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Dedekind or Klein ?

The black&white psychedelic picture on the left of a tessellation of the hyperbolic upper-halfplane, was called the Dedekind tessellation in this post, following the reference given by John Stillwell in his excellent paper Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.

But is this correct terminology? Nobody else uses it apparently. So, let’s try to track down the earliest depiction of this tessellation in the literature…

Stillwell refers to Richard Dedekind‘s 1877 paper “Schreiben an Herrn Borchard uber die Theorie der elliptische Modulfunktionen”, which appeared beginning of september 1877 in Crelle’s journal (Journal fur die reine und angewandte Mathematik, Bd. 83, 265-292).

There are a few odd things about this paper. To start, it really is the transcript of a (lengthy) letter to Herrn Borchardt (at first, I misread the recipient as Herrn Borcherds which would be really weird…), written on June 12th 1877, just 2 and a half months before it appeared… Even today in the age of camera-ready-copy it would probably take longer.

There isn’t a single figure in the paper, but, it is almost impossible to follow Dedekind’s arguments without having a mental image of the tessellation. He gives a fundamental domain for the action of the modular group $\Gamma = PSL_2(\mathbb{Z}) $ on the hyperbolic upper-half plane (a fact already known to Gauss) and goes on in section 3 to give a one-to-one mapping between this domain and the complex plane using what he calls the ‘valenz’ function $v $ (which is our modular function $j $, making an appearance in moonshine, and responsible for the black&white tessellation, the two colours corresponding to pre-images of the upper or lower half-planes).

Then there is this remarkable opening sentence.

Sie haben mich aufgefordert, eine etwas ausfuhrlichere Darstellung der Untersuchungen auszuarbeiten, von welchen ich, durch das Erscheinen der Abhandlung von Fuchs veranlasst, mir neulich erlaubt habe Ihnen eine kurze Ubersicht mitzuteilen; indem ich Ihrer Einladung hiermit Folge leiste, beschranke ich mich im wesentlichen auf den Teil dieser Untersuchungen, welcher mit der eben genannten Abhandlung zusammenhangt, und ich bitte Sie auch, die Ubergehung einiger Nebenpunkte entschuldigen zu wollen, da es mir im Augenblick an Zeit fehlt, alle Einzelheiten auszufuhren.

Well, just try to get a paper (let alone a letter) accepted by Crelle’s Journal with an opening line like : “I’ll restrict to just a few of the things I know, and even then, I cannot be bothered to fill in details as I don’t have the time to do so right now!” But somehow, Dedekind got away with it.

So, who was this guy Borchardt? How could this paper be published so swiftly? And, what might explain this extreme ‘je m’en fous’-opening ?

Carl Borchardt was a Berlin mathematician whose main claim to fame seems to be that he succeeded Crelle in 1856 as main editor of the ‘Journal fur reine und…’ until 1880 (so in 1877 he was still in charge, explaining the swift publication). It seems that during this time the ‘Journal’ was often referred to as “Borchardt’s Journal” or in France as “Journal de M Borchardt”. After Borchardt’s death, the Journal für die Reine und Angewandte Mathematik again became known as Crelle’s Journal.

As to the opening sentence, I have a toy-theory of what was going on. In 1877 a bitter dispute was raging between Kronecker (an editor for the Journal and an important one as he was the one succeeding Borchardt when he died in 1880) and Cantor. Cantor had published most of his papers at Crelle and submitted his latest find : there is a one-to-one correspondence between points in the unit interval [0,1] and points of d-dimensional space! Kronecker did everything in his power to stop that paper to the extend that Cantor wanted to retract it and submit it elsewhere. Dedekind supported Cantor and convinced him not to retract the paper and used his influence to have the paper published in Crelle in 1878. Cantor greatly resented Kronecker’s opposition to his work and never submitted any further papers to Crelle’s Journal.

