Category: stories

the secret revealed…

Published March 26, 2008 by lievenlb

Often, one can appreciate the answer to a problem only after having spend some time trying to solve it, and having failed … pathetically.

When someone with a track-record of coming up with surprising mathematical tidbits like John McKay sends me a mystery message claiming to contain “The secret of Monstrous Moonshine and the universe”, I’m happy to spend the remains of the day trying to make sense of the apparent nonsense

Let j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion
at oo of the elliptic modular function.

Compute sum(c[k]^2,k=1..24) modulo 70

I expected the j-coefficients modulo 70 (or their squares, or their partial sums of squares) to reveal some hidden pattern, like containing the coefficients of Leech vectors or E(8)-roots, or whatever… and spend a day trying things out. But, all I got was noise… I left it there for a week or so, rechecked everything and… gave up

Subject:   Re: mystery message
From:  lieven.lebruyn@ua.ac.be
Date:  Fri 21 Mar 2008 12:37:47 GMT+01:00
To:    mckayj@Math.Princeton.EDU

i forced myself to recheck the calculations i did once after receiving your mail.
here are the partial sums of squares of j-coefficients modulo 70 for the first 
100 of them

[0, 46, 26, 16, 32, 62, 38, 3, 53, 13, 63, 39, 29, 59, 45, 10, 60, 40, 30,
 10, 40, 26, 6, 56, 42, 22, 68, 48, 48, 64, 64, 45, 25, 15, 31, 31, 67,
 47, 7, 21, 51, 31, 31, 61, 21, 1, 17, 12, 2, 16, 46, 60, 20, 10, 54, 49,
 63, 63, 53, 29, 29, 23, 13, 13, 27, 27, 17, 7, 67, 43, 43, 52, 42, 42,
 16, 6, 42, 42, 42, 36, 66, 32, 62, 52, 66, 66, 0, 25, 5, 5, 35, 21, 11,
 11, 57, 57, 61, 41, 41]

term 24 is 42...
i still fail to see the significance of it all.
atb :: lieven.

A couple of hours later I received his reply and simply couldn’t stop laughing…

From:  mckay@encs.concordia.ca
Subject:   Re: mystery message
Date:  Sat 22 Mar 2008 02:33:19 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

I apologize for wasting your time. It is a joke
depending, it seems, on one's cultural background.

See the google entry:

Answer to Life, the Universe, and Everything

Best, John McKay

Still confused? Well, do it!

3 Comments

Monstrous Easter Egg Race

Published March 23, 2008 by lievenlb

Here’s a sweet Easter egg for you to crack : a mysterious message from none other than the discoverer of Monstrous Moonshine himself…

From:  mckayj@Math.Princeton.EDU
Date:  Mon 10 Mar 2008 07:51:16 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

The secret of Monstrous Moonshine and the universe. 

Let  j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion 
at oo of the elliptic modular function.

Compute sum(c[k]^2,k=1..24) modulo 70

Background: w_25 of page x of the preface of Conway/Sloane book SPLAG 

Also in Chapter 27:
The automorphism group of the 26-dimensional Lorentzian lattice
The Weyl vector w_25 of section 2.

Jm

I realize that all of you will feel frustrated by the fact that most university libraries are closed today and possibly tomorrow, hence some help with the background material.

SPLAG of course refers to the cult-book Sphere Packings, Lattices and Groups.

26-dimensional Lorentzian space $\mathbb{R}^{25,1} $ is 26-dimensional real space equipped with the norm-map

$|| \vec{v} || = \sum_{i=1}^{25} v_i^2 – v_{26}^2 $

The Weyl vector $\vec{w}_{25} $ is the norm-zero vector in $\mathbb{R}^{25,1} $

$\vec{w}_{25} = (0,1,2,3,4,\ldots,22,23,24,70) $ (use the numerical fact that $1^2+2^2+3^2+\ldots+24^2=70^2 $)

The relevance of this special vector is that it gives a one-line description for one of the most mysterious objects around, the 24-dimensional Leech Lattice $L_{24} $. In fact

$L_{24} = \vec{w}^{\perp}/\vec{w} $ with $\vec{w}^{\perp} = { \vec{x} \in \Pi_{25,1}~:~\vec{x}.\vec{w}=0 } $

where $\Pi_{25,1} $ is the unique even unimodular lattice in $\mathbb{R}^{25,1} $. These facts amply demonstrate the moonshine nature of the numbers 24 and 70. Apart from this, the previous post may also be of use.

