copies of the original Greek texts!
George Baloglu recommends this book:
6) Dimitri Gutas, Greek Thought, Arabic Culture: The GraecoArabic
Translation Movement in Baghdad and Early 'Abbasid Society
(2nd4th/8th10th 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,
http://www.maa.org/publications/maareviews/scienceintranslationmovementsofknowledgethroughculturesandtime
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
singlehandedly 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 Syriacspeaking scholars of
Greek in the Persian city of Jundishapur, and Arabic and
Pahlavispeaking Muslim scholars of Syriac, including the
Nestorian Hunayn Ibn Ishak (809873) 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), 177219.
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've followed recent issues of
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 farreaching conjectures tying together seemingly
unrelated objects in number theory, algebraic geometry, and the
theory of automorphic forms. To motivate it, recall the classical
KroneckerWeber theorem which describes the maximal abelian extension
Q^{ab} of the field Q of rational numbers (i.e., the maximal extension
of Q whose Galois group is abelian). This theorem states that Q^{ab}
is obtained by adjoining to Q all roots of unity; in other words,
Q^{ab} is the union of all cyclotomic fields Q(1^{1/N}) obtained
by adjoining to Q a primitive Nth root of unity
1^{1/N}
The Galois group Gal(Q(1^{1/N})/Q) of automorphisms of Q(1^{1/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(1^{1/N})
which sends
1^{1/N}
to
1^{m/N}.
Therefore we obtain that the Galois group Gal(Q^{ab}/Q), or,
equivalently, the maximal abelian quotient of Gal(Qbar/Q),
where Qbar 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 padic
integers, Z_p*, where p runs over the set of all primes. Thus,
we obtain that
Gal(Q^{ab}/Q) = product_p Z_p*.
The abelian class field theory gives a similar description for the
maximal abelian quotient Gal(F^ab/F) of the Galois group Gal(Fbar/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(F^ab/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 Q_p of padic numbers and the field R of real numbers, and
the quotient A(F)*/F* is isomorphic to
R+ x product_p Z*_p.
where R+ is the multiplicative group of positive real numbers.
Hence the group of its components is
product_p Z*_p
in agreement with the KroneckerWeber theorem.
One can obtain complete information about the maximal abelian
quotient of a group by considering its onedimensional
representations. The above statement of the abelian class field
theory may then be reformulated as saying that onedimensional
representations of Gal(Fbar/F) are essentially in bijection with
onedimensional 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 *ndimensional
representations* of Gal(Fbar/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
GL(n,A(F))/GL(n,F)
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 ndimensional 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 finitedimensional representations,
equipped with the structure of the tensor product. Therefore
looking at ndimensional 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(Qbar/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 twodimensional 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 ndimensional 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 qexpansion 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 (n1)storder Taylor series
padic integers Taylor series
padic numbers Laurent series
Adeles for the rationals Adeles for the rational functions
Fields Onepoint spaces
Homomorphisms to fields Maps from onepoint spaces
Algebraic number fields Branched covering spaces of the complex plane

Quote of the week:
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

Addendum: I thank Fabien Besnard for some suggestions on how to improve
this Week's Finds. Bruce Smith, Noam Elkies, and Miguel AlvarezCarrion
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
> secondhand 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 notes" 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 Xray 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,
http://www.pbs.org/wgbh/nova/archimedes/palimpsest.html
11) Heather Rock Woods, Placed under Xray gaze, Archimedes
manuscript yields secrets lost to time, Stanford Report, May 19, 2005,
http://newsservice.stanford.edu/news/2005/may25/archimedes052505.html
12) Erica Klarreich, Glimpses of genius: mathematicians and
historians piece together a puzzle that Archimedes pondered,
Science News 165 (2004), 314. Also available at
http://www.sciencenews.org/articles/20040515/bob9.asp
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, they 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, http://magazine.byu.edu/?act=view&a=43
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 570degree swell of molassesthick 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 halfmoons, others
folded into vshapes.
After their discovery the mortality rate for the scrolls continued to climb
as wouldbe 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 twoply
tissue," says Jeffrey Fish of Baylor University.
The current unrolling methoddeveloped by a team of Norwegian conservators
involves applying a gelatinbased 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 highquality 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,
http://www.papyrology.ox.ac.uk/multi/procedure.html
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 CarrionAlvarez 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.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html