Clearly, Borchardt was involved in the dispute and it is plausible that he ‘invited’ Dedekind to submit a paper on his old results in the process. As a further peace offering, Dedekind included a few ‘nice’ words for Kronecker

Bei meiner Versuchen, tiefer in diese mir unentbehrliche Theorie einzudringen und mir einen einfachen Weg zu den ausgezeichnet schonen Resultaten von Kronecker zu bahnen, die leider noch immer so schwer zuganglich sind, enkannte ich sogleich…

Probably, Dedekind was referring to Kronecker’s relation between class groups of quadratic imaginary fields and the j-function, see the miracle of 163. As an added bonus, Dedekind was elected to the Berlin academy in 1880…

Anyhow, no visible sign of ‘Dedekind’s’ tessellation in the 1877 Dedekind paper, so, we have to look further. I’m fairly certain to have found the earliest depiction of the black&white tessellation (if you have better info, please drop a line). Here it is

It is figure 7 in Felix Klein‘s paper “Uber die Transformation der elliptischen Funktionen und die Auflosung der Gleichungen funften Grades” which appeared in may 1878 in the Mathematische Annalen (Bd. 14 1878/79). He even adds the j-values which make it clear why black triangles should be oriented counter-clockwise and white triangles clockwise. If Klein would still be around today, I’m certain he’d be a metapost-guru.

So, perhaps the tessellation should be called Klein’s tessellation??
Well, not quite. Here’s what Klein writes wrt. figure 7

Diese Figur nun – welche die eigentliche Grundlage fur das Nachfolgende abgibt – ist eben diejenige, von der Dedekind bei seiner Darstellung ausgeht. Er kommt zu ihr durch rein arithmetische Betrachtung.

Case closed : Klein clearly acknowledges that Dedekind did have this picture in mind when writing his 1877 paper!

But then, there are a few odd things about Klein’s paper too, and, I do have a toy-theory about this as well… (tbc)

3 Comments

the secret revealed…

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

Blackle

There seems to be a slight chance that the next US-administration may (finally) be joining the rest of the civilized world and sign the Kyoto-treaty. Here’s an appeal to Flock and other webbrowsers : please add blackle.com to our Search Engine Preferences!

The idea is simple : you Google as you’d do anyway but … you save a lot of energy. Via PD2 (for Pseudonymous Daughter 2).

4 Comments

block google analytics cookies

When more than 200.000 websites are using google analytics, a one-man action proposed last time to de-activate GA on neverendingbooks is pretty useless. Perhaps a better alternative is :

When you use google analytics on your site, announce this and add a link to CustomizeGoogle BLOCK GOOGLE ANALYTICS COOKIES. Your visitors can then decide whether or not to block google analytics cookies only.

Or link to this post, as I’ll give a detailed step-by-step instruction so that even web-newbies can protect themselves agains google analytics stalking cookies.

1. Surf using a Firefox-clone Because CustomizeGoogle “is a Firefox extension that enhances Google search results by adding extra information (like links to Yahoo, Ask.com, MSN etc) and removing unwanted information (like ads and spam)” it only works on Firefox-like browsers. Screenshots below use Flock, the mac “social web browser”.

2. Go to CustomizeGoogle BLOCK GOOGLE ANALYTICS COOKIES and click on the Install Now! link (upper right)

3. A warning message will appear saying “Flock prevented this site from asking you to install software on your computer”. As we do want to install, click on the Edit Options button (on the left).

4. A pop-up window appears and click on the allow-button.

5. Click again on the Install Now! link from CustomizeGoogle. A pop-up will appear asking you to confirm installation of the extension. Click on the install-buttom (right below).

6. Installation complete! But you have to restart before you can use it. Click on the restart button and Flock will do it.

7. Select under Flock/Tools CustomizeGoogle Options.

8. A pane pops up with plenty of configurable options.

9. Select “Privacy” and mark the ‘Don’t send any cookies to GoogleAnalytics’ option.

10. Done! You can now surf to any of the 200.000+ google-analytics-powered sites without being stalked!

Using this feature makes you more anonymous. But your visit on a single webpage can still be logged. This way, both Google and the owner of the website knows that someone visited a webpage, but it’s difficult to track all pages you’re visiting. And it’s really really hard for Google to track that you visited both Website A and Website B.