3 Comments

Finding Moonshine

Published March 2, 2008 by lievenlb

On friday, I did spot in my regular Antwerp-bookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me…

To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more, the more I convinced myself that it was a mere resampling of Symmetry and the Monster, The equation that couldn’t be solved, From Error-Correcting Codes through Sphere Packings to Simple Groups and the diary-columns du Sautoy wrote for a couple of UK-newspapers about his ‘life-as-a-mathematician’…

Still, I took the book home, made a pot of coffee and started reading the first chapter. And, sure enough, soon I had to read phrases like “The first team consisted of a ramshackle collection of mathematical mavericks. One of the most colourful was John Horton Conway, currently professor at the University of Princeton. His mathematical and personal charisma have given him almost cult status…” and “Conway, the Long John Silver of mathematics, decided that an account should be published of the lands that they had discovered on their voyage…” and so on, and so on, and so on.

The main problem I have with du Sautoy’s books is that their main topic is NOT mathematics, but rather the lives of mathematicians (colourlful described with childlike devotion) and the prestige of mathematical institutes (giving the impression that it is impossible to do mathematics of quality if one isn’t living in Princeton, Paris, Cambridge, Bonn or … Oxford). Less than a month ago, I reread his ‘Music of the Primes’ so all these phrases were still fresh in my memory, only on that occasion Alain Connes is playing Conway’s present role…

I was about to throw the book away, but first I wanted to read what other people thought about it. So, I found Timothy Gowers’ review, dated febraury 21st, in the Times Higher Education. The first paragraph below hints politely at the problems I had with Music of the Primes, but then, his conclusion was a surprise

The attitude of many professional mathematicians to the earlier book was ambivalent. Although they were pleased that du Sautoy was promoting mathematics, they were not always convinced by the way that he did it.

I myself expected to have a similar attitude to Finding Moonshine, but du Sautoy surprised me: he has pulled off that rare feat of writing in a way that can entertain and inform two different audiences – expert and non-expert – at the same time.

Okay, so maybe I should give ‘Finding Moonshine’ a further chance. After all, it is week-end and, I have nothing else to do than attending two family-parties… so I read the entire book in a couple of hours (not that difficult to do if you skip all paragraphs that have the look and feel of being copied from the books mentioned above) and, I admit, towards the end I mellowed a bit. Reading his diary notes I even felt empathy at times (if this is possible as du Sautoy makes a point of telling the world that most of us mathematicians are Aspergers). One example :

One of my graduate students has just left my office. He’s done some great work over the past three years and is starting to write up his doctorate, but he’s just confessed that he’s not sure that he wants to be a mathematician. I’m feeling quite sobered by this news. My graduate students are like my children. They are the future of the subject. Who’s going to read all the details of my papers if not my mathematical offspring? The subject feels so tribal that anyone who says they want out is almost a threat to everything the tribe stands for.
Anton has been working on a project very close to my current problem. There’s no denying that one can feel quite disillusioned by not finding a way into a problem. Last year one of my post-docs left for the City after attempting to scale this mountain with me. I’d already rescued him from being dragged off to the City once before. But after battling with our problem and seeing it become more and more complex, he felt that he wasn’t really cut out for it.

What is unsettling for me is that they both questioned the importance of what we are doing. They’ve asked that ‘What’s it all for?’ question, and think they’ve seen the Emperor without any clothes.
Anton has questioned whether the problems we are working on are really important. I’ve explained why I think these are fundamental questions about basic objects in nature, but I can see that he isn’t convinced. I feel I am having to defend my whole existence. I’ve arranged for him to join me at a conference in Israel later this month, and I hope that seeing the rest of the tribe enthused and excited about these problems will re-inspire him. It will also show him that people are interested in what he is dedicating his time to.

Du Sautoy is a softy! I’d throw such students out of the window…

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valentines_night@intensive_care

Published February 17, 2008 by lievenlb

Not your idea of a romantic evening out? Neither it’s mine, but then, sometimes shit happens…

Blogging and monitoring this server’s status are no priorities at the moment, so please switch to RSS-syndication, if you haven’t done it already.

ps. all’s fine now and, I’ll be back.

6 Comments

Archimedes’ stomachion

Published February 11, 2008 by lievenlb

The Archimedes codex is a good read, especially when you are (like me) a failed archeologist. The palimpsest (Greek for ‘scraped again’) is the worlds first Kyoto-approved ‘sustainable writing’. Isn’t it great to realize that one of the few surviving texts by Archimedes only made it because some monks recycled an old medieval parchment by scraping off most of the text, cutting the pages in half, rebinding them and writing a song-book on them…

The Archimedes-text is barely visible as vertical lines running through the song-lyrics. There is a great website telling the story in all its detail.