2 Comments

another numb3rs screenshot

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.

2 Comments

top 5 analytics tips

Google analytics is collecting data on this site for over a month now. Perhaps it is time to share a few lessons Ive learned over this period.

UPDATE : I have de-activated all google analytics plugins on this site. I may re-activate them for short periods later but ONLY to detect problems or to check on plugin performance. I will announce this in the sidebar with a ‘this site contains google analytics script code’-sticker. I encourage people using google analytics stalking code to do the same.

1. Aim at the generic visitor, not the specific

Analytics offers an amazing amount of data, in debatable detail. For example, via the map-overlay one can zoom in to specific towns and communities. When combined with other data, such as new/returning visitors etc., this quickly becomes borderline stalking. So, repress that urge to check-out whether someone you know is reading your blog, and how often, and how long, and how deep, and … Use analytics only to get an idea of what the generic visitor does with your site.

2. There is a world outside your blog

Don’t get obsessed by analytics-data and, certainly, do not feel that you have to react to it, all the time! For example, below the evolution of the number of visitors coming here over the last month

The dramatic fall in attendance from last friday until this monday might have worried me (when obsessed). However, a quick check gave a similar drop for the number of new visitors and the number of search-engine referrals. Probably, people were, at that time, more interested in the stock-market-crisis than in this blog… Besides, most people visiting here come from the US and I learned that they had a Luther King vacation day, so perhaps lots of them enjoyed a prolonged week-end, away from their computer.

3. Detect & correct major, lasting changes

So, forget about temporary blips. However, when a certain trend has every indication of becoming permanent, it might be the moment to check out what is going on. Below the bounce rate-evolution over the last month

This is what you might call a lasting drop! Fortunately this time I knew what was going on, because of the actions described in the bounce rate post. But, when you detect a similar drop in certain stats it is time to figure out the causes. Perhaps you de-activated by mistake a certain plugin (see below), or something is wrong with your server, or…

4. Check-out plugin performance

There are tons of WordPress plugins, some useful, some less. So it is best to check whether activating a plugin has the desired effect. For example, you should be able to detect installation of an SEO-plugin (for, Search Engine Optimalization) in the Traffic Sources/Search Engines graph, installing and using a tagging-plugin should give you more referrals from Technorati and look-a-likes, etc.

5. Don’t take it too seriously

You can use analytics data just for fun! For example, do a quiz, show visitor-data and ask for global events explaining the graph (as above). Or notice quirks in your data. For example, here the time-on-site graph over the last month

My generic visitor seems to have a cyclic attention-span…

7 Comments

yahoo pipes on iTouch

The next thing on my tech-to-do-list : learn all about Yahoo Pipes :

Pipes is a powerful composition tool to aggregate, manipulate, and mashup content from around the web. Like Unix pipes, simple commands can be combined together to create output that meets your needs. Here are a few popular ways the service can be used:
– create your ultimate custom feed by combining many feeds into one, then sorting, filtering and translating them.
– geocode your favorite feeds and browse the items on an interactive map.
– remix your favorite data sources and use the Pipe to power a new application.
– build custom vertical search pages that are impossible with ordinary search engines.
– power widgets/badges on your web site.
– consume the output of any Pipe in RSS, JSON, KML, and other formats.

I’ve posted before on setting up your own lifestream, or your own planet, or scraping feeds, or subscribing to my brain, or … whatever. The good news is : all these ideas are now superseded by Pipes!