Contrary to what the books claims I don’t think we will have to rewrite maths history. Didn’t we already know that the Greek were able to compute areas and volumes by approximating them with polygons resp. polytopes? Of course one might view this as a precursor to integral calculus… And then the claim that Archimedes invented ordinal calculus. Sure the Greek knew that there were ‘as many’ even integers than integers… No, for me the major surprise was their theory about the genesis of mathematical notation.

The Greek were pure ASCII mathematicians : they wrote their proofs out in full text. Now, here’s an interesting theory how symbols got into maths… pure laziness of the medieval monks transcribing the old works! Copying a text was a dull undertaking so instead of repeating ‘has the same ratio as’ for the 1001th time, these monks introduced abbreviations like $\Sigma $ instead… and from then on things got slightly out of hand.

Another great chapter is on the stomachion, perhaps the oldest mathematical puzzle. Just a few pages made in into the palimpsest so we do not really know what (if anything) Archimedes had to say about it, but the conjecture is that he was after the number of different ways one could make a square with the following 14 pieces

People used computers to show that the total number is $17152=2^8 \times 67 $. The 2-power is hardly surprising in view of symmetries of the square (giving $8 $) and the fact that one can flip one of the two vertical or diagonal parts in the alternative description of the square

but I sure would like to know where the factor 67 is coming from… The MAA and UCSD have some good pages related to the stomachion puzzle. Finally, the book also views the problema bovinum as an authentic Archimedes, so maybe I should stick to my promise to blog about it, after all…

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Oxen of the Sun

Published February 7, 2008 by lievenlb

The Oxen of the Sun (of the Problema Bovinum) is one of the most difficult chapters in Joyce’s Ulysses. Ulysses is the 1904 version of Homer’s Odyssey so the Oxen appear also in his Book XII :

And thou wilt come to the isle Thrinacia. There in great numbers feed the kine [cattle] of Helios and his goodly flocks, seven herds of kine and as many fair flocks of sheep, and fifty in each.

Homer must have suffered from an acute form of innumeracy as the minimal solution to the cattle problem gives as the total number the smallest integer greater than

$\frac{25194541}{184119152} (109931986732829734979866232821433543901088049+ $

$50549485234315033074477819735540408986340
\sqrt{4729494})^{4658} $

a number whose actual digits take up 47 pages, one of the most useless pieces of mathematical wall-paper!

censured post : bloggers’ block

Published February 6, 2008 by lievenlb

Below an up-till-now hidden post, written november last year, trying to explain the long blog-silence at neverendingbooks during october-november 2007…

A couple of months ago a publisher approached me, out of the blue, to consider writing a book about mathematics for the general audience (in Dutch (?!)). Okay, I brought this on myself hinting at the possibility in this post

Recently, I’ve been playing with the idea of writing a book for the general public. Its title is still unclear to me (though an idea might be “The disposable science”, better suggestions are of course wellcome) but I’ve fixed the subtitle as “Mathematics’ puzzling fall from grace”. The book’s concept is simple : I would consider the mathematical puzzles creating an hype over the last three centuries : the 14-15 puzzle for the 19th century, Rubik’s cube for the 20th century and, of course, Sudoku for the present century.

For each puzzle, I would describe its origin, the mathematics involved and how it can be used to solve the puzzle and, finally, what the differing quality of these puzzles tells us about mathematics’ changing standing in society over the period. Needless to say, the subtitle already gives away my point of view. The final part of the book would then be more optimistic. What kind of puzzles should we promote for mathematical thinking to have a fighting chance to survive in the near future?

While I still like the idea and am considering the proposal, chances are low this book ever materializes : the blog-title says it all…

Then, about a month ago I got some incoming links from a variety of Flemish blogs. From their posts I learned that the leading Science-magazine for the low countries, Natuur, Wetenschap & Techniek (Nature, Science & Technology), featured an article on Flemish science-blogs and that this blog might be among the ones covered. It sure would explain the publisher’s sudden interest. Of course, by that time the relevant volume of NW&T was out of circulation so I had to order a backcopy to find out what was going on. Here’s the relevant section, written by their editor Erick Vermeulen (as well as an attempt to translate it)

Sliding puzzle For those who want more scientific depth (( their interpretation, not mine )), there is the English blog by Antwerp professor algebra & geometry Lieven Le Bruyn, MoonshineMath (( indicates when the article was written… )). Le Bruyn offers a number of mathematical descriptions, most of them relating to group theory and in particular the so called monster-group and monstrous moonshine. He mentions some puzzles in passing such as the well known sliding puzzle with 15 pieces sliding horizontally and vertically in a 4 by 4 matrix. Le Bruyn argues that this ’15-puzzle (( The 15-puzzle groupoid ))’ was the hype of the 19th century as was the Rubik cube for the 20th and is Sudoku for the 21st century.
Interesting is Le Bruyn’s mathematical description of the M(13)-puzzle (( Conway’s M(13)-puzzle )) developed by John Conway. It has 13 points on a circle, twelve of them carrying a numbered counter. Every point is connected via lines to all others (( a slight simplification )). Whenever a counter jumps to the empty spot, two others exchange places. Le Bruyn promises the blog-visitor new variants to come (( did I? )). We are curious.
Of course, the genuine puzzler can leave all this theory for what it is, use the Java-applet (( Egner’s M(13)-applet )) and painfully try to move the counters around the circle according to the rules of the game.

Some people crave for this kind of media-attention. On me it merely has a blocking-effect. Still, as the end of my first-semester courses comes within sight, I might try to shake it off…

vacation reading (2)

Published February 5, 2008 by lievenlb

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.

2 Comments

quotes of the day

Published February 1, 2008 by lievenlb

Some people are in urgent need of a vacation, myself included…

From the paper Transseries for beginners by G.A. Edgar, arXived today :

Well, brothers and sisters, I am here today to tell you: If you love these formulas,
you need no longer hide in the shadows! The answer to all of these woes is here.
Transseries.

In a comment over at The Everthing Seminar

Shouldn’t dwarfs on the shoulders on giants be a little less arrogant?

by Micromegas.
Well, I’d rather enter a flame war than report about it. But, for some reason I cannot comment at the EverythingSeminar, nor at the SecretBloggingSeminar. Is this my problem or something to do with wordpress.com blogs? If you encountered a similar problem and managed to solve it, please let me know.

UPDATE (febr. 2) : my comment did surface after 5 days. Greg fished it out of their spam-filter. Thanks! I’ll try to comment at wordpress.com blogs from now on by NOT linking to neverendingbooks. I hope this will satisfy their spam-filter…

11 Comments

another numb3rs screenshot

Published January 28, 2008 by lievenlb

Ever since my accidental ‘discovery’ of the word CEILIDH written on a numb3rs blackboard, I keep an eye on their blackboards whenever I watch a new episode, and try to detect terms I might know. Here’s todays screen-shot

I had to choose one frame from a minute long shot (the ‘link’ on the left hand side is more recognizable in other frames). Anyway, here’s what I thought to recognize : a link, a quaternion-algebra over a number field,

$\begin{pmatrix} -1,-3 \\ \mathbb{Q}\sqrt{-2} \end{pmatrix} $ to be precise, and a $B^{max}_{order} $ in it. So did someone construct link-invariants from maximal orders in quaternion algebras? I surely didn’t know, but when in doubt there is always… google. I searched for ‘link invariant quaternion algebra maximal order’ and the third hit on page 2 gave me a pdf-file of a paper which seemed to have the relevant terms in it.

The paper is Automorphic forms and rational homology 3–spheres by Frank Calegari and Nathan Dunfield. Bingo! Their first figure is the ‘link’ drawn on the blackboard

which actually turns out to be a graph… This couldn’t be a coincidence, so as in the ceilidh-story, there had to be a connection with one of the authors. ‘Calegari+numb3rs’ didnt look promising but ‘Dunfield+numb3rs’ returned the hit Crime and Computation from CalTech News.

Krumholtz’s star turn as a math genius belies his dismal record as an algebra student. He explained that in preparation for his role he hung around Caltech last fall, “wandering the hallways and campus for two to three weeks,” to soak up the academic ambiance. To plumb character motivation he talked to a real-life youthful math guy, Caltech’s 30-year-old professor Dunfield.

In a phone interview shortly after the show’s television debut Dunfield recalled spending about an hour with Krumholtz, who plays 29-year-old Charlie. “He wanted to know what it’s like to do mathematics and work in academia, what types of things his character would likely be concerned about, like tenure or other issues.”

The professor, who was so un-starstruck that he hadn’t even made a point of watching the premiere, added, “He wanted to know, why would somebody choose to become a mathematics professor. Would they have to love math?” What was his response? The professor said he does not recall.

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