Pipes is a free online service that lets you remix popular feed types and create data mashups using a visual editor. You can use Pipes to run your own web projects, or publish and share your own web services without ever having to write a line of code. You make a Pipe by dragging pre-configured modules onto a canvas and wiring them together in the Pipes Editor. Once you’ve built a Pipe, you’ll be able save it on our server and then call it like you would any other feed. Pipes offers output in RSS 2.0, RSS 1.0 (RDF), JSON and Atom formats for maximum flexibility. You can also choose to publish your Pipe and share it with the world, allowing other users to clone it, add their own improvements, or use it as a subcomponent in their own creations.

This is the essential message to get : yahoo-pipes allows you to remix the web, filtering out all noise! And the good news is

  1. There are plenty of public pipes around to get you going, and
  2. Pipes has an iTouch-friendly interface (see above left). All you have to do is to Safari to iphone.pipes.yahoo.com and use them.

Here are a few public-pipes you can use out of the box!

  • iPhone / iPod Touch: The Most Comprehensive Feed Ever!, doing what it promises : giving you the best iTouch-posts without having to roam for them.
  • JSON Geocoder, returning lat/lon/address info from the the given address.
  • Uber Blog Search, Search all the blogosphere with one query. Hits Google, Ask, Technorati, and icerocket then returns the unique results. Below the web-interface giving the results for ‘noncommutative’…

and finally, one of my favorites, implementing to some extend the Lifestream-idea (iTouch-interface above left)

  • lifefeed – virable, Easily Aggregate your social whereabouts great for blogs profiles and more! Aggregates Your Feeds From: -Digg -Last.fm -Twitter -Flickr -Del.icio.us and your very own blog Adopt and Improve, enjoy!

I’ll promise to spend some time soon to set up my very own pipes and make them available…

5 Comments

return of the cat ceilidh

I couldn’t believe my eyes. I was watching an episode of numb3rs, ‘undercurrents’ to be precise, and there it was, circled in the middle of the blackboard, CEILIDH, together with some of the key-exchange maps around it…

Only, the plot doesn’t involve any tori-crypto… okay, there is an I-Ching-coded-tattoo which turns out to be a telephone number, but that’s all. Still, this couldn’t just be a coincidence. Googling for ‘ceilidh+numbers‘ gives as top hit the pdf-file of an article Alice in NUMB3Rland written by … Alice Silverberg (of the Rubin-Silverberg paper starting tori-cryptography). Alice turns out to be one of the unpaid consultants to the series. The 2-page article gives some insight into how ‘some math’ gets into the script

Typically, Andy emails a draft of the
script to the consultants. The FBI plot
is already in place, and the writers want
mathematics to go with it. The placeholder “math” in the draft is often nonsense or
jargon; the sort of things people with no
mathematical background might find by
Googling, and think was real math. Since
there’s often no mathematics that makes
sense in those parts of the script, the best
the consultants can do is replace jargon
that makes us cringe a lot with jargon that
makes us cringe a little.

From then on, it’s the Telephone Game.
The consultants email Andy our suggestions (“replace ‘our discrete universes’
with ‘our disjoint universes'”; “replace
the nonsensical ‘we’ve tried everything
-a full frequency analysis, a Vignere
deconstruction- we even checked for
a Lucas sequence’ with the slightly less
nonsensical ‘It’s much too short to try
any cryptanalysis on. If it were longer
we could try frequency analyses, or try
to guess what kind of cryptosystem it is
and use a specialized technique. For example, if it were a long enough Vigenere
cipher we could try a Kasiski test or an
index-of-coincidence analysis’). Andy
chooses about a quarter of my sugges-
tions and forwards his interpretation
of them to the writers and producers.
The script gets changed, and then the
actors ad lib something completely dif-
ferent (‘disjointed universes’: cute, but
loses the mathematical allusion; ‘Kasiski
exam’ : I didn’t mean that kind of ‘test’).

She ends her article with :

I have mixed feelings about NUMB3RS. I still have concerns about the violence, the depiction of women, and the pretense
that the math is accurate. However, if NUMB3RS could interest people in the power of mathematics enough for society
to greater value and support mathematics teaching, learning, and research, and
motivate more students to learnthat would be a positive step.

Further, there is a whole blog dedicated to some of the maths featuring in NUMB3RS, the numb3rs blog. And it was the first time I had to take a screenshot of a DVD, something usually off limits to the grab.app, but there is a simple hack to do it…

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Top 5 wp-plugins to improve your bounce-rate

The bounce rate indicates how many web-visitors leave the site without visiting any other pages before a certain session timeout elapses. That is :

High bounce rate means the site must be horrible to site visitors and most likely they would never return again. They are not even interested to check other content of the site. First impression counts.

After installing google analytics some weeks ago, I noticed a worrying high bounce rate : close to 80%… At first I thought this was due to the fact that all iTouch fans left the moment they saw a mathematical symbol, but further analysis proved me wrong : iTouch fans study posts here a lot longer than the average mathematician. But then, what was the reason? Is this site really so horrible to look at? is the content of such poor quality?

Anyway, I’ve tried out a couple of tricks, with surprising effect. The bounce rate dropped from 80% to under 3% and best of all, it appears to stay that low. Here, the google-analytics bounce-rate evolution of the last 3 weeks

So, what kind of magic voodoo did I perform on january 6th? I’ve installed a couple of WordPress Plugins and changed the upper part of the entry-page. My basic assumption is that people leave a site when they come to it for specific information (for example via a search engine), do not find the info immediately and don’t want to spend too much time looking for it. So, I wanted to have all tools to find content on this site right in front of the potential new-comer. Here are the 5 major changes to the header part and the plugins Ive used.

1 : Rather than having a monthly-archive in the sidebar providing no more info than the number of posts in a particular month, create a proper archive page where visitors can find the titles and links to all posts in reverse chronological ordering. I did use the Smart Archives Plugin.

2 : Even better : have a drop-down archive right under the header-picture so that visitors can scroll down the list of all posts without having to load another page. Ive used the Awsom Archive Plugin.

3 : Let visitors see in a glance what your blog is all about by having a tag cloud under the header. I didnt feel like tagging 300 old posts, so I used the Simple Tags Plugin to do it all for me.

4 : Have a welcome message near the top to aid new visitors (especially when you have, like me, 77.51% of them around). The message disappears after their third visit. It’s a wonderful idea, made possible by the What Would Seth Godin Do-Plugin.

5 : Write series of posts and have links to the other parts available at the top of the new post. Likely, people are going to check out the other parts for more information. Rather than hard-coding the links by hand, Ive used the In Series Plugin.

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quick iTouch links

MacBookAir? Is this really the best Apple could come up with? A laptop you can slide under the door or put in an envelop? Yeez… Probably the hot-air-book is about as thick as an iTouch. The first thing I did was to buy a leather case to protect the vulnerable thing, making it as thick as a first generation iPod… (needless to say, when my MacBookPro breaks down, ill replace it with a MacBookAir, clearly!)

Ranting about MacWorlds : Wired has a great article on last year’s event. Steve Job’s iPhone presentation is something that will be part of the collective memory when it comes to 2007-recollections. Few people will have realized that the Apple-team didnt have a working prototype a few weeks before… Here’s The Untold Story: How the iPhone Blew Up the Wireless Industry. A good read!

If you plug in your jailbroken iTouch, you will be asked wether you want to upgrade to 1.1.3, something we all feared for a long time and so it takes just nanoseconds to hit the cancel-button. But, there is good news! Rupert Gee reports that you can downgrade to 1.1.1 and redo jailbreak. I won’t try it for some time, but still…

In the unlikely event that you come here being a mathematician, here’s what I did with my iTouch today. Ive downloaded the Connes-Marcolli talks on Renormalization and Motives part 1, part 2, part 3, part 4, part 5, part 6, part 7 and part 8 at work. They are in mp4-format so you can load them into iTunes and onto your iTouch!!! Weather is not favorable for outdoor-cycling at the moment, so I used the home-trainer, put the iTouch in front of me and, boy, was I educated…